Assignment 6, Fall 2000

STATISTICS 403 - Assignment 6

Due Tuesday, October 31, 2000

Two machines are used to measure a vibration property of a rubber product. The data below are the vibrational force (kg) of samples from each machine. The items entering each machine are randomly selected from the production process that produces items that are similar in their vibrational characteristics. Any differences apparent in the data are thus assumed to be due to the machines.
- (a) Referring to the boxplot above, compare the two machines in terms of vibrational force.
- (b) Below is Minitab output for an analysis of the data using a two-sample t-procedure. What does this analysis tell you about the distribution of vibration for the two machines?
  T-Test of the Mean
```
Machine     N     Mean    StDev    SE Mean
1          21     29.8     15.5      3.4   
2          18     44.1     14.6      3.4

95% CI for mu (1) - mu (2): (  -24.1,  -4.5)
T_Test mu (1) = mu (2) (vs not =): T = -2.95  P= 0.0055  DF =37
Both use Pooled StDev = 15.1
```
- (c) Below is Minitab output for an analysis of the data using an analysis of variance procedure. What does this analysis tell you about the distribution of vibration for the two machines?
  One-way Analysis of Variance
```
Analysis of Variance for Force
Source      DF         SS         MS        F         P
Machine      1       1982       1982     8.71     0.005
Error       37       8417        227
Total       38      10399

Level        N       Mean       StDev
1           21       29.77      15.49
2           18       44.07      14.59    Pooled StDev = 15.08
```
- (d) How do the two procedures compare? I want you to tell me about the conclusions drawn from each analysis. I also want you to tell me about the similarity of P-values, test statistics and estimates of population standard deviations.
- (e) Use the Wilcoxon-Mann-Whitney statistic to test to see if the two machines have different distributions. Use the normal approximation to obtain a P-value.
- (f) Use the Kruskal-Wallis statistic to test to see if the two machines have different distributions.
- (g) How do the two procedures compare? I want you to tell me about the conclusions drawn from each analysis. I also want you to tell me about the similarity of P-values and test statistics.
A graduate in exercise physiology wants to compare four training methods suitable for marathon runners. Forty-nine individuals interested in training for a marathon are randomly divided into four groups of roughly equal sizes. Each group was then randomly assigned to a particular training method. At the end of three months, the participants are to be timed on a trial marathon run. For various reasons, not all participants complete the training course or trial marathon run. Below are times (hr:min) for the 40 participants grouped according to training methods 1, 2, 3 and 4.
- (a) Use the Kruskal-Wallis test statistic to test the hypothesis the that median times for the four groups are the same against the alternative that some groups are different.
- (b) Using an experiment wide error rate (significance level) of 0.06, which training methods are statistically different?
- (c) Construct a 95% confidence interval on the difference in median times for training methods 2 and 3. An assumption of normality is not appropriate, although the assumption that the distributions have equal shapes is.
- (d) Extend Mood's median test to this problem. That is, find the combined (all four groups) sample median and construct a 2 by 4 table of counts. Calculate a value of a statistic and compute an appropriate P-value (be careful to get the degrees of freedom right).