INDEX:

To facilitate lookup, the instructions are divided into the following categories:

         I.   Data Manipulation - Entering data, sorting data, clearing lists, friendly values from graphs.
        II.  Single-Variable Statistics - Histogram by hand, simple histogram with the calculator, choosing 
             your own classes when using the calculator, frequency polygon, cumulative frequency (Ogive), percentile
             graph, relative frequency polygon, cumulative relative frequency graph, histogram from grouped data, frequency and
             cumulative frequency graphs from grouped data, box and whisker plot, discrete probability distribution, coefficient of  
             variation, finding standard deviation, finding standard deviation from grouped data.
        III.  Two Variable Statistics – scatter plot, regression analysis, finding r, r², a, and b in correlation using a calculator, 
              finding r, r², a, and b in correlation using a formula.
       IV.  Aids in doing statistics by hand
        V.  Permutations, combinations, factorials, random numbers.
       VI.  Normal Distribution - Area under a normal curve, Finding Z values, Graphing a curve,  WINDOW
             settings for graphing a curve, Probability Distribution Function using normalpdf(, Graphing the
             Normal Distribution Using normalpdf(, normalcdf(, ZInterval,
       VII. Other Distributions -  TInterval, invT Finding a t-value given α and df, Chi-squared Distribution, binomialpdf,
               binomialcdf.
      VIII.  Hypothesis testing - mean and z-test (data), mean and z-test (statistics), mean and t-test (data),
              mean and t-test (statistics).
        IX.  Simple program for calculating InverseT with a TI-83Plus.
        X.  Statistics of two Populations - confidence interval for two dependent population, confidence interval for two 
              independent populations (Data and Stats),

RELEASE DATE:  10/1/06         DATE LAST REVISED:  10/1/08

Back to home page.

NOTE:  Copying  limitations and printing hints are at the end of this document.

FORWARD:  It seems that at the ends of the spectrum of opinions about using calculators there are two polar
opposites:  Use a calculator to the maximum or don't use it for anything except arithmetic.  I have tried to take
into consideration the broad spectrum and include methods for both use of a calculator only and use of the
calculator to take some of the drudgery of arithmetic out of the use of the computation formulas.

NOW A WORD ABOUT MY USE OF LISTS:  Lists are a  powerful tool for doing statistics. In most computation
formulas, lists can be substituted for the variables in most applications.   When doing list arithmetic such as
multiplication, addition, and subtraction and storing the result in another list, the operation can be done from
either the list screen or the home screen.  Fro example L₁*L₂ will do the same thing at the list screen as
L₁*L₂→L₃ at the home screen.  (The arrow is a result of pressing STO.)  But when using a function such as
sum( , the operation must be done from the home screen.  So, I will be using both the home screen and the list
screen to do list operations in this document.

I.  Data Manipulation
     (NOTE:  In some instances you may want to clear a list or lists before you start entering data.  You  
     can overwrite data already in a list, but remember that if the old list was longer than the new one,
     you must delete the remaining old data an item at a time.  The easiest way to clear one of the tabular
     lists, L ₁ -L ₆ is to place the cursor on the name above the list and press CLEAR; then ENTER.  You
     can also clear a number of lists or any list as follows: a) Press STAT, 4 (ClrList).  This will paste "ClrList"
     to the home screen.  Press 2nd; then the button for the list number you want to clear, for example
     L₁ ; then press ENTER.  If you use more than one list separate the lists by a comma.
      1)  Entering Data:
          a)  Press STAT; then ENTER.  A table for entering data will appear.
          b)  To enter data, just place the cursor where you want to enter the data and press the correct
               numbers.  You don't have to erase old data if there is already data in the list, but if the old list
               is longer than the new list, you will need to delete the remaining old data items.  Just place
               the cursor over the data and press DEL.       

    2)  Putting Data in Order:
           a)  Press STAT, 2 (SortA).  This will paste SortA to the home screen. 
           b)  Press 2nd, L₁ (or whatever list you want to sort); then press ENTER; then return to your
                tables to view the sorted data. Note that you can also sort data in descending order with
                SortD.

   3)  Friendly Values on Graphs Using TRACE:
      Many times when you use the TRACE function, you will get an x-value such as 2.784532.  If you
      change the x-range in the WINDOW function to be a multiple of 4.7, the x-values will be "friendlier"
      values that can be more easily plotted by hand.  Usually the easiest way to do this is to press ZOOM,
      4, for ZDecimal and use  Zoom In or Zoom Out to adjust the window size if it's not satisfactory.  That's
      fine if are satisfied with a symmetric window.  If you need an asymmetric window, you get the friendly
      values by pressing WINDOW and setting the window parameters by hand.  Let's take a value and say
      that after a stat plot we get some "unfriendly" values and we press WINDOW and get X-min = .6 and
      X-max = 8.2.  If we change X-min to 0 and X-max to 2*4.7 = 9.4; then we will have friendlier values when
     using TRACE.

II.  Single-Variable Statistics  
     1)  Doing a Frequency Distribution Histogram by Hand:
           a)  Use items 1 and 2 in Section I above to enter and sort your data.
           b)  Find the class width as follows:
                  (1)  Let S represent the smallest data number (The first number in your sorted list.), L be 
                        the largest number (The largest number in the sorted list.), and C be the number of
                        classes you've chosen. Find the class width, W, with the formula W = (L-S)/C.
           c)  Determine the limits of the classes by adding the class width to each successive class. 
                Don't forget that the lower class limit is counted as part of the class width.
           d)  Determine the number of data points in each class as follows:
                (1)  If your data is in L₁, go to that list. Make sure your data is sorted in ascending order;  
                      then scroll down to the last number that falls within the upper limit of the first class.  At
                      the bottom of the list your will see L1(#), where # is the number of data items in your first
                      class. 
                (2)  Scroll down to the last item of the second class and subtract the number of items in the
                      first class from the number that appears in L1(#). Continue this until you come to the
                      end of the list.  Note that if you also want cumulative frequency, just write down the
                      numbers as you progress. 
            e)  Subtract 0.5 from each lower class limit of the first class to get the lower boundary of the
                 first class.  Add the class width to get successive boundaries.
             f)  Alternatively, you could do the histogram described below and use the data classes and
                values from that histogram.

      2)  Doing a Histogram with the TI-83 Plus:
           This procedure describes how to do  a simple histogram for which the calculator selects the class 
           width and, therefore, the number of classes..  First you need to get your data into lists. 
           a)  First go to the graphing screen by pressing the Y= button and deselecting any  functions so 
                that they won't be displayed with your graph.
                Now, go to the list and enter data as follows:
           b)  Press [STAT], [ENTER]
           c)  Then enter the numbers in L1.  (Or whatever list you choose.)
           d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
           e)  Cursor to the  icons opposite Type, select the third icon, histogram,  and press [ENTER] to 
                highlight the histogram icon.
           f)  Enter L₁ (or whatever list your data is in) opposite Xlist, by pressing 2nd, L_1.  Make sure there's a
               1 opposite Freq if you have ungrouped data.
          g)  Press [ZOOM]; then 9 (ZoomStat)  and the histogram will appear on the screen.
          h)  To find the numbers for the limits of the classes and the number of items in the class, press
                [TRACE]; then use  the cursor to move across the tops of the bars in the histogram and read the  
                various numbers. .

    3)  Selecting Your Own Class Widths for the Histogram Generated by the Calculator.
          a)  Enter your data into List L_1.  If your data is not in order, you can sort it by pressing STAT,  
               selecting SortA(, then entering the list name of the data (often L₁).  As an example, you might
               have this displayed on your screen: SortA(L₁.  Now, press ENTER, and your data will be sorted.
          b)  Now, from the sorted data, determine the class width and lower boundary of the lowest class as
               described under "Doing a Frequency Distribution by Hand" above. 
                Now do this:
          c)  Press [2nd], [STAT PLOT], highlight 1, and press [ENTER].
          d)  Highlight ON on the next screen; then highlight the histogram symbol.  Make sure L₁ is entered 
                for  the Xlist.  Note that if you do not have tables data ( where frequencies are given),  use  the default
               value, 1, for the  frequency.
           e)  Press [WINDOW],  and enter the lower boundary of the lowest class as Xmin  and your 
                 chosen  class width in Xscl.  Note that once the lower boundary  and class width are set, the 
                 upper limit is automatically determined.  Set ymin at zero.
            f)  Press [GRAPH] and the histogram will appear.  You can use [TRACE] to display the value
                  of the boundary limits and frequencies of a particular bar on the histogram.
           g)  If the graph extends above the top edge of the screen, Press WINDOW and increase the Ymax
                value.  I also usually set Ymin to -1. 

