13)  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)  From the list screen, highlight the title of L₃ and press 2ND, L₁, x, 2ND, L₂..  You will now have L₁*L₂
                   at the bottom left of the list screen.
             b)  Press ENTER and you will have the individual values stored in list L_3.
             c)  To get the sum of these values,  do this.
                    CAUTION:  DO NOT store sums in the lists if the particular list is going to be used in a succeeding arithmetic
                    operation.  Instead, do these calculations from the home screen.
                      (1)  Move the cursor down to the first blank space in L₃.   Press 2nd, LIST; cursor to MATH, and press 5. 
                             The expression sum( will be displayed at the bottom of  the list screen. 
                      (2)  Press 2ND, L₃ , ).You will have sum(L₃) at the bottom of the list screen.
                      (3)  Press ENTER and the sum of those values will be displayed as the last item of L₃. 
             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 L₄,
              do the following:
             a)  From the list screen, place the cursor on the title for list L₄ , press 2ND, L₁, x², ,x, 2ND, L₂.  You will
                   have L₁²*L₂ at the bottom left of the lists.
             b)  Press ENTER and the individual values will be entered in list L_4.
             c)  To get the sum of these values do the following:
                      (1)  Caution:  Do not store sums in lists if you plan to use this list in another arithmetic operation.  Place the cursor
                             in the first blank space in L₄, then press 2nd, LIST; cursor to MATH, and press 5.  The expression
                             sum( will be displayed at the bottom left of the LIST screen.
                      (2)  Press 2ND, L_4, )..  You will have sum(L₄) at the bottom of the list screen. 
                      (3)  Press ENTER and the sum of those values in L₄ will be displayed as the last entry in L_4.. 
              d)  Now we want to subtract the value for µ² from the last value obtained and that will be the variance.  You can always do that
                     by hand but if you want to be a little more creative, do it this way.  First press 2ND, QUIT to go to the home screen.
                     Suppose that your sum for L_3, µ, and L₄, Σx² P(x), are in rows 6.   Press 2ND, L_4, (, 6, ), -, 2ND, L_3, (, 6, ).  You should now have this:
                      L_4,(6 ) -L_3,( 6).  Press ENTER and the variance will be displayed.
               e)  To calculate the standard deviation from the variance in the list assuming that the variance is in L₃(7),  move the cursor down one
                     space to L₃₍8)and press 2ND, √, 2nd, L₃,(,7,) and press ENTER.  The standard deviation will be displayed in L₃(8).
               f)  Of course if you calculated the standard deviation from the home screen, if you have just calculated the variance, press 2ND, √,
                    2nd, ANS, ENTER. 
            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 let the calculator do it all.   Just as information, the total
           expression for the variance using this method would be this:  sum(L₁²*L₂) - (sum(L₁ *L₂))² .

    14)  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 move the cursor to the first blank space at the end of the data.
            b)  Press 2nd, LIST and move the cursor to MATH.
            c)  Press 7 to paste stdDev( to the bottom of the list  screen.
            d) Press 2nd, L₁, ), and then press the divide symbol.
             e)  Press 2nd, LIST, move the cursor to MATH, and press 3 for mean(.
              f)  Press 2nd, L₁ , close the parentheses and then press ENTER to display the CV as the last number in L₁.
              NOTE: If you want the answer in percent, multiply the answer by 100. If you're going to use this list for other calculations, be sure to delete the CV value before performing any operations.

   15)  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.
       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 and then press ENTER.  The standard
            deviation and several other statistics will be displayed.
         e)  To calculate  the variance, merely re-enter the value for the  standard deviation, S_x  ,
              and square it^,   Note:  If you round off the standard deviation, you may have a slightly different answer than
              you would if you had calculated the variance separately by hand.  To avoid that, enter all decimal places for S_x and
             square that value.  If you don't like entering long numbers, you can do this:  Press VARS, 5, 3, ENTER, x² ,
             ENTER.          

