SECTION 16.3 Exercises

Question 16.47

16.47 French tourism economy.

Ski resort activities make up nearly 20% of the French tourism economy. The French ski resort economy is under pressure to remain competitive in light of new entrants to the ski resort market, which are less expensive (for example, Slovenia and Montenegro). A research study was conducted to assess the productivity and efficiency of French ski resorts.15 The study examined the productivity of 64 French ski resorts with the Luenberger Productivity Indicator (LPI) over a two-year time frame. LPI as an overall measure of productivity is commonly used by economists because it can be decomposed into the usual constituents of productivity growth: technological chang and efficiency change. A positive LPI indicates an increase in productivity, while a negative LPI indicates a decrease in productivity. The researcher of this study wished to investigate the relationship between the ski resorts’ size and productivity. Ski resorts in the study were classified as being “large” (level 1), “medium” (level 2), or “small” (level 3).

ski

16-28

  1. What are the null hypothesis and the alternative hypothesis? Explain why a nonparametric procedure would be appropriate in this setting.
  2. Use the Kruskal-Wallis test to compare LPI across the three size classifications of ski resorts. Write a brief statement of your findings.

16.47

(a) H0: LPIs have the same distribution in all groups. Ha: LPIs are systematically higher in some groups than in others. The standard deviations for the three sizes are very different. (b) , There are significant differences in LPI across different resort sizes.

Question 16.48

16.48 Evaluating an educational product.

Case 14.2 (pages 733 and 734) considers the evaluation of a new educational product designed to improve children’s reading comprehension. Three methods (Basal, SRTA, and Strat) are evaluated on three groups of 22 children. Your company markets educational materials aimed at parents of young children. The response variable is a measure of reading comprehension called COMP that was obtained by a test taken after the instruction was completed. Use the Kruskal-Wallis test to compare the three methods.

eduprod

Question 16.49

16.49 Loss of vitamin C in bread.

Does bread lose its vitamins when stored? Here are data on the vitamin C content (milligrams per 100 grams of flour) in bread baked from the same recipe and stored for one, three, five, or seven days. The 10 observations are from 10 different loaves of bread.16

bread

Condition Vitamin C (mg/100 g)
Immediately after baking 47.62 49.79
One day after baking 40.45 43.46
Three days after baking 21.25 22.34
Five days after baking 13.18 11.65
Seven days after baking 8.51 8.13

The loss of vitamin C over time is clear, but with only two loaves of bread for each storage time, we wonder if the differences among the groups are significant.

  1. Use the Kruskal-Wallis test to assess significance, then write a brief summary of what the data show.
  2. Because there are only two observations per group, we suspect that the common chi-square approximation to the distribution of the Kruskal-Wallis statistic may not be accurate. The exact -value (from the SAS software) is . Compare this with your -value from (a). Is the difference large enough to affect your conclusion?

16.49

H0: Vitamin C values have the same distribution in all groups. Ha: Vitamin C values are systematically higher in some groups than in others. , . The data do not show differences in vitamin C across different conditions. Although the test is not significant, looking at the data suggests that there is vitamin C lost over time. (b) With the new P-value from SAS, we would reject the null hypothesis and conclude there are systematically higher vitamin C values in some groups than in others, showing the loss of vitamin C over time.

Question 16.50

16.50 Exercise and bone density.

Many studies have suggested that there is a link between exercise and healthy bones. Exercise stresses the bones and this causes them to get stronger. One study examined the effect of jumping on the bone density of growing rats.17 Ten rats were assigned to each of three treatments: a 60-centimeter “high jump,” a 30-centimeter “low jump,” and a control group with no jumping. Here are the bone densities (in milligrams per cubic centimeter) after eight weeks of 10 jumps per day.

bone

Group Bone density (mg/cm3)
Control 611 621 614 593 593 653 600 554 603 569
Low jump 635 605 638 594 599 632 631 588 607 596
High jump 650 622 626 626 631 622 643 674 643 650
  1. The study was a randomized comparative experiment. Outline the design of this experiment.
  2. Make side-by-side stemplots for the three groups, with the stems lined up for easy comparison. The distributions are a bit irregular but not strongly non-Normal. We would usually use analysis of variance to assess the significance of the difference in group means.
  3. Do the Kruskal-Wallis test. Explain the distinction between the hypotheses tested by Kruskal-Wallis and ANOVA.
  4. Write a brief statement of your findings. Include a numerical comparison of the groups as well as your test result.

Question 16.51

16.51 Decay of polyester fabric.

In Exercise 16.17 (page 16-13), the breaking strengths (in pounds) of strips of polyester fabric buried in the ground were considered for two time points. Breaking strength is a good measure of the extent to which the fabric has decayed. Here are the breaking strengths for several lengths of time.18

poly2

Time Breaking strength
2 weeks 118 126 126 120 129
4 weeks 130 120 114 126 128
8 weeks 122 136 128 146 131
16 weeks 124 98 110 140 110
  1. Find the standard deviations of the four samples. They do not meet our rule of thumb for applying ANOVA. In addition, the sample buried for 16 weeks contains an outlier. We will use the Kruskal-Wallis test.

    16-29

  2. Find the medians of the four samples. What are the hypotheses for the Kruskal-Wallis test, expressed in terms of medians?
  3. Carry out the test and report your conclusion.

16.51

(a) The standard deviations are 4.60, 6.54, 9.04, 16.09. (b) The medians are 126, 126, 131, 110. H0: The median breaking strength for all groups are the same. Ha: Some medians are higher in some groups than in others. (c) , . The data do not show median breaking strength differences across different times; there appears to be no decay.

Question 16.52

16.52 Food safety: Fairs, fast food, restaurants.

CASE 16.2 Case 16.2 (page 16-10) describes a study of the attitudes of people attending outdoor fairs about the safety of the food served at such locations. It contains the responses of 303 people to several questions. The variables in this data set are (in order):

fsafety

subject hfair sfair sfast srest gender

The variable “sfair” contains responses to the safety question described in Case 16.2. The variables “srest” and “sfast” contain responses to the same question asked about food served in restaurants and in fast-food chains. Explain carefully why we cannot use the Kruskal-Wallis test to see if there are systematic differences in perceptions of food safety in these three locations.