  4.  Constructing a Frequency Polygon from Ungrouped Data:
          After graphing the histogram, you can use TRACE to get the data for the frequency polygon and a cumulative
          frequency graph if you wish. 
          a)  Press TRACE and use the arrow to move across the histogram bars.  Record the values for x-min, x-max, and "n"
                on a sheet of paper in tabular form.
          b)  Add one-half the class width to each x-min value and record those values.   Store these values in a list, for example
                L₂ if you have your histogram data in L₁.  Store the corresponding values of "n" in L₃.
          c)  Press 2nd, STAT PLOT, ENTER.  If "On" is not highlighted; then select it and press ENTER.
          d)  Highlight the second icon on the first row; then enter L₂ opposite Xlist and L₃ opposite Ylist.
          e)  Press ZOOM, 9 and the graph will appear on the screen.
          NOTE:  Some teachers or texts prefer return-to-zero graphs.  If your course requires that, do the following after step b)  
          above:
          A.  Calculate a midpoint of a new class preceding the first class and another midpoint after the last class.  These
               values will be entered into L₂.  To do that place the cursor at the first item in L₂, press INS and replace the zero that 
               appears with your the first midpoint you calculated. Go to the bottom of the L₂ list and enter the second value you
               calculated.
         B.  Now you want to enter zero in L₃ opposite each of these new midpoints.  Place the cursor at the top of L₃ and press
              INS.  A zero will be added.  Now cursor to the bottom of the list and enter a zero opposite the last new midpoint
              that you entered in L₂.
         C. Proceed with step c) above. 

     5.  Constructing a Cumulative Frequency Chart (Ogive) Graph:
          a)  Enter the Xmax values that you recorded above in a list.  For example, L₄  if you still have data in the  other  lists.
          b)  Now, store the cumulative frequency data in L ₅ as follows:  Press 2nd, LIST, cursor to OPS, and press 6.  cumSum(
               will be posted to the home screen. 
          c)  With the cursor after the parenthesis, press 2nd, L_3, ), STO, 2nd, L_5, ENTER.  You will now have cumSum(L₃)→L₅
               pasted to the screen.
          d)  Press 2nd, STAT PLOT, highlight "On" if necessary and press ENTER
          e)  Highlight the second icon on the first row; then enter L₄ opposite Xlist and L₅ opposite Ylist.
               NOTE:  If you did a return-to-zero graph for the frequency polygon, go to the list and delete the last
               midpoint and zero in L₄ and L₅ respectively.
          f)  Press ZOOM, 9 and the graph will appear on the screen.

     6)   Relative Frequency polygon and Cumulative Relative Frequency (Ogive) Graphs: 
             Do these exactly  as in the frequency polygon and cumulative frequency graph above except that after storing
             the data (step b) for the frequency polygon) do this step:  Press 2nd, L₃ /N,  STO, 2nd, L₃ .  This will convert
             the data in L₃ to relative frequency.

     7)  Histogram Using Grouped Data:
           a)  Enter the midpoints of the classes into L₁ and the corresponding frequencies into L₂ .
           b)  Press 2nd, STAT PLOT, ENTER.
           c)  If "On" is not highlighted, select it and press ENTER.
           d)  Move the cursor to the histogram symbol and press ENTER; then enter L₁ opposite Xlist and L₂ opposite Ylist.
           e)  Press ZOOM, 9 and the histogram will be displayed. 
           Note:  If you want to select your own classes do this before pressing ZOOM 9 in step "e" above.
              1)  Press WINDOW and enter the lowest boundary value opposite Xmin
                    and  the class width opposite Xscl.  You may also want to change Ymin to something like zero or -1 so that
                    histogram will not be so far above the baseline.
              2)  Press GRAPH and the histogram will be displayed.

     8)  Frequency Polygon Using Grouped Data:
           Do this exactly like the histogram, except select the line graph icon, the second icon.  If you've already done the   
           histogram,  just change the icon and press GRAPH.

     9)  Cumulative Frequency (Ogive) Graph from Grouped Date:
           a)  Enter the upper class limits in a list, for example, L₃ if you have data in the first two lists.
           b)  If you have the frequency in L₂ , do the following: 
                A)   Press 2nd, LIST, cursor to OPS, and press 6.  cumSum(  will be posted to the home screen. 
                B)  With the cursor after the parenthesis, press 2nd, L_2, ), STO, 2nd, L₄ .  You will now have
                      cumSum(L₂) →L₄   pasted to the screen.  Press ENTER.
          c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
          d)  Highlight the second icon, and enter L₃ opposite Xlist and L₄ opposite Ylist.
          e)  Press ZOOM, 9 and the graph will be displayed.

    10)  Relative Frequency and Cumulative Relative Frequency Graphs for Grouped Data:
            Do these exactly as in the frequency polygon and cumulative frequency graph above except that  after storing
            the data for the frequency polygon do this step:  Press 2nd, L₄ /N,  STO, 2nd, L₄ .  This will convert the data
            in L₄ to relative frequency. This assumes that the frequency data is stored in L₄ .
            N is the total number of data points. 

    11)  Percentile Graphs:
            This graph is fairly similar to the Ogive graph.  We will do this in two groups of steps:  Preparing data
            and plotting data.
            Preparing Data:
            a)  Enter upper boundaries in L₁ and the corresponding frequencies in L2.  If you want the graph to start
                  at zero, enter the first lower boundary with zero for the frequency.
           b)  Press 2nd, QUIT to get out of the List.
           c)  Press 2nd, LIST, cursor to OPS, and press 6 to paste cumSum( to the home screen.
           d)  Press 2nd, L₂ , ), ÷ .  You now should have cumSum(L₂)/ on the home screen.
           e)  Press 2nd, LIST, cursor to MATH and press 5 to paste sum( to the screen.
           f)  Press 2nd, L₂, ).  You now should have cumSum(L₂)/Sum(L₂) on the home screen.
           g)  Press x (the multiply symbol), 100, STO, 2nd, L₃.  You now should have
                cumSum(L₂)/Sum(L₂) *100→L₃ pasted to the home screen. 
           h)  Press ENTER and the data will be stored in L₃ .
          Plotting the Data:
           i)  Press 2nd, STAT PLOT, ENTER
           j)  Select the second icon and enter L₁ opposite Xlist and L₃ opposite Ylist.
           k)  Press ZOOM, 9 and your graph will be displayed.
           l)  You can find the exact percentiles of the boundaries by using TRACE, and approximate percentiles of
                other x-values by using the cursor.
        
    12) Box and Whisker Plot
         a)  First go to the graphing screen by pressing the Y= button.  Deselect any  Y= functions so that 
               they won't be entered on your graph.  If you choose, clear the list as described at the beginning
              of this document.
         b)  Press [STAT], [ENTER] to go to the list tables.
         c)  Enter your  numbers in L1.  (Or whatever list you choose.)
         d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn on  Plot 1.
         e)  Opposite the word Type, cursor  to the icon that represents a box-and-whisker plot, icon 5, and 
              press [ENTER] to highlight the box plot icon.  (See the note at the end of this topic for when to
              use icon 4.)
         f)  Enter the list you put the data in, usually L₁, in the Xlist, by pressing 2nd, L_1. or whatever list
             you chose.  
         g)  Press [ZOOM]; then 9 (ZoomStat)  and the box-and-whisker plot will appear on the screen.
         h)  To find the numbers for the limits of the quartiles, press [TRACE]; then use  the cursor to move
               across the diagram and obtain the values for quartiles or the beginning and ending values.
              NOTE:  If you have one or two outliers (numbers much larger than the rest) you may want to use  
             icon 4.  This will not include the outliers in the last whisker, but will plot them as separate points
             after the end of the last whisker.
 
 13) Box and Whisker Plot by Hand
       You can save yourself considerable calculation if you use the calculator to find Q₁, Median, and Q₃
       when doing a box-and-whisker plot by hand.  To find those values do the following:
       a)  Press STAT, cursor to CALC  and press ENTER.  "1-Var Stats" will be displayed on the home 
            screen.
       b)  If your data is in list L₁ just press ENTER.  Otherwise press 2nd and the list name where your
            data is stored.
       c)  Cursor down and you will find Q ₁ , Q₃ , and Med listed.  "Med" is the median.
 