  B.  Calculating  Numbers to Plug into a Computation 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 themselves to learn the details of the  
      process.  The following gives 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 ,  ∑x² and ∑x to plug into the formula.
   a)  Enter the data in the table as indicated previously in this document. 
   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 and Sx = 11.73923.
  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 show the detailed calculations, skip to filling in the formula in step “f.”  Otherwise, continue
 with the next step.
   d)  Now we’ll need an x² column.   Place the cursor on the title L₂, press 2^nd, L₁, x², ENTER.  The squares of the
          numbers in L₁ will be displayed in L₂.  You can enter into your table  the numbers that you found for n, ∑x, and
         ∑x² from the 1-Var Stats function.
     e)  Now, we want to use the numbers that we previously recorded to plug into the variance
        formula.   So, from the home screen enter (7492-216²/7)/(6).  You can either merely enter these numbers or your
          worksheet or test sheet and square the standard deviation you found above and enter for the answer, or you can
           do more time and work to enter the numbers in your calculator and find the variance.
     f)  If you entered the numbers in the calculator, press ENTER and you should have 137.8…, which is the variance.
     g)  To find  the standard deviation, press 2ND, √ , 2ND, Ans, ENTER,  or you can just record the standard deviation that
           and you recorded above.  In either case, you will have 11.73...

16.  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 terms.  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) Place the cursor on the title of L₁; then press (, 2ND, L₁, + 2ND, L₂,), ÷,
                  2 .  You should have (L₁ + L₂)/2 at the bottom left of the tables.  Press     
                  ENTER and the midpoints will be stored in L₁.
      d)  Enter the frequencies in L₂ as described under Entering Data immediately above.
            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.  
      f)  Press ENTER, and the standard deviation along with several other statistics will be
             displayed.  The sample standard deviation is 14.868….
       g)  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 Computation Formula:
     The standard deviation and variance for grouped data 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  and store them in L₁ 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) Place the cursor on the title of L₁; then press (, 2ND, L₁, + 2ND, L₂,), ÷,
                  2 .  You should have (L₁ + L₂)/2 at the bottom left of the tables.  Press     
                  ENTER and the midpoints will be stored in L₁.
 Now let’s calculate the required numbers.
b) 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. 
c)  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=∑xf=3780,
     ∑x²=∑x²f=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 from the Lists screen. .    
d)  Calculate xf and store it in L₃ by placing the cursor over the title for L₃ pressing  2ND, L₁, *, 2ND, L₂._.
       You should have L_1*L₃ at the bottom left of the tables.  Press ENTER and the products will be stored
       in list L_3.
e)  Calculate x²f by placing the cursor on the title for L₄  and  pressing 2ND, L_1, x² , * , 2ND,  L₂. 
     You should now have L₁² *L₂  at the bottom left of the tables. 
f)  Press ENTER and the results will be stored in list L_4. 
 g)  You don’t need to calculate Σf.  That is the value for “n” that you previously recorded.  
 h)   You don’t need to calculate Σxf.  That is the value for ∑ x that you previously recorded.
 i)  At this point you can either just record the formula with the numbers plugged in on your work sheet
     or test sheet or you can do the extra work to do the calculation with your calculator.  To get the
     answer without putting the numbers in your calculator, merely square the standard deviation, which
     you previously recorded. If you're going to do it with the calculator, do the next steps.
     Now, you want to plug the appropriate numbers into the formula for the variance. From the  
     home screen enter (296600-3780²/50)/(49)
 j)  Press ENTER and you should have 221.06, which is the variance.
 k)  If you want the standard deviation, you can just use the one you previously recorded or you can calculate
       calculate by pressing 2ND, √ , 2ND, Ans, ENTER, and you will have 14.868...
      Note that if you calculated the standard deviation first, just square that value to get the variance.

