Suppose that a random sample of size n 5 was selected from

Give as much information as you can about the P-value for an upper-tailed F test in each of the following situations. a. df1
4, df2
15, F
5.37 b. df1
4, df2
15, F
1.90 c. df1
4, df2
15, F
4.89 d. df1
3, df2
20, F
14.48 e. df1
3, df2
20, F
2.69 f. df1
4, df2
50, F
3.24

Give as much information as you can about the P-value of the single-factor ANOVA F test in each of the following situations. a. k
5, n1
n2
n3
n4
n5
4, F
5.37 b. k
5, n1
n2
n3
5, n4
n5
4, F
2.83 c. k
3, n1
4, n2
5, n3
6, F
5.02 d. k
3, n1
n2
4, n3
6, F
15.90 e. k
4, n1
n2
15, n3
12, n4
10, F
1.75

Employees of a certain state university system can choose from among four different health plans. Each plan differs somewhat from the others in terms of hospitalization coverage. Four samples of recently hospitalized individuals were selected, each sample consisting of people covered by a different health plan. The length of the hospital stay (number of days) was determined for each individual selected. a. What hypotheses would you test to decide whether average length of stay was related to health plan? (Note: Carefully define the population characteristics of interest.) b. If each sample consisted of eight individuals and the value of the ANOVA F statistic was F
4.37, what conclusion would be appropriate for a test with a
.01? c. Answer the question posed in Part (b) if the F value given there resulted from sample sizes n1
9, n2
8, n3
7, and n4
8.

The experiment described in Example 15.4 also gave data on change in body fat mass for men (Growth Hormone and Sex Steroid Administration in Healthy Aged Women and Men, Journal of the American Medical Association [2002]: 22822292). Each of 74 male subjects who were over age 65 was assigned at random to one of the following four treatments: (1) placebo growth hormone and placebo steroid (denoted by P P), (2) placebo growth hormone and the steroid testosterone (denoted by P S), (3) growth hormone and placebo steroid (denoted by G P), and (4) growth hormone and the steroid testosterone (denoted by G S). The accompanying table lists data on change in body fat mass over the 26-week period following the treatment that are consistent with summary quantities given in the article Change in Body Fat Mass (kg) Treatment P  P P  S G  P G  S 0.3 3.7 3.8 5.0 0.4 1.0 3.2 5.0 1.7 0.2 4.9 3.0 0.5 2.3 5.2 2.6 2.1 1.5 2.2 6.2 1.3 1.4 3.5 7.0 0.8 1.2 4.4 4.5 1.5 2.5 0.8 4.2 1.2 3.3 1.8 5.2 0.2 0.2 4.0 6.2 1.7 0.6 1.9 4.0 1.2 0.7 3.0 3.9 Change in Body Fat Mass (kg) Treatment P  P P  S G  P G  S 0.6 0.1 1.8 3.3 0.4 3.1 2.9 5.7 1.3 0.3 2.9 4.5 0.2 0.5 2.9 4.3 0.7 0.8 3.7 4.0 0.7 4.2 0.9 4.7 2.0 0.6 n 0.117 0.121 0.117 0.119 0.100 0.933 3.112 4.605 s 1.139 1.443 1.178 1.122 s 2 1.297 2.082 1.388 1.259 Also, N
74, grand total
158.3, and Carry out an F test to see whether true mean change in body fat mass differs for the four treatments. 15

The article Compression of Single-Wall Corrugated Shipping Containers Using Fixed and Floating Text Platens (Journal of Testing and Evaluation [1992]: 318320) described an experiment in which several different types of boxes were compared with respect to compression strength (in pounds). The data at the top of page 796 resulted from a single-factor experiment involving k
4 types of boxes (the sample means and standard deviations are in close agreement with values given in the paper). Do these data provide evidence to support the claim that the mean compression strength is not the same for all four box types? Test the relevant hypothesis using a significance level of .01.