14)  Discrete Probability Distribution
        Let's take a simple example to demonstrate this:  Suppose a word is flashed on a screen several
       times while people are  trying to recognize the word.  The list below indicates what percentage of the
       group required a given number of flashes to recognize the word.  
              No. of Flashes     1        2        3     4       5
              Percent                27      31     18     9      15
              P(x)                     .27     .31     .18    .09   .15
         In summary, the method is to enter the number of flashes into list L₁ and the corresponding P(x)  
         values  into L₂ as the frequency.  The details are as follows:
         a)   Enter the number of flashes in list L₁ and the corresponding P(x) values in L₂ opposite the
                number of flashes.  (How to enter data in a list is covered at the beginning of this document.)
         b)  Press STAT, cursor to CALC and press ENTER.  1-Var Stats will be displayed on the home
               screen.
         c)  Press 2nd, L₁, press the comma,  then 2nd, L₂ .  You should now have 1-Var Stats L₁, L₂ on the
              home screen.
         d)  Press ENTER and the values for the mean (expected value), standard deviation and other
               statistics will be displayed.
          e)  If you need the variance, merely re-enter the value for the  standard deviation, σ_x  , and square it^,

  15)  Doing a Discrete Probability Distribution by Hand
         Many teachers still see value in cranking out the numbers for these statistics, so here are methods
            to take some of the drudgery out of doing the arithmetic.
         The mean can be obtained by the following formula: mean = Σxp(x).
            To obtain the individual values and store them in list L₃, do the following:  (The x-values should
             should be stored in L₁  and the p(x) values in L₂.)
             a)  Press 2ND, L₁, x, 2ND, L₂, STO, 2nd, L₃.  You will now have L₁*L₂→L₃ pasted to the home
                  screen.
             b)  Press ENTER and you will have the individual values stored in list L₃ and displayed on the
                   home screen.
             c)  To get the sum of these values,  do this.
                      (1)  Press 2nd, LIST; cursor to MATH, and press 5.  The expression sum( will be pasted to
                              the home screen. 
                      (2)  Press 2ND, L₁ ,x, 2ND, L₂ , ), STO, 2ND, L₃.  You will have sum(L₁ *L₁)→L₃ pasted
                             to the home screen.
                      (3)  Press ENTER and the sum of those values will be displayed.  Obviously if you only
                             need the mean and not the details of the arithmetic, do only part c.
             You can obtain the variance and standard deviation by first solving for the variance using
             the formula:  Σx² P(x) - µ² where µ is the mean obtained as above.  To obtain the individual
             values of the first term,  x² P(x). and store them in list L₃, do the following:
              a)  Press 2ND, L₁, x², ,x, 2ND, L₂ , STO, 2ND, L₃.  You will have L₁²*L₂→L₃ pasted to the home
                   screen.
              b)  Press ENTER and the individual values will be entered in list L₃ and pasted to the home
                    screen.
              c)  To get the sum of these values do the following:
                      (1)  Press 2nd, LIST; cursor to MATH, and press 5.  The expression sum( will be pasted to
                              the home screen. 
                      (2)  Press 2ND, L₁ ,x² ,x, 2ND, L₂ , ), STO, 2ND, L₃.  You will have sum(L₁²*L₂)→L₃ pasted
                             to the home screen.
                      (3)  Press ENTER and the sum of those values will be displayed and stored in L₃.  Obviously
                             if you only need the sum of the values in the first term  and not the details of the arithmetic,
                            do only part c.
              d)  Now subtract the value for µ² from the last value obtained and that will be the variance.
               e)  To obtain the standard deviation, take the square root of the variance as follows:
                        (1)  If you have just calculated the variance do press 2ND, √, 2nd, ANS, ENTER.  Otherwise,
                               insert the value for the variance in place of ANS.
            NOTE:   Obviously, if you only want to obtain the values for the  these three parameters,  you can
            use this method, but it is much easier to use method 14 above.   Just as information, the total
           expression for the variance using this method would the this:  sum(L₁²*L₂) - (sum(L₁ *L₂))² .

16)  Calculation of Coefficient of Variation from List Data:
         The coefficient of variation, CV=s/x-bar, is a simple arithmetic calculation if you have the mean
         and standard deviation.  But calculations from a list are a little more involved.  Here's an easy way
         to do it.
         a)  Store the data in a list, for example L₁, and press 2nd, QUIT to leave the lists.
         b)  Press 2nd, LIST and move the cursor to MATH.
         c)  Press 3 to paste mean( to the home screen.
         d) Press 2nd, L₁, ), and then press the divide symbol.
         e)  Press 2nd, LIST, move the cursor to MATH, and press 7.
         f)  Press 2nd, L₁ and then press ENTER to display the CV.

17.  Finding the Standard Deviation and Variance of Ungrouped Data:

A.  Calculated by the Calculator Only    
     a)  Entering Data:
          1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a    
               list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
          2)  To enter data, just place the cursor where you want to enter the data and press the 
               correct numbers, then press ENTER.  You don't have to erase old data if there is  
               already data in the list, but if the old list  is longer than the new list, you will need to
               delete the remaining old data items.  Just place the cursor over the data and press
               DEL. 
      b)  Suppose that you have the sample of data listed immediately below and you want to find   
            the standard deviation and variance.
            Data:  22, 27, 15, 35, 30, 52, 35
       c)  Enter the data in list L₁ as described under Entering Data immediately above, then press 
            2^nd , QUIT to leave the tables.
       d) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats”
            should be pasted to the home screen. If the data is in L₁, just press ENTER, otherwise  
            press 2^nd and the list number where the data is stored.
             

  B.  Calculating  Numbers to Plug into a Formula::
     The standard deviation can be found easily by using 1-Var Stats as described above, but  
      many teachers require that students do the calculations by hand to learn the details of the  
      process.  The following give a method for using the TI-82, TI-83 Plus, or TI-84 for doing much  
     of the arithmetic required and obtaining numbers to plug into the formulas.
     Suppose that students did sit-ups according the table shown below.    

The variance computation formula is as follows:   s² = [(Σx² -(Σx)²)/n)]/(n-1), where s² is the variance . So,  we will need  ∑x² and ∑x to plug into the formula.
   a)  Enter the data in the table as indicated previously in this document.  Press 2nd, QUIT to get  
      out of the lists.
   b) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats”
      should be pasted to the home screen. If the data is in L₁, just press ENTER, otherwise 
      press 2^nd and the list number where the data is stored.
   c)  Copy n=7, ∑x = 216, and ∑x² =7492.
  NOTE:  You now have enough data to plug into the formula and solve for the variance and standard deviation.  If you are not required to do the detailed calculations, ship to filling in the formula in step “f.”  Otherwise, continue with the next step.

    d)  Now we’ll need an x² column.   Press 2^nd, L₁, x², STO, L₂.   You should have
         L₁² →L₂ on the home screen.
    e)  Press ENTER and the numbers will be displayed on the home screen and stored in list L₂.
    f)  Now, we want to use the number that we previously recorded to plug into the variance
        formula.   So, at the home screen enter (7492-216²/7)/(6). 
     g)  Press ENTER and you should have 137.8…, which is the variance.
     h)  To find  the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you will have
           11.39...

18.  Finding the Variance and Standard Deviation of Grouped data.
    A.  Calculated by the Calculator Only:
      a)  Entering Data:
          1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a    
               list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
          2)  To enter data, just place the cursor where you want to enter the data and press the 
               correct numbers and press ENTER.  You don't have to erase old data if there is already  
               data in the list, but if the old list  is longer than the new list, you will need to delete the  
               remaining old data tems.  Just place the cursor over the data and press DEL. 
      b)  Suppose that you have the sample of data listed in the table below and you want to find
            the standard deviation and variance.
        

       c)  Enter the class midpoints in list L₁.  You can either do the midpoints by hand or calculate   
            and store them in list L₁ as follows:
            (1) Store the lower boundaries in list L₁ and the upper boundaries in L₂. 
            (2) Press 2ND, QUIT to get out of the list editor and press (, 2ND, L₁, + 2ND, L₂,), divide
                  symbol, 2 STO, L₁.  You should have (L₁ + L₂)/2→ L₁ on the home screen.  Press     
                  ENTER and the midpoints will be stored in L₁.
       d)  Enter the frequencies in L₂ as described under Entering Data immediately above, then
            press  2^nd , QUIT to leave the tables.
            Now we’ll calculate the required statistics.
       e) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats”
            should be pasted to the home screen. Press 2^nd, L₁ ; then press the comma and finally
            press 2^nd, L₂.  
        e)  Press ENTER, and the standard deviation along with several other statistics will be
             displayed.  The sample standard deviation is 14.868….
         f)  To find the variance, just square the standard deviation by entering the number, pressing
             the x² button, and then ENTER. 
 

  B.  Calculating  from Grouped Data to Plug into a Formula:
     The standard deviation and variance for grouped are similar to ungrouped data except that the
     x-values are replaced by the midpoints of the classes.  Let's assume some sort of grouped  
     data as indicated by the first and third columns below.
    

