**Homework 5 – Solutions by Lan Lan**

This homework is based on the
following example:

Bearings for high-speed
turbine engines can be made of many different types of compounds. The times to fatigue failure (in units of
millions of cycles) for high-speed bearings were measured for five different
types of compounds. Compounds I and II
are made of an alloy containing lead. Compounds III, IV, and V are made of an alloy
containing iron. Compounds I and III
were made by US companies and compounds II, IV and V were made by foreign
companies. The data are listed below:

Compound |
Failure
Time |
Compound |
Failure
Time |
Compound |
Failure
Time |
Compound |
Failure
Time |
Compound |
Failure
Time |

I |
3.03 |
II |
3.19 |
III |
3.46 |
IV |
5.88 |
V |
6.43 |

I |
5.53 |
II |
4.26 |
III |
5.22 |
IV |
6.74 |
V |
9.97 |

I |
5.6 |
II |
4.47 |
III |
5.69 |
IV |
6.9 |
V |
10.39 |

I |
9.3 |
II |
4.53 |
III |
6.54 |
IV |
6.98 |
V |
13.55 |

I |
9.92 |
II |
4.67 |
III |
9.16 |
IV |
7.21 |
V |
14.45 |

I |
12.51 |
II |
4.69 |
III |
9.4 |
IV |
8.14 |
V |
14.72 |

I |
12.95 |
II |
5.78 |
III |
10.19 |
IV |
8.59 |
V |
16.81 |

I |
15.21 |
II |
6.79 |
III |
10.71 |
IV |
9.8 |
V |
18.39 |

I |
16.04 |
II |
9.37 |
III |
12.58 |
IV |
12.28 |
V |
20.84 |

I |
16.84 |
II |
12.75 |
III |
13.41 |
IV |
25.46 |
V |
21.51 |

The goal of this homework is
to compare the average failure times for the bearings made from the 5 different
compounds.

1.
Use PROC MEANS to
calculate the average times to fatigue failure for each of the 5
compounds. Which compound had the
longest average time to fatigue failure?
The shortest time?

*The output of PROC MEANS shows compound
V has the highest time to fatigue failure (mean = 14.7) while compound II has
the lowest fatigue failure time (mean = 6.05).*

2.
Use PROC
UNIVARIATE to construct side-by-side box plots.
Based on these plots, do you notice any obvious differences between the
5 compounds?

*By checking the box plot, compound V has
higher times and compound II has lower times.
Also, the range between the 25 ^{th} percentile and the 75^{th}
percentile are different for each compound.
Compounds I and V have bigger ranges and
compound II has a smaller range.*

3.
Use PROC GLM to
compute the analysis of variance for these data. Based on the overall F-test, is there a
significant difference between the treatment means for the 5 different
compounds? State the null and
alternative hypotheses for the test, and report the F-statistic and p-value.

*Ho:
mI = mII = mIII = mIV = mV*

*Ha:
not all means are equal*

*F = 5.02 and p-value = 0.002*

*At the 5% significance level, reject the
null hypothesis and conclude that there IS a significant difference between the
average failure times for the 5 compounds.*

4.
Based on what you
know about the compounds, come up with 2 pre-planned comparisons for the
means. Define the contrasts associated
with these comparisons and use the contrast statement in PROC GLM to test your
hypotheses. Be sure to state the null and
alternative hypotheses for your test, report the p-value, and state your conclusion
in non-statistical terms.

*Are the average failure times for alloys
containing lead significantly different than for alloys containing iron?*

*Ho:
(mI + mII)/2 = (mIII + mIV + mV)/3*

*Ha: (mI + mII)/2 not equal (mIII + mIV + mV)/3*

*F = 4.3 and p-value = 0.0439*

*At the 5% significance level, reject the
null hypothesis and conclude that there is a significant difference between the
average failure times between lead alloys and iron alloys.*

*Are the average failure times for
compounds made by US companies significantly different than for compounds made
by foreign companies?*

*Ho:
(mI + mIII)/2 = (mII + mIV + mV)/3*

*Ha: (mI + mIII)/2 not equal (mII + mIV + mV)/3*

*F = 0.16 and p-value = 0.689*

*Fail to reject the null hypothesis and
conclude that there is no difference between the average failure times for US
made and foreign made compounds.*

5.
Now, use multiple
comparison methods to compare all possible pairs of treatment means. Using PROC GLM, do the multiple comparisons
using the F-protected LSD, Tukey, and Duncan
methods. For each method, which means
are declared to be the same, and which methods are declared to be significantly
different.

*LSD: Same - I&V /
I, III, & IV / II, III, & IV*

* Different - I&II / II
& V / III & V
/ IV & V*

*Tukey**: Same – I, IV, & V / I,
II, III, & IV*

* Different – V & III /
V & II*

*Duncan**: Same – I & V /
II, III, & IV / I, III, & IV*

* Different – V & II /
I & II / V & III
/ V & IV *

6.
Based on your
findings write a short (one paragraph) summary of your results, and make
recommendations about which compound(s) you would use if your goal was to
create ball-bearings with the longest time to fatigue failure.

*Compounds I and V clearly had the highest
times to fatigue failure. Also, there
appears to be no difference between the compounds made by US companies and
foreign companies. There is a difference
between the lead alloy and iron alloy; and, the iron alloy compounds had higher
average fatigue failure times than the lead alloy compounds. If I were to recommend compounds to use to
create ball-bearings with the longest time to fatigue failure, I would recommend
compounds I and V.
Compound V could be the best, while compound I is also acceptable. The decision between the two could be based
on other matters such as price and desired material.*