17)  Weighted Average: 
          Suppose you have some scores with the weights indicated: 
          Score      Weight
           83              .3
           85              .3
           85              .5
           89              .3
           90              .7
           a)  Press STAT, ENTER and enter the scores in list L₁ and the weights in L₂.
           b) Press STAT, move the cursor to CALC, and press ENTER to paste "1-Var Stats"  to the home screen. 
           c)  If your scores and weights are in lists as indicated above, press ENTER and the weighted average will be
                given as x-bar (x with a bar over it.).  If your data are in other lists, enter those lists separated by a comma and
                press ENTER.

18)  Median of Grouped data:
       Consider the following table.

a)  To find the median class, enter the midpoints in L₁ and the frequencies in L₂ .
b)  Press STAT, move the cursor to highlight CALC and press ENTER.  1-Var Stats will be displayed on the home screen.
c)  If the data are in lists L₁ and L² just press ENTER.  If they are in other lists, you must enter the lists.  For example,  
     press 2^ND, L₂, comma, 2^ND, L₃ and then press ENTER
d)  Press ENTER and scroll down to Med=19.5.  That is the median of the class 15- 
     24. So 15-24 is the median class.
e)  Enter the appropriate data into the following formula:
Median = L + I *(N/2 - F)/f
Where
L = lower boundary of the interval containing the median.
I = width of the interval containing the median.
N = total number of respondents.
F = cumulative frequency of those below the median class.
f = number of cases in the median class.
f)  When you are finished entering, you should have this:
    14.5+10(5000/2-750)/2005
g)  Press ENTER and you should get 23.228… Notice that the answer is different form
     the value of 19.5 given by the calculator. That value of 19.5 was chosen by merely finding the  midpoint 
     of the median class.

 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.  It is not absolutely necessary to sort your data, but the TRACE will sometimes operate
             in a confusing manner without sorting.  So, I recommend sorting.  To sort, press[STAT], select 2,  
             SortA( for ascending order.  SortA(  will be posted to the home screen.  Press
           [2nd], L₁, 2nd, L2, [ENTER].
      d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
      e)  Move the 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.  You can use TRACE and the arrows
            to move along and read the data pairs.

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 don't want to graph,
            continue with these instructions.   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),  enter the lists separated by
            commas if the lists are not in L₁ and L_2.  If the numbers are in L₁ and L_2, you need not enter the list names.
       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.  LinReg Y1. (If the numbers
             are in other lists, the lists followed by commas are also included
       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.  Pressing ZOOM, 0 (zero), for ZoomFit will get you a preliminary window setting.
      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  graph as indicated previously.

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 Computation Formula:  
      We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
      use "2-Var Stats" to obtain the values for such expressions as ΣxΣy and Σx² to enter on our worksheet
      or test sheet.   Then we will use LinReg(a+bx to find the values for "r" and "r²." 
      This last procedure will eliminate the necessity for entering the numbers for the formulas into our calculators
      to get the final answers.
     
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:
       At this point you will save yourself a lot of time if you calculate r and r² with the calculator.  To do that,
       press STAT, move the cursor to CALC and press 8 for LinReg(a+bx.  Press ENTER if the lists are in L₁ and
       L₂.  If they are not in those lists, you will need to enter the lists separated by a comma.  Press ENTER and
      r and r² will be displayed along with other statistics.  If you are required to show your work, you will need
      to write the numbers and on your paper.   Record the following:
        d)  r= (nΣxy –ΣxΣy)/[(√(nΣx²- (Σx)²)(√(nΣy²- (Σy)²)]
                = (8*62750-521*960)/(√(8*33979-521²)(√(8*116900-960²))
               =.7979…..
           If you chose to put the numbers in the calculator, you might want to read the following:
       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.
        g)  Of course, r² is just the square of Ans, or you can just copy if from the stats calculation.  

6)  Finding the Values a and b for the Best-Fit Equation Using a Computation Formula:
      We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
      use "2-Var Stats" to obtain the values for such expressions as ΣxΣy and Σx² which we can enter in the
      formula on our worksheet or test sheet.  Then, we will use the calculator function LinReg(a+bx to find
      the values for "a" and "b" without doing the arithmetic on our calculators.  Finally, for those who are
      allowed to use a simper method than the arithmetic intensive formula, I will suggest the use of matrices
     and the calculator function rref( for finding the final answers. intensive substitution method used in many textbooks.