The formula for the grouped data variance is this:
           s² =( Σx²  -(Σxf)² /Σf)/(Σf-a) 
a) You can either do the midpoints by hand or calculate and store them in list L₁ as follows:
     (1) Store the lower boundaries in list L₁ and the upper boundaries in L₂. 
     (2) Press 2ND, QUIT to get out of the list editor and press (, 2ND, L₁, + 2ND, L₂,), divide
          symbol, 2 STO, L₁.  You should have (L₁ + L₂)/2→ L₁ on the home screen.  Press ENTER
          and the midpoints will be stored in L₁.
b)  Press STAT, ENTER to go to the lists and store the frequencies in list L₂.  After you have     
     finished entering the frequencies and midpoints, press 2^nd, QUIT to leave the lists.
 Now let’s calculate the required numbers.
c) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats”
    should be pasted to the home screen. Press 2^nd, L₁ ; then press the comma and finally
    press 2^nd, L₂. 
d)  Press ENTER and several statistics along with the standard deviation will be displayed.   
     Record the standard deviation, Sx =14.868 for a reference.  Also record ∑x=3780,
     ∑x²=296600, and n=50.  You’ll need these values later. 
      Notice that the value for ∑f is listed as n in the calculator and ∑xf is listed as ∑x and ∑x²f is  
      listed as ∑x². 
NOTE:   You now have enough numbers to plug into the formula and solve for the variance.  
If you are not required to do the detailed calculations to fill in the table, skip to item “j” below. 
Otherwise continue with the next step.    
e)  Calculate xf and store it in L₃ by pressing  2ND, L₁, *, 2ND, L₂, STO, L_3. You should have
      L_1*L₂→L₃  on the home screen.  Press ENTER and the products will be stored in list L₃  and will  
      be displayed on the home screen.
f)  Calculate x²f by pressing 2ND, L_1, x² , * , 2ND,  L₂, STO, 2ND, L₄ .  You should now have
     L₁² *L₂→L₄ on the home screen. 
g)  Press ENTER and the results will be stored in list L₄  and will be displayed on the home
      screen.
 h)  You don’t need to calculate Σf.  That is the value for “n” that you previously recorded.  
 i)   You don’t need to calculate Σxf.  That is the value for ∑ x that you previously recorded.
 j)  Now, you want to plug the appropriate numbers into the formula for the variance. From the  
     home screen enter (296600-3780²/50)/(49)
 k)  Press ENTER and you should have 221.06, which is the variance.
 l)  If you want the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you will have 14.868...

 III. Two-variable Statistics
 1)   Scatter Plot
       First you need to get your data into lists. 
       a)  Go to the graphing screen by pressing the Y= button and deselecting any  functions so that 
             they won't be entered on your graph.  If you want to clear the lists before entering data, see the
             note at the beginning of this document.
       b)  Press [STAT], [ENTER] to go to the list tables.
       c)  Enter the data-point numbers ( the x-values)  in L1 and the corresponding values (y-
             values)   in L2.  (If your data is not in order you can sort in order by pressing [STAT], select 2,  
             SortA( or SortD( for descending order.  SortA( , or SortD( will be posted to the home screen.  Press
           [2nd], L₁, 2nd, L2, [ENTER].  BE CAREFUL!  If your data in L₂ is not in ascending order when 
           correlated to L_1, then your data in L₁ and L₂ will not be correlated correctly after sorting.
      d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
      e)  Cursor to the scatter diagram, the first icon opposite Type,  and press [ENTER] to highlight the
            scatter diagram icon.
      f)  Enter L₁ in the Xlist, and L₂ in the Ylist (do this by pressing 2nd and the appropriate list button);  
            then select the type marker you prefer.  (I like the + symbol. ).
      g)  Press [Zoom], 9 and the scatter plot will appear on the screen.

2)  Plotting  x-y line chart
      Do that the same as the scatter plot in item 1 above except that when you select the type, choose the 
      second icon for the line symbol rather that the scatter-diagram  icon.

3)  Regression Analysis:
      Assume that you have the following information on the heights and weights on a group of young
     women:

       First you need to get your data in lists.  You can do that from the home screen, but if you have any   
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and
            the corresponding values in L2.
      b)  After you have finished entering data, Press[STAT]. 
      c)   Cursor to CALC and press <8>, [ENTER] (Where <8> is just the number 8 from the keyboard.)
            Note that if you want to use QuadReg or some other analysis, press the number to the left of that
            entry. LinReg (a+bx) will appear on the screen if you chose 8.    
       d) If you want to graph the equation of the best-fit line, ship to item “e” below.  If you have your data  
           in the L₁ and L₂ as described above, just press ENTER.  If  you have your data in other lists, you’ll  
           need to  enter the lists by pressing 2^nd, press the list number for x_, comma, 2^nd, press the list number
           for y; then press ENTER.  In either case a, b, r², and r will be displayed.  Note that if r and r²  are not 
           displayed, press 2^nd , CATALOG, D, ; then scroll to DiagnosticON and press ENTER.  
            ANSWER:  If you pressed ENTER you should have these values:  a=-186.47.., b=4.705…,
           r² =.63366…, and r=.7979…
      e)  If you want to graph the equation,  then immediately after LinReg(a=bx),  press  [2nd], [L1] [,],  
           [2nd],  [L2], [, ].  ( Note that the commas outside the brackets are just separators so you can tell what
           I'm doing. They do not appear in the syntax.
       f)  Now, you want to store this as a Y-variable, say, Y1. So, do it this way:  Press [VARS], Cursor to
            Y-VARS, [ENTER], [ENTER].  You should now have this on your screen.  QuartReg L1, L2, Y1
       g)  Press [ENTER].   After a few seconds a long equation with coefficients having several decimal 
            places will appear on the screen.
      h)  To graph that, you could just press GRAPH.  Depending on your data values, you may need to 
            adjust the WINDOW to get a good display. 
      i)  Note that if you have already done the regression equation without storing it in a Y-variable, you 
           can do that as follows:
          1)  Press Y=; then VARS; then 5 (Statistics).
          2)  Cursor over to EQ and press 1 (or ENTER). The regression equation will be stored in the Y1=
               position.  You can then press GRAPH to graph it.

4)  Plotting a graph with the scatter plot and the regression equation on the same axis.
     First you need to do the regression graph as described above in item 3.  Now, you want to put the 
     scatter plot on the screen with the graph. To do this:
     a) Press [2nd], [STAT PLOT] and press [ENTER], ENTER to turn Plot 1 on.
     b) Cursor to the scatter diagram for Type (the first icon) and press [ENTER] to highlight the scatter  
         diagram.
     c) Enter L1 in the Xlist, and L2 in the Ylist; then select the type marker you prefer. (I like a + ).
     d) Press ZOOM, 9 (for ZoomStat) and the scatter plot and best-fit graph will appear on the screen.
     e)  You can press [TRACE] to display the x-y values of the data points, or press the down arrow to 
           jump  to points on the line.
Note that if your data has several decimal places and you'd rather have fewer, you can make the data
 friendlier by making the x-distance (xmax-xmin) a multiple or sub-multiple of 9.4. 

5)  Finding the Correlation Values r and r² Using a Formula: 
                  Assume that you have the following information on the heights and weights on a group of young women:

       First you need to get your data in lists.  You can do that from the home screen, but if you have any   
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and
            the corresponding values in L2.
            NOTE:  The formula for “r” is this:  (nΣxy –ΣxΣy)/[(√nΣx²- (Σx)²)(√nΣy²- (Σy)²)].  So, you will  
            need  Σx, Σy, ΣxΣy, Σx², Σy^2,, and n.  You can get all of these by using the 2-Var Stats
            function.  Use that as follows:
      b)  With the data in lists L₁ and L₂ press STAT, move the cursor to CALC, and press 2.  The
           expression 2-Var Stats, should be displayed on the screen. 
      c) If the data are in L₁ and L₂, press ENTER and the necessary values will be displayed.  If the
          data are not in those lists, you will have to enter the list numbers where the data are stored.  
          Notice that you will need to scroll down to get some of the values on the screen.  Record the
          values for these parameters:  Σx=521,  Σx²=33979, n=8, Σy=960, Σy²=116900, Σxy=62750.
      NOTE:  Just a few words on entering the data in the calculator:  All denominators and
      numerators with  more than one term must be enclosed in parentheses.  On the TI-83 Plus or
     TI-84,  a square root expression must be enclosed in parentheses.  Example:  √(nΣx²- (Σx)²).
        Now let’s plug the numbers into the equation for r:
        d)  r= (nΣxy –ΣxΣy)/[(√nΣx²- (Σx)²)(√nΣy²- (Σy)²)]
                = (8*62750-521*960)/(√(8*33979-521²)(√(8*116900-960²))
               =.7979…..
       e)  Some students seem to have difficulty accurately entering a long expression such as in item "d." 
            Those students can do the calculation without loss of accuracy by using the following method.
          1)  Enter the numerator in the calculator and store it in variable N.  In this manner: 
               8*62750-521*960, STO, ALPHA, N. 
          2)  Calculate the denominator and store it in two separate variables M and D. In this manner
               √(8*33979-521² )  , STO, ALPHA, M; then √(8*116900-960²), STO, ALPHA, D.
          3)  N÷(M*D), ENTER.  You'll get the same answer as above.