The article The Soundtrack of Recklessness: Musical Preferences and Reckless Behavior Among Adolescents (Journal of Adolescent Research [1992]: 313331) described a study whose purpose was to determine whether adolescents who preferred certain types of music reported higher rates of reckless behaviors, such as speeding, drug use, shoplifting, and unprotected sex. Independently chosen random samples were selected from each of four groups of students with different musical preferences at a large high school: (1) acoustic/pop, (2) mainstream rock,(3) hard rock, and (4) heavy metal. Each student in these samples was asked how many times he or she had engaged in various reckless activities during the last year. The following table lists data and summary quantities on driving over 80 mph that is consistent with summary quantities given in the article (the sample sizes in the article were much larger, but for the purposes of this exercise, we use n1
n2
n3
n4
20): Number of Times Driving Over 80 mph by Musical Preference MainAcoustic/ stream Hard Heavy Pop Rock Rock Metal 2334 3243 4134 1213 3323 3413 3343 3223 2422 2424 1434 3435 2244 2335 2233 3224 2235 2344 3122 4343 n 20 20 20 20 2.50 2.70 2.75 3.55 s .827 .979 .967 .887 s 2 .6839 .9584 .9351 .7868 x Also, N
80, grand total
230.0, and Carry out an F test to determine if these data provide convincing evidence that the true mean number of times driving over 80 mph varies with musical preference.

Suppose that a random sample of size n
5 was selected from the vineyard properties for sale in Sonoma County, California, in each of three years. The following data are consistent with summary information on price per acre (in dollars, rounded to the nearest thousand) for disease-resistant grape vineyards in Sonoma County (Wines and Vines, November 1999). 1996 30,000 34,000 36,000 38,000 40,000 1997 30,000 35,000 37,000 38,000 40,000 1998 40,000 41,000 43,000 44,000 50,000 a. Construct boxplots for each of the three years on a common axis, and label each by year. Comment on the similarities and differences. b. Carry out an ANOVA to determine whether there is evidence to support the claim that the mean price per acre for vineyard land in Sonoma County was not the same for the three years considered. Use a significance level of .05 for your test.

The article An Analysis of Job Sharing, Full-Time, and Part-Time Arrangements (American Business Review [1989]: 3440) reported on a study of hospital employees from three different groups. Each employee reported a level of satisfaction with his or her work schedule (1
very dissatisfied to 7
very satisfied), as shown in the accompanying table. Sample Sample Group Size Mean 1. Job sharers 24 6.60 2. Full-time employees 24 5.37 3. Part-time employees 20 5.20a. What are numerator and denominator dfs for the F test? b. The test statistic value is F
6.62. Use a test with significance level .05 to decide whether it is plausible that true average satisfaction levels are identical for the three groups. c. What is the value of MSE?

High productivity and carbohydrate storage ability of the Jerusalem artichoke make it a promising agricultural crop. The article Leaf Gas Exchange and Tuber Yield in Jerusalem Artichoke Cultivars (Field Crops Research [1991]: 241252) reported on various plant characteristics. Consider the accompanying data on chlorophyll concentration (gm/m2 ) for four varieties of Jerusalem artichoke: Variety BI RO WA TO Sample mean .30 .24 .41 .33 Suppose that the sample sizes were 5, 5, 4, and 6, respectively, and also that MSE
.0130. Do the data suggest that true average chlorophyll concentration depends on the variety? State and test the appropriate hypotheses using a significance level of .05.

It has been reported that varying work schedules can lead to a variety of health problems for workers. The article Nutrient Intake in Day Workers and Shift Workers (Work and Stress [1994]: 332342) reported on blood glucose levels (mmol/L) for day-shift workers and workers on two different types of rotating shifts. The sample sizes were n1
37 for the day shift, n2
34 for the second shift, and n3
25 for the third shift. A single-factor ANOVA resulted in F
3.834. At a significance level of .05, does true average blood glucose level appear to depend on the type of shift?