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 the following
          parameters:  x¯=65.125, Σx=521,  Σx²=33979,
          n=8, Σy=960, y-bar=120, Σy²=116900, Σxy=62750
       At this point you will save yourself a lot of time if you calculate "a" and" b" with the calculator.  To do that,
       press STAT, move the cursor to CALC and press 8 for LinReg(a+bx.  Press ENTER if the lists are in L₁ and
       L₂.  If they are not in those lists, you will need to enter the lists separated by a comma.  Press ENTER and
      "a" and "b" will be displayed along with other statistics.  If you are required to show your work, you will need
      to write the numbers and on your paper.  
      c)  Plug these numbers into the formula and then enter the expression in 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 following  formula on your worksheet,  or in the calculator  if you're going to calculate the values.:
         b=(nΣxy –ΣxΣy)/(nΣx²- (Σx)²).
          Now record these numbers on your worksheet or in the calculator  if you're going to calculate the values.:
           =(8*62750-521*960)/(8*33979-521²)
           =4.7058…..
     e)  Now, record this formula on your worksheet: 
          a= y-bar –b(x-bar)
          Now just enter these numbers on your worksheet or enter them in your calculator if you're going to calculate the value.
            =120-4.7058 *65.125
           =-186.465…

  7.  Testing the Correlation Coefficient:
        Suppose that we have the data given in the table below and we want to test the correlation coefficient at a significance level
        of 1%.  Further suppose we believe that the correlation coefficient is positive. The null hypothesis is that the correlation is 0
       

      a)  If the data are not already in the lists, press STAT, ENTER and entr the x-values in list L₁ and the
            y-values in list L₂.
       b)  Press STAT, move the cursor to TESTS , and scroll down to LinRegTTest.  Press ENTER.
       c)  On the screen that appears, move th e cursor down to >0 and press ENTER.
       d)  Move the cursor to Calculate and press ENTER.
       e)  Among the items displayed is P=.009218...which is less that 0.01.  So, we reject the null hypothesis
             and conclude that the correlation coefficient is positive.  

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.  If you want to sort data in an independent and dependent list, L₁ and L2_,
             for example, use SortA(L₁,L₂).

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.  Note that for grouped data, ∑xf is listed on the
          calculator as ∑x and ∑x² f is listed as ∑x² .

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, first press STATS, ENTER and highlight the list name L₃.
          Now, press 2nd,  L₁, x (Multiply symbol),  2nd, L_2.  Then press ENTER.
     c)  Obviously, x-y can be obtained by merely substituting the subtraction symbol for the
          multiplication symbol in the steps 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, first press STATS, ENTER and highlight the list name L₃.
          Now,  press 2nd,  L_1, x² (the square symbol; 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, highlight the list name where you
          want the data stored; then enter the operation as described above.  Finally, press 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, highlight the list name where you
          want the data stored; then enter the operation as described above.  Finally, press 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.

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.
      e)  Alternately, you can go to the lists, hightlight the name where you want the numbers stored and then enter
           the randint(1,50,10) as described above.

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 0 and a standard deviation of 1. 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.  normalcdf(:  Sample taken from a normal distrubution:
          Suppose a sample of 35 is taken from the population above (μ=70 and σ=4.5).  What is the probability
          that the mean is greater than 72?
          a)  Complete item a) in the general method above.
           b)   Enter numbers so that your display is the following:  normalcdf(72, 1E9,70,4.5/√(35) 
                Enter "E" by pressing 2nd, EE (The comma key.)
           c)  Press ENTER and you'll get .00427... which is, of course, 0.427 per percent.

     5.   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.
               Note that if you reset the window, you may need to activate the expression again.  To do that, press
               2nd, ENTER, ENTER.

       6.  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.

       7.   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.