6)  Finding the Values a and b for the Best-Fit Equation Using a Formula:
      Assume that you have the following information on the heights and weights on a group of young women:

      The formula for “b” is this:  (nΣxy –ΣxΣy)/(nΣx²- (Σx)²).  So, you will need to record the values
      for .  x-bar, y-bar, Σx, Σy, ΣxΣy, Σx², Σy², and n.. You can get all of these by  using the 2-Var Stats function.  
     Use that as  follows:
      a)  With the data in lists L₁ and L₂ press STAT, move the cursor to CALC, and press 2.  The
           expression 2-Var Stats, should be displayed on the screen. 
      b)  Press ENTER and the necessary values will be displayed.  Notice that you will need to
           scroll down to get some of the values on the screen.  Record the values for these
          parameters: ).  So, you will need to record these values: x-bar=65.125, Σx=521,  Σx²=33979,
          n=8, Σy=960, y-bar=120, Σy²=116900, Σxy=62750
      c)  Plug these numbers into the formula and then enter the expression your calculator. 
           Just a few notes on entering the data in the calculator:  All denominators and numerators
          with more than one term must be enclosed in parentheses.  On the TI-83 Plus or TI-84,  a
          square root expression must be enclosed in parentheses.  Example:  √(nΣx²- (Σx)²)
   d)  Enter the values in the calculator  for this formula:
         b=(nΣxy –ΣxΣy)/(nΣx²- (Σx)²).
           =(8*62750-521*960)/(8*33979-521²)
           =4.7058…..
     e)  Now, calculate the value for a from the formula:
          a= y-bar –b(x-bar)
            =120-4.7058 *65.125
           =-186.465…
  f)  Some students seem to have difficulty accurately entering a long expression such as in item "d." 
          Those students can do the calculation without loss of accuracy by using the following method.
          1)  Enter the numerator in the calculator and store it in variable N.  In this manner: 
               8*62750-521*960, STO, ALPHA, A. 
          2)  Enter the denominator and store it in variable D.  8*33979-521² , STO, ALPHA, D.
          3)  Enter  N÷D and press ENTER.  You'll get the same answer as above.

 IV.  Aids in doing statistics by hand.
       General:  Often in book problems in school you'll need to do a lot of calculations by hand.  These  
         techniques will save you a lot of arithmetic.
 1.  Arranging Data In Order.  (This is the same as item 2 in section I above, which I will repeat here.)
     a)  Enter the data in one of the lists as indicated in Section I.    
     b)  Press STAT, 2 (SortA).  This will paste SortA to the home screen. 
     c)  Press 2nd, L₁ (or whatever list you want to sort); then press ENTER.  "Done" will be displayed
          on the home screen, indicating your data has been sorted. Note that you can also sort data in 
          descending order with SortD.

2.  Finding Mean (x-bar), ∑x, or ∑x² , σ, Median, Q₁, Q₃ for Grouped or Ungrouped Data.
    For Ungrouped Data:
    a)  After entering your data in the list as described in item 1 of Section I, above, press STAT, and
         cursor over to CALC, and press ENTER. "1-Var Stats" will be pasted to the home screen.
    b)  Enter the list name you want to operate on by pressing 2nd; then the list number, for example L_1.
     c)  Press ENTER.
    d)  A number of results will be displayed on the home screen.
     NOTE:  You can also find these values for discrete random variable statistics by entering the values
                 of the variable in L₁ , for example, and the corresponding data values in L₂.
   For Grouped data:
    a)  Find the midpoints of each group and enter those values in L₁; then enter the corresponding frequencies
        L₂.  Entering data in a list is described in item 1 of Section I, above.
        
    b)  Press STAT, cursor over to CALC, and press ENTER. "1-Var Stats" will be pasted to the home screen.
     c)  Press 2nd, L₁, 2nd, L₂; then press ENTER.
    d)  Various statistics will be displayed on the home screen.

3.  Finding products such as xy or (x-y):
     a) Assume that your x-data is in L₁ and your y-data is in L₂.  Then obtain the product by pressing
        2nd, L₁; x (multiply symbol), 2nd, L₂, ENTER.
     b)  If you want the data stored in a list, L₃ for example, before pressing ENTER in item a, press 2nd,
          L₁, STO, 2nd, L_3.  Then press ENTER.
     c)  Obviously, x-y can be obtained by merely substituting the subtraction symbol for the
          multiplication symbol in atep a) above.

4.  Squaring operations such as elements of lists.
     a)  To square the elements of a data set, first enter the data in a list, for example L₁.
     b)  Press 2nd, L₁; then the x² symbol, ENTER.  The squared elements will be displayed.
     c)  If you want to store the squared data in a list, for example L₃, then before pressing ENTER in
          item b above, press 2nd, STO, 2nd,  L₃.  Then press ENTER.
     d)  If you want to multiply corresponding elements of two lists and square each result; then your
         expression should be like this:  (L₁ * L₂)² .

5.  Find x-x¯ (Sorry, I have no symbol for the mean, so I displaced the bar.) from the data in
     list  L₁.
     a)  Enter 2nd, L₁, -, 2nd, LIST.  Note that" -" is a minus sign not a negative sign.
     b)  Cursor to MATH and press 3.  You should now have "L₁-mean(" pasted to the home screen.
     c)  Press 2nd, L₁, ENTER.  The result will be displayed on the home screen. 
     d)  If you want to store the results in a list, for example L₃, then before ENTER in item "c" above, press
         STO, 2nd, L₃; then ENTER

 6.  Finding (x-x¯ )²  
      a)  Press (, 2nd, L₁, -, 2nd, LIST.
      b)  Cursor to MATH and press 3.  You should now have "(L₁-mean(" pasted to the home screen.
        c)  Press 2nd, L₁,),),x² .  The expression ((L₁-mean(L₁))² should now be displayed on the screen.
          Press ENTER and the results will be displayed on the home screen.
      d)  If you want to store the results in a list, for example L₃, before pressing ENTER in item "c"
           above, press STO, 2nd, L₃; then ENTER.

7.  Finding (Σx)² and Σx²
     Some computation formulas for the standard deviation require (Σx)² .  To find that, do the following:
      a)  Enter your data in a list as described at the beginning of this document.  Press 2nd, QUIT to get
          out of the list. Press ( to enter a parenthesis on the home screen.
      b)  Press 2nd, LIST, and cursor over to MATH.
      c)  Press 5.  "(sum(" should be entered on the home screen.
      d)  Press 2nd, L₁ or whatever list your data is stored in.
      e)  Press ), ), x² .  You now should have (sum(L₁))² on your home screen.
       f)  Press ENTER and the results will be displayed on the screen.
       g) Σx² can be found by using the "1-Var Stats" function under STATS, CALC, but you can also
          find it by entering "sum L₁² "

8.  Notice that you may also do several other operations by pressing 2nd, STAT; then moving the cursor to
    MATH and entering the list name that you wish to operate on.

     k)  Press ENTER and you should have 138.1428, which is the variance.
     l)  If you want the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you will have 11.75...

V.  Permutations, combinations, factorials, random numbers:
 1. Finding Permutations.
    a)  Suppose we want the permutations (arrangements) of  8 things 3 at a time, enter 8 on the home 
         screen.
    b) Press MATH and cursor over to PRB and press 2, (nPr). You will have 8 nPr pasted to the screen.
    c)  Enter 3 and press ENTER.  You will get 336.

 2. Finding Combinations:.
     a)  Suppose we want the combinations (groups) of  8 things 3 at a time, enter 8 on the home screen.
     b) Press MATH and cursor over to PRB and press 3. (nCr). You will have 8 nCr pasted to the screen.
     c)  Enter 3 and press ENTER.  You will get 56.

 3. Finding Factorials.
     a)  Suppose we want 5 factorial (5!).  From the home screen press 5.
     b) Press MATH and cursor over to PRB and press 4 (!)). You will have 5! pasted to the screen.
     c)  Press ENTER and you answer, 120, will be displayed.

 4.  Randomly generated data sets:
      Sometimes problems use a randomly generated set of data. Suppose we want to generate 10 
      random numbers between 1 and 50 and store them in List 1.  The proper syntax is randint(lower,
      upper, how many).  That can be obtained as follows:
      a)  Press MATH, cursor over to PRB and press the number 5. randint( will appear on the screen.
      b)  Enter 1, 50, 10, so that your screen displays randint(1,50,10).  Press ENTER
      c)  Now, if you want to cause these numbers to be stored in L1, before pressing ENTER in item b,
          press STO;2nd, L₁. The  entries, randint(1,50,10)->L₁, will appear on the screen.
      d)  Press ENTER and the numbers generated will appear on the screen and will be stored in list L₁.

VI.  Normal Distribution:
      Note:  In this section, a general method will be outlined; then a specific example will be worked.  The same
      problem will be used in several of the examples.

       General, normalcdf(:  This function returns the value of the area between two values of the random variable
          "x."  This can be interpreted as the  probability that a randomly selected variable will fall within those two
          values of "x," or as a percentage of the x-values that will lie within that range.  The syntax for this function is
          normalcdf( lower bound, upper bound, μ, σ.  If the mean and standard deviation are not given, then the
          calculation defaults to the standard normal curve with a mean of 1 and a standard deviation of 0. I use the
          values -1E9 and 1E9 for left or right tails.  The E in obtained by pressing 2nd, EE.  This can be used to solve
          such problems as the following: P(x<90), P(x>100), or P(90<x<120).  If µ and σ are omitted, the default
             distribution allows the solution of the following: P(z<a), P(z>a), or
          P(a<z<b).

   1.  normalcdf(: Area under a curve between two points with μ (mean) and σ (std. dev.) given.
        a)  Press 2nd, DISTR, 2.  The term "normalcdf(" will appear on the home screen.
        b)  Enter the number for the left boundary, right boundary, μ, and σ in that order.  You do not need
             to close the parentheses, but it's okay if you do.
        c)  Press ENTER and the value of the area between the two points will be displayed. Notice that
             you do not explicitly convert the points to z-values as in the hand method.
          Ex. 1:  Assume a normal distribution of values for which the mean is 70 and the std. dev. is 4.5.
         Find the probability that a value is between 65 and 80, inclusive.
          a)  Complete item a) above.
          b)   Enter numbers so that your display is the following:  normalcdf(65,80,70,4.5.
          c)  Press ENTER and you'll get 0.85361 which is, of course, 85.361 percent.

    2.  normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std. dev.) given.  
         Ex. 2:  In the above problem, determine the probability that the value is less than 62.
           a)  Complete item a) in the general method above.
           b)   Enter numbers so that your display is the following:  normalcdf(-1E9, 62,70,4.5.  Notice that
               the "-" is a negative sign, not a minus sign.  Enter "E" by pressing 2nd, EE (The comma
               key.)
           c)  Press ENTER and you'll get 0.03772 which is, of course, 3.772 per percent.

     3.  normalcdf(: Area under a curve to the right of a point with μ (mean) and σ (std. dev.) given.
         Ex. 3:  In the above problem, determine the probability that a value is greater than or equal to 75.
           a)  Complete item a) in the general method above.
           b)   Enter numbers so that your display is the following:  normalcdf(75, 1E9,70,4.5. 
                Enter "E" by pressing 2nd, EE (The comma key.)
           c)  Press ENTER and you'll get 0.13326 which is, of course, 13.326 per percent.

     4.   ShadeNorm(:  Displaying a graph of the area under the normal curve.
           General:  This function draws the normal density function specified by µ and σ and shades the area
           between the upper and lower bounds.  This is essentially a graph of normalcdf(.  It will display the
          area and upper and lower bounds.  Not including µ and σ defaults to a normal curve.  The following
          instructions, "a" through "c," are general instruction  to follow.
           a)  First turn off any Y= functions that may be active.  Do this by moving the cursor to a
                highlighted = sign and pressing ENTER.
           b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
               home screen.  Enter the correct parameters depending on whether the problem is like 1, 2,
               or 3 above.  
           c)  Press ENTER, and the graph may be visible on the screen.  You will almost certainly need 
                to reset the Window parameters by pressing WINDOW and changing Xmin, Xmax, Ymin, and
               Ymax settings to get a decent display. As a first approximation, set Xmin at 5 standard  
               deviations below the mean and Xmax at 5 above the mean. (See the following example.)  Start out with
               a Ymax about   0.3 and go from there.  You can set the Ymin at 0, or if you wish, set it at about
               negative one-fiftieth of Ymax.  You may need to fine tune from there.
            Ex 1:  Draw the graph of example 2 above.
              a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.01, Ymax = 0.1.  You can reset the 
                   scales as you choose to eliminate the broad baseline. 
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
                  home screen.
              c)  Enter parameters so that your display looks like this:  ShadeNorm(-1E9, 62, 70, 4.5.
              d)  Press ENTER and a reasonable looking graph should appear on the screen.

       5.  invNorm(:  Inverse Probability Calculation: 
            Find the number x, in a normal distribution such that a number is less than x with a given 
            probability. The syntax for this is invNorm(area, [μ, σ]).  The part in brackets indicates that there
            is a default for those values.  The default is the standard curve with mean=0 and standard deviation. is  1.
           Ex. 1:   In Ex. 1 immediately above, find the number x, such that a randomly selected number will be below
             that number with a 90% probability. 
            a)  Press 2nd, DISTR, 3 to select invNorm(.
            b)  Enter parameters so that your display looks like this: invNormal(.90,70,4.5.
            c)  Press ENTER and your answer will be 75.766.
            Ex. 2:  Given a normal distribution with a mean of 100 and standard deviation of 20.  Find a value X_o such
               that the given x-value is below X_o is .6523.  That is P(X<X_o) = .6523.
                 a)  Press 2nd, DISTR, 3 to place "invNORM(" on the home screen.
                 b)  Enter information so that the entry looks like the following:  invNORM(.6523,100, 20. 
                 c)  Press ENTER and your answer will be 107.83.
              Ex. 3:  What is the lowest score possible to be in the upper 10% of the class if the mean is 70 and the
                standard deviation is 12?
                 a)  Press 2nd, DISTR, 3. to place "invNORM(" on the home screen.
                 b)  Enter information so that the entry looks like the following:  invNORM(1-.1,70, 12.  Your answer will
                     be 85.38 or 86 rounded off.

       6.   ShadeNorm(:  Window Settings for Graphing (shading) the Inverse Probability area:
            General:  If you are accustomed to graphing using the standard WINDOW settings called by
              ZOOM, 6, then you're in for a big surprise if you use those settings for graphing the normal
              curve.  So, before you display the ShadeNorm( function, press WINDOW and set the values
              as follows:
               a)  Xmin = μ - 4σ. Round of to the next integer.
               b)  Add the same number to the mean that you subtracted from the Xmin to get Xmax.
               c)  Xscl= Set at the standard deviation.
               d)  Ymin=0.  Some people like to set this at a small negative number, but if you have
                    problems with a wide range of std. devs. you'll have to keep changing it.  I set it at 0; then 
                    I'm done with it. 
               e)  Ymax= As a first approximation, set this at 0.4/σ.
                f)  Yscl= Most of the time the y-axis is not displayed, so I usually just set it at 0.01 and
                    leave it there.

        7.  ShadeNorm(:  Graphing (shading) the Probability area:
            Ex. 1:  Obviously if you wanted to graph the example immediately above, you could use the
            ShadeNorm( using the lower bound of -1E9 and the upper bound of 75.766.  You would do that
            as follows: 
             a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.005, Ymax = 0.1.  You can reset the 
                   scales as you choose to eliminate the broad baseline. 
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
                  home screen.
              c)  Enter parameters so that your display looks like this:  ShadeNorm(-1E9, 75.766, 70, 4.5.
              d)  Press ENTER and a reasonable looking graph should appear on the screen.
               Note that if you wanted to shade the region where the probability would be greater than 90%,
               you would choose 75.766 for the lower boundary and 1E9 as the upper bound.
            
            Ex. 2:  Suppose you wanted to graph a distribution and shade the area between the points 40 and 54,
              with a mean of  46 and a std. dev. of 8.5
            a)  Press WINDOW and set Xmin=12, Xmax=80, Ymin=-.005, Ymax = 0.06.  You can reset the 
                   scales as you choose to eliminate the broad baseline. 
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
                  home screen.
              c)  Enter parameters so that your display looks like this:  ShadeNorm(40, 54, 46, 8.2.
              d)  Press ENTER and a reasonable looking graph should appear on the screen.  The area
                   under the curve, 0.603198, will be displayed on the screen along with the upper and lower
                   bounds.

        8.   normalpdf(:  Probability Distribution Function using normalpdf( :
             General:  This function is used to find the fraction, and therefore also the percentage, of the
                distribution that corresponds to a particular value of x.  The syntax of this function is
                normalpdf(X, μ, σ
             A) Finding the Percentage of a Single Value:
              Ex. 1:  Suppose that the mean of a certain distribution is 60 and the standard deviation is 12. 
              What percentage of the population will have the value 50?
                 a)  Press 2nd, DISTR, 1 to paste normalpdf( to the home screen.
                 b)  Enter data so that your display is as follows:  normalpdf(50,60,12.
                 c)  Press ENTER and your answer should be .02317 which is about 2.3 percent.
             B)  Graphing the distribution:  
               Ex. 1:  Suppose that the mean of a certain distribution is 60 and the standard deviation is 12. 
                Investigate percentages for several x-values.
                 a)  First press WINDOW and set Xmin 12 (mean minus 4 std. dev.).  Set Xmax at the same
                      number of units above the mean, i.e., 108.
                 b)  Press Y= and select the Y1= position; then press 2nd, DISTR, 1 to paste normalpdf( to  
                      the Y1= position.
                 c)  Enter data so that the entry after Y1= looks line this:  normalpdf(X, 60,12.
                 d)  Press ZOOM, 0 to select ZoomFit and the curve should appear on the screen.
                 e)  Press TRACE and you can move along the curve and read the values for different x-
                     values.  If you want a specific value, perhaps to get rid of the x-value decimals, just enter
                     that number and press ENTER.