        8.  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, 8.5.
              d)  Press ENTER and a reasonable looking graph should appear on the screen.  The area
                   under the curve, 0.56562, will be displayed on the screen along with the upper and lower
                   bounds.

        9.   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.

          10. 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., known 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 (98.225, 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, (6 on a TI-84) 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 included a program of
                     my own at the end of this document.  It may not be the greatest, but it's simple and 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 (8 for TI-84), 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...

          5.  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.

               6.  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( ( or alternately press ALPHA, B) 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( (or press ALPHA, A) 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 MATH 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.
          Simple Program for Calculating  InverseT:
             I have written   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.  The Program is in the Appendix at the end of this document.      

IX.  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, highlight the list name, L₃ example, where you want to store the data.
   Now, store the paired differences in list L3 as follows:
     c)  Press 2nd, L1, minus sign, 2nd, L2.
    d)  You should now have L1 - L2 at the bottom  on the lists screen.  Press ENTER and the differences
           will be stored in list L₃.
  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.
           To go to the confidence interval screen do this:
      b)  Press STAT, move the cursor to TESTS, and press 0 (zero).
      c)  On the screen that appears, move the cursor to Data and press ENTER.
       d)  Opposite "List 1," press 2nd, L₁ and opposite "List2," press 2nd, L₂.
       e)  Set the confidence level you choose opposite "C-Level."
       f)   Highlight "No" opposite "Pooled" if  there are no assumptions about the variations.
        g)   Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be 
              displayed.

X.  Other Tests and Inferences:

       1.  One-way ANOVA:
            Suppose that you are trying to determine a better way to motivate learning and come up with the following scores in response
      to different types of motivations:
     

    Determine if  one of the methods is better that the others.. 
         The null hypothesis,  H_o, will be that all of the means are equal.  Suppose that we want to determine if they are by
         99 % confidence level. 
   a)  Press STAT, ENTER, and enter the each group of data in lists L₁ through L₄. To clear old data from a list, place the cursor on
         the list title, press CLEAR; then ENTER.  DO NOT press DEL to clear a list.
   b)  The syntax for ANOVA is ANOVA(List 1, List 2, List 3, List 4, ...List n).  So,  press STAT, move the cursor to TESTS, and move
         the cursor down that list to ANOVA(.  That's item H: on the TI-84.  Press ENTER and ANOVA( will be displayed on the home
         screen.
     c)  Inter list numbers so that you have the following display: ANOVA(L₁, L₂, L₃, L₄.  You can either close the parentheses or not.
    d)  Press ENTER and various results will be displayed.  One of these is P=.004764....  So, H_o is rejected. 

2. Chi-Square Test for Independence:
     Suppose that we have the observed (those indicated by O= ) values in the following table and we want to know if independence
    is indicated at the α=0.01 level.
   

     First we need to calculate the expected values, i.e., those already entered as E= in the table.  The expected value is this:
   P(cell) * Sample Size.
   The probability, P, of a cell is calculated using the rows and column totals.  For example the probability cell 5 is as follows:
   P(5) = P(brand 3) and P(Group 1)
           =40/200 * 80/200
   Now, we multiply that by the sample size of 200:
   E=40/200 * 80/200*200
      =16
  This can be simplified to the following:
  E=[(Row Total)(Column Total)]/(Sample Size)
  Now we want to test for independence, and we will first enter the observed values in Matrix [A] and the expected values in Matrix [B].
  a)  Press 2ND, MATRIX, move the cursor to EDIT and press ENTER to edit matrix [A].
  b)  Enter 4 x 2 for the matrix configuration and then enter the observed values.  Press 2ND, QUIT to leave this matrix.
  c)  Press 2ND, MATRIX and move the cursor to EDIT.  Then press 2 to edit matrix [B].
  d)  Enter 4 x 2 for the matrix configuration and then enter the  expected values in the matrix.  Press 2ND, QUIT to end the
        matrix editing.
  e)  Press STAT,  move the cursor to TEST, and select Χ² -Test from that list and press ENTER.
  f)  The Calculate screen will be displayed with the matrices indicated for storing observed and expected.
  g)  Move the cursor to Calculate  and press ENTER.  Χ² =24.64 and p=1.79467 will be displayed. 
  h)  Since P is smaller than the α-value of 0.01, we reject the null hypothesis of independence and conclude that the preferences are dependent.
  i)  Alternatively, we could have compared the statistic of Χ=24.64 with the critical value for α=0.01.  Using a df = (4-1)(2-1) and consulting a
        table, we would have found Χ² =11.34.  Since the test statistic is larger, we would reach the same conclusion as above.