           9. ZInterval:  This gives the range within which the population mean is expected to fall with a desired
               confidence level.  The sample size should be > 30 if the population standard devation is not
               known.                                                                           
              Ex. 1:  Suppose we have a sample of 90 with sample mean x¯  = 15.58 and s = 4.61.  What is the 95%
                confidence level interval?
                   a)  Press STAT, cursor to TESTS, and press 7.
                   b)  On the screen that appears, cursor to "Stats" on the ZInterval screen and press ENTER.
                   c)  Enter data opposite positions as follows:  σ: 4.61, x¯ :15.58, n:90, and C-Level: .95.
                   d)  Cursor down to Calculate, press ENTER, and the interval (14.628, 16.532) will appear along with
                        the values for "n" and the mean.
              Ex. 2:  Suppose that you have a set of 35 temperature measurements and you want to know with a 95%
                         confidence level what limits the population mean of temperature measurement will fall within.
                   a)  First you need to enter the data in a list, say L_1, by pressing STAT, ENTER, and entering your data
                        in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
                   b)  Press STAT, cursor to TEST and press 7 to get the ZInterval screen.
                   c)  Cursor to "Data" and press ENTER.
                   d)  Next you need to know the sample standard deviation.  To enter that opposite σ, do this:  Press 2nd, LIST,
                        move the cursor to MATH and press 7.  The expression stdDev( will be pasted opposite σ.
                         e)  Press 2nd, L₁ , or whatever list you have your data in. When you move the cursor the value will be entered.
                    f)  Enter information as follows:  List: Press 2nd, L₁, Freq: 1, C-Level: .95.
                   g)  Cursor to Calculate and press ENTER.  The same type data will be displayed as in Ex. 1 above.

VII. Other Distributions and Calculations:

         1. TInterval:  If the sample size is <30, then the sample mean cannot be used for the population mean,  and
             the ZInterval cannot be used.  However, if the distribution is essentially normal, i.e., know to be normal
             form other sources or has only one mode and is essentially symmetrical, then the Student t Distribution
             can be used.
              Ex. 1:  Suppose you take ten temperature measurements with sample mean x¯  = 98.44 and s = .3.
                What is the 95% confidence level interval?
                   a)  Press STAT, cursor to TESTS, and press 8.
                   b)  On the screen that appears, cursor to "Stats" and press ENTER.
                   c)  Enter data opposite positions as follows:  x¯ :98.44, S _x : .3   n:10, and C-Level: .95.
                   d)  Cursor down to "Calculate", press ENTER, and, after a few seconds, the interval (98.228, 98.655)
                        will appear along with the values for  "n" and the mean.
              Ex. 2:  Suppose that you have a set of 10 temperature measurements and you want to know with a 95%
                         confidence level what limits the population mean of temperature measurement will fall within.
                   a)  First you need to enter the data in a list, say L_1, by pressing STAT, ENTER, and entering your data
                        in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
                   b)  Press STAT, cursor to "TEST" and press 8 to get the TInterval screen.
                   c)  Cursor to "Data" on the TInterval screen and press ENTER.
                   d)  Enter information as follows:  List: Press 2nd, L₁, Freq: 1, C-Level: .95.
                   e)  Cursor to "Calculate" and press ENTER.  After a few seconds, the interval (xx.xxx, xx.xx)
                        will appear along with the values for  "n," the mean, and sample standard deviation.

            2.  Student's t Distribution:  The Student's t Distribution is applied similar to the normal probability function, but it
                 can be applied to where there are less than 30 data points, for example: P(t> 1.4|df = 19).  The last part means
                 that the number of degrees of freedom ( one less that the number of data points) is 19.
               Ex. 1:  Find the probability that t> 1.4 give that you have 20 data points. 
                 a)  Press 2nd, DISTR,  5, to paste tcdf( to the home screen.
                 b)  Enter data so that your display is as follows:  tcdf(1.4, 1E9,19.
                 c)  Press ENTER and your answer should be .0888...

           3.  invT: Finding a t-value Given α and df:
               If you are working a problem using the t-value, there are different options depending on your needs and whether
               you're using a TI-83 Plus or a TI-84 Silver Edition.
                 TI-84 Plus Silver Edition:  This calculator has an invT, so do the following:
                    (1)  Press 2nd, DISTR, 4, and invT( will be pasted to the screen.
                    (2)  Enter α or 1-α, depending on whether you have a left or right tail; then enter the degrees of freedom, df.
                    (3)  Press ENTER and the value for "t" will be displayed.  Note that you may need to divide α by 2 if you
                          have not already made that adjustment. 
                 TI-83 Plus:  This calculator does not have an invT, so you can do either of two procedures:
                    (1)  Look up the t-value in your book.  This is by far the easier. 
                    (2)  If you have an α that's not in the table or don't have a table, you can do this:
                          Suppose you want the t-value for α=.1 for a left-tailed test.
                          (a)  Press MATH, 0, and the solver will be pasted to the screen. 
                          (b)  Press the UP arrow so that the equation is displayed.
                          (c)  Press 2nd, DISTR, 5 and tcdf( will be pasted in as a formula.
                          (d)  Enter data so that your entry will look like this: tcdf(-1E9, X, 10) - .100 and press ENTER.
                          (e)  Press the UP arrow and enter -1 opposite X.
                          (f)   Press ALPHA, SOLVE, and the value for "t" will be displayed opposite X after about 20 seconds. 
                           Suppose you want the t-value for α=.1 for a right-tailed test.
                           The steps are exactly the same except for these.
                           (d)  Enter data so that your entry will look like this: tcdf(-1E9, X, 10) - .900 and press ENTER.
                           (e)  Press the UP arrow and enter 1 opposite X.
                  Use a Calculator Program: 
                    There are several program posted on the Web, for example, at www.ticalc.org .  I will also be posting a
                    program that I have written some time soon.  It may not be the greatest, but it works.

            4.  The Chi-squared Distribution:  The χ² Distribution is implemented similar to the Student's t
                 Distribution. 
                Ex. 1:  Assume that you want to find P(χ² > 24|df=20) the same as in the above Student's t Distribution.
                 a )  Press 2nd, DISTR,  7, to paste χ²cdf( to the home screen.
                  b)  Enter data so that your display is as follows:  ^χ2cdf(24, 1E9,19.
                  c)  Press ENTER and your answer should be .1961...

          7.  Binomial Distribution, binonpdf(:
             Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12. 
             Determine the probability that 4 of them are defective.
                a)  Press 2ND, DISTR, move the cursor down to A:binompdf( and press ENTER.
                b)  Enter numbers so that your entry is like this: binompdf(12, .05, 4.
                c)  Press ENTER and 0.00205 will be displayed.

               8.  Binomial Distribution, binoncdf(:
               Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12. 
               Determine the probability that 4 or more of them are defective.
               First I'll show a very easy way that gives only the answer; then I'll show a method that takes more time, but
               provides much more intermediate results.
                Short Way:
                a)  Press 1, and then - , the subtraction sign.
                b)  Press 2ND, DISTR, move the cursor down to B:binomcdf( and press ENTER.
                c)  Enter numbers so that the display looks like this:  binomcdf(12, .05, 3.
                d)  Press ENTER and the answer, .0022364 will be displayed.
                 Longer Way:
                 a) Press 2ND, DISTR; then move the cursor to A:binompdf( and press ENTER.
                 b)  Enter information so that your display looks like this:  binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}).  Be sure
                       to use braces rather than parentheses.
                  c)  Press STO, 2ND, L₁ to tell the calculator which list to store the individual values in. 
                  Now, we want to also get the sum of all of these.  Do that as follows:
                   d)  Press ALPHA, : (the decimal point key); then 2ND, LIST, move the cursor to MATH, and press 5.  The expression
                         binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) : sum( should now  be displayed on the home screen.
                   e)  Press 2ND, L_1,.  You should now have this expression:  binompdf(12, .05, {4, 5,6,7,8,9,10,11,12})  sum( L1).
                   f)  Press ENTER,  and the answer, .0022364, will be displayed.  If you need the individual numbers,
                        they are in list L₁.  Just press STAT, ENTER to see them.
                    Ex 2:  Suppose in the above example you want to know the probability of 3 and fewer.
                      a)  Press 2ND, DISTR, move the cursor down to B:binomcdf( and press ENTER.
                      b)  Enter numbers so that the display looks like this:  binomcdf(12, .05, 3.
                     c)  Press ENTER and the answer, .997763... will be displayed.
                    Ex 3:  Suppose that, on average, one out of ten apples in a fruit stand is unacceptable.  What is the probability that
                          8, 9, or 10 of a set of 11 such apples are acceptable?
                      a)  Press 2ND, LIST; move the cursor to MATH and press 5 to paste sum( to the home screen.
                      b)  Press 2ND, DISTR, ALPHA, A.  You will now have sum(binomialPdf( posted to the home screen.
                      c)  Enter data so that you have sum(binomialPdf(11, .9, {8,9,10}))  on the home screen.  Be sure to use braces
                           rather than parentheses enclosing the numbers 8, 9, 10. 
                      d)  Press ENTER and .667...will be displayed.

VIII.  Hypothesis Testing:
      1.  Testing for Mean  and z Distribution with Data:
           a)  Enter the data into L₁ or whatever list you choose.
           b)  Press STAT and move the cursor over to TESTS.
           c)  Press 1 or ENTER for Z-Test.
           d)  Move the cursor to Data and press ENTER.
           e)  Opposite µ_o, enter the mean for the null hypothesis.
            f)  Opposite σ, if you are using the sample standard deviation and it is not given, do the following: Press 2nd,
                LIST, move the cursor to CALC and press 7.  stdDev(, will now be displayed opposite σ.  Now, enter you
                list number where the dats is stored by pressing 2nd, and the list number, for example L_{1 .}  
           g)  Enter L₁ opposite List and 1 opposite Freq.
           h)  Select the proper condition for the alternative hypothesis.
            i)  Move the cursor to Calculate and press ENTER.
            j)  If you want to use the calculator to find the z-value or critical value, see those procedures below.

       2.  Testing for Mean and z Distribution with Statistics: 
           a)  Press STAT and move the cursor over to TESTS.
           b)  Press 1 or ENTER for Z-Test.
           c)  Move the cursor to Stats and press ENTER.
           d)  Opposite µ_o, enter the mean for the null hypothesis.
           e)  Enter the given values for σ, x-bar, and n.
           f)  Select the proper condition for the alternative hypothesis.
           g)  Move the cursor to Calculate and press ENTER.  The z-value, p-value and some other statistics will
                be displayed.     

      3)  Finding a z-vlaue for a particular confidence level:
           Suppose you want the z-value for a particular α, e.g., 5%. Do this:
           a)  Press 2nd, DISTR, 3 for invNorm(. 
           b)  Enter α for a left-tailed or 1-α for a right-tailed and press ENTER.
               c)  The z-value will be displayed. 

       4)  Finding critical values of x. 
           Suppose you have a mean of 5.25, standard deviation of .6 and you want the critical number for an α
               of 5%. 
             a)  Press 2nd, DISTR, 3, and invNorm( will be pasted to the home screen.
             b)  Enter numbers so that your entry looks like this:  invNorm(.05, 5.25, .6.  For a left tail, enter the value
                     for α and for a right tail enter 1-α..
             c) Press ENTER and the inverse will be displayed.  

       5.  Testing for Mean  and t Distribution with Data:
           a)  Enter the data into L₁ or whatever list you choose.
           b)  Press STAT and move the cursor over to TESTS.
           c)  Press 2 for T-Test.
           d)  Move the cursor to Data and press ENTER.
           e)  Opposite µ_o, enter the mean for the null hypothesis.
           f)  Enter L₁ opposite List and 1 opposite Freq.
           g)  Select the proper condition for the alternative hypothesis.
            h)  Move the cursor to Calculate and press ENTER.
            i)  If you are working a problem using the p-value test, read the p-value and compare it with α or α-1 as appropriate.
            j) If you are working a problem using the t-value test, you will need to know the critical values for the level of
               significance, α, that you have chosen.  There are different options depending on your needs and whether
               you're using a TI-83 Plus or a TI-84 Silver Edition. See "invT: Finding a t-value Given α and df:" in section VII of
               this document for the details of these options.
                

       6.  Testing for Mean and T Distribution with Statistics: 
        
           a)  Press STAT and move the cursor over to TESTS.
           b)  Press 2 or ENTER for T-Test.
           c)  Move the cursor to Stat and press ENTER.
           d)  Opposite µ_o, enter the mean for the null hypothesis. 
           e)  Enter the given values for σ, x-bar, and n. If you don't know x-bar you can enter it by placing the cursor opposite
                the symbol for mean; then press 2nd, LIST, cursor to MATH, and press 3; then press ENTER. Enter L₁ and
                press ENTER.
           h)  Select the proper condition for the alternative hypothesis.
            i)  Move the cursor to Calculate and press ENTER.
            j)  If you are working a problem using the p-value test,  read the p-value and compare it with α or α-1 as appropriate.
            k) If you are working a problem using the t-value test, you will need to know the critical values for the level of
               significance, α, that you have chosen.  There are different options depending on your needs and whether
               you're using a TI-83 Plus or a TI-84 Silver Edition. See "invT: Finding a t-value Given α and df:" in section VII of
               this document for the details of these options.

     IX.    Simple Program for Calculating  InverseT:

        This is a simple program for those who want to find t-values with a calculator.   Because    the TI-83Plus has a fairly slow clock speed, a solution may take 20 seconds or so.  When you enter the program,, you can add more letters to the menu items if you prefer.  I have  abbreviated them  to save memory space in my calculator.

        Using the Program:

          a) After you’ve entered the program, highlight the program name and press ENTER.        
          b) The program will ask for the confidence level,  α, and then the degrees of freedom,  df.  For this program, α   
               is not divided by 2 when doing a two-tailed test. Remember that for a
             
         c)  You will then be presented with a menu to select either right-tail, left-tail, or 2-tail.  Select the one appropriate by 
               either pressing the appropriate number or highlighting the number and pressing ENTER.  The answer will be   
              displayed in approximately 20  seconds.

 PROGRAM:
: ”FKIZER 91906”
: INPUT “DF=”, D
: Menu(“SELECT”, Lft TL”, 1, “RT TL”, 2, “2-TL”, 3)
: Lbl 1
: solve(tcdf(-1E9, X, D) – A, X, -1.7) →T
: Goto 4
: Lbl 2
: solve(tcdf(-1E9, X, D) –(1- A), X, 1.7) →T
: GoTo 4
: Lbl 3
: solve(tcdf(-1E9, X, D) – A/2, X, 1.7) →T
: Disp abs(T
:Lbl 4
:Disp T

X.  Statistics of two Populations:   
      1.  Confidence Interval for Two Dependent Populations:
  Enter the data from population 1 into L₁ and the data from population 2 into L₂.  Do this as follows:
     a)  Press STAT, ENTER, and enter the data in the displayed lists.
     b) After entering the data, press 2nd, QUIT to go to the home screen.
   Now, store the paired differences in list L3 as follows:
     c)  From the home screen, press 2nd, L1, minus sign, 2nd, L2.
    d)  Press STO, 2nd, L3.   You should now have L1 - L2 → L3 on the home screen.
  Now, find the confidence level as follows:
     e)  Press STAT, move the cursor to TESTS, and press 8 for TInterval.
     f)   On the screen that appears, move the cursor to "Data" and press ENTER; then enter 1 opposite Freq
          and press ENTER.
    g)   Enter the confidence level you want opposite C-Level, for example .95.
    h)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be
             displayed.   

2.  Confidence Interval for Two Dependent Populations (Stats):
      If you do not have data, but have the mean, standard deviation, and n, use this procedure.
     a)  Press STAT, move the cursor to TESTS, and press 8 for TInterval.
     b)   On the screen that appears, move the cursor to "Stats" and press ENTER.
     c)  Enter the sample mean, standard deviation, and the number of data points opposite "n.".
     d)   Enter the confidence level you want opposite C-Level, for example .95.
      f)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be
             displayed.   

 3.  Confidence Interval for Two Independent Populations (Stats):
       a)  Press STAT, move the cursor to TESTS, and press 0 (zero).
       b)  On the screen that appears, move the cursor to Stats and press ENTER.
       c)  Enter the sample means, standard deviations, and number of data points, n, for each sample.
       d)  Set the confidence level you choose opposite "C-Level."
       e) Highlight "No" opposite "Pooled" if there are no assumptions about the variations.
       f)  Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be 
           displayed.

4.  Confidence Interval for Two Independent Populations (Data):
        Enter the data from population 1 into L₁ and the data from population 2 into L₂.  Do this as follows:
     a)  Press STAT, ENTER, and enter the data in the displayed lists.
     b) After entering the data, press 2nd, QUIT to go to the home screen.
      To go to the confidence interval screen do this:
      c)  Press STAT, move the cursor to TESTS, and press 0 (zero).
      d)  On the screen that appears, move the cursor to Data and press ENTER.
       f)  Opposite "List 1," press 2nd, L₁ and opposite "List2," press 2nd, L₂.
       g)  Set the confidence level you choose opposite "C-Level."
       h)   Highlight "No" opposite "Pooled" if  there are no assumptions about the variations.
        i)   Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be 
              displayed.

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Page Activated: 8/21/06