3. Х² Goodness of Fit:
     The TI-83 Plus does not have a GOF function, so I will first give the procedure for the  TI-84; then I will give a procedure for the
     TI-83 Plus.   
     Suppose that a cell phone vendor wants to test the colors of the cases of cell phones to see if customers have a color preference.  
    A sample is taken and the data in the following table is collected.  The vendor wants a confidence level of 95%.  That is,  α =.05.

  The expected values are calculated by adding all of the observed values and dividing by 5, the number of categories.
  H_O: Customers show no color preference.
  H_1:  Customers show a color preference.
   a) Press STAT and enter the observed values in L₁ and the expected values in L₂.
   b) Press STAT, move the cursor to TESTS, and cursor down to D: Х² GOF – TEST and press ENTER.     
       Alternately, you can press ALPHA, D to activate that procedure.
  c)  On the screen that appears, make sure that L₁ is opposite Observed and L₂ is opposite Expected.
 d)  Enter 4 opposite df.  The value of df is one less than the number of categories.
  e)  Move the cursor to Calculate and press ENTER. 
   f)  The value P= .01882... will be displayed.  Since this value is less than α=.05, we reject the null   
        hypothesis.
Calculation for TI-83 Plus:
  What we are actually going to do is first find the sum of the values listed opposite CONTRB on the TI-84.
  This sum will give us the value for X².  We will then use the X² cdf to find the value for p.
   a) Press STAT and enter the observed values in L₁ and the expected values in L₂.
   b)  Press 2^ND, QUIT to go to the home screen.  
   c)  Press 2^ND, LIST and move the cursor to MATH.  Press 5 to paste sum( to the home screen.
  d)  Enter information so that you have the following:  sum((L₁-L₂)²/L₂). Press ENTER and you should get   
        the answer of 11.809 for X².
Calculate the p-value:
  f)  Press 2^ND, DISTR, move th cursor to X²cdf( and press ENTER.
 g)  Enter information so that you have the following: X²cdf(11.809, 1E9, 4).  The “E” is made by pressing  
     2^ND; then the comma key.
 h)  Press ENTER and the value .018829…, the same value as with the TI-84, should be displayed.

XII:  Appendix:

 PROGRAM for calculating InverseT
: ”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
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 for sorting data into classes:
NOSCAL
:FKIZER090210
:SortA(L₁)
:min(L₁)-->S
:dim(L₁)-->Q
:max(L₁)-->M
:int(M/W)+1-->dim(L₂)
:Input "CLS WDTH  ",W
:0-->T:1-->X:W-->F:0-->C
:ClrHome
:Lbl 1
:While L₁(X)≥S and L₁≤F
:T+1-->T
:X+1-->X
:If X>Q
:Then
:T-->L₂(C+1)
:Goto 2
:End
:End
:C+1-->C
:T-->L₂(C)
:0-->T
:S+W-->S
:F+W-->F
:Goto 1
:Lbl 2
:L₂
After you’ve entered the program, use it in this manner.
a)  First enter the data in list L_1. The data need not be in any order.  
b)  To execute the program, highlight the program name and press ENTER.        
c) The program will ask for the class width, CLS WDTH.  Enter the class width and press ENTER.
d)  The numbers for the classes will be stored in list L₂ and that list will be displayed after execution.  Note that you can
       move the numbers after the ellipses (the three dots) with the cursor arrows.  When finished press CLEAR to stop the
       program. .