In the introduction to this chapter, we considered a study comparing three groups of college students (soccer athletes, nonsoccer athletes, and a control group consisting of students who did not participate in intercollegiate sports). The following information on scores from the Hopkins Verbal Learning Test (which measures immediate memory recall) was Soccer Nonsoccer Group Athletes Athletes Control Sample size 86 95 53 Sample mean score 29.90 30.94 29.32 Sample standard deviation 3.73 5.14 3.78 In addition, Suppose that it is reasonable to regard these three samples as random samples from the three student populations of interest. Is there sufficient evidence to conclude that the mean Hopkins score is not the same for the three student populations? Use a
.05.

The article Utilizing Feedback and Goal Setting to Increase Performance Skills of Managers (Academy of Management Journal [1979]: 516526) reported the results of an experiment to compare three different interviewing techniques for employee evaluations. One method allowed the employee being evaluated to discuss previous evaluations, the second involved setting goals for the employee, and the third did not allow either feedback or goal setting. After the interviews were concluded, the evaluated employee was asked to indicate how satisfied he or she was with the interview. (A numerical scale was used to quantify level of satisfaction.) The authors used ANOVA to compare the three interview techniques. An F statistic value of 4.12 was reported. a. Suppose that a total of 33 subjects were used, with each technique applied to 11 of them. Use this information to conduct a level .05 test of the null hypothesis of no difference in mean satisfaction level for the three interview techniques. b. The actual number of subjects on which each technique was used was 45. After studying the F table, explain why the conclusion in Part (a) still holds

An investigation carried out to study the toxic effects of mercury was described in the article Comparative Responses of the Action of Different Mercury Compounds on Barley (International Journal of Environmental Studies [1983]: 323327). Ten different concentrations of mercury (0, 1, 5, 10, 50, 100, 200, 300, 400, and 500 mg/L) were compared with respect to their effects on average dry weight (per 100 seven-day-old seedlings). The basic experiment was replicated four times for a total of 40 dryweight observations (four for each treatment level). The article reported an ANOVA F statistic value of 1.895. Using a significance level of .05, test the null hypothesis that the true mean dry weight is the same for all 10 concentration levels

The accompanying data on calcium content of wheat are consistent with summary quantities that appeared in the article Mineral Contents of Cereal Grains as Affected by Storage and Insect Infestation (Journal of Stored Products Research [1992]: 147151). Four different storage times were considered. Partial output from the SAS computer package is also shown. Storage Period Observations 0 months 58.75 57.94 58.91 56.85 55.21 57.30 1 month 58.87 56.43 56.51 57.67 59.75 58.48 2 months 59.13 60.38 58.01 59.95 59.51 60.34 4 months 62.32 58.76 60.03 59.36 59.61 61.95 Dependent Variable: CALCIUM Sum of Mean Source DF Squares Square F Value Pr>F Model 3 32.13815000 10.71271667 6.51 0.0030 Error 20 32.90103333 1.64505167 Corrected Total 23 65.03918333 R-Square C.V. Root MSE CALCIUM Mean 0.494135 2.180018 1.282596 58.8341667 a. Verify that the sums of squares and dfs are as given in the ANOVA table. b. Is there sufficient evidence to conclude that the mean calcium content is not the same for the four different storage times? Use the value of F from the ANOVA table to test the appropriate hypotheses at significance level .05.

In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125-cc motorcycle, five plugs of each brand were tested, and the number of miles (at a constant speed) until failure was observed. A partially completed ANOVA table is given. Fill in the missing entries, and test the relevant hypotheses using a .05 level of significance. Source of Sum of Mean Variation df Squares Square F Treatments Error 235,419.04 Total 310,500.76

The partially completed ANOVA table given in this problem is taken from the article Perception of Spatial Incongruity (Journal of Nervous and Mental Disease [1961]: 222), in which the abilities of three different groups to identify a perceptual incongruity were assessed and compared. All individuals in the experiment had been hospitalized to undergo psychiatric treatment. There were 21 individuals in the depressive group, 32 individuals in the functional other group, and 21 individuals in the brain-damaged group. Complete the ANOVA table. Carry out the appropriate test of hypothesis (use a
.01), and interpret your results. Source of Sum of Mean Variation df Squares Square F Treatments 76.09 Error Total 1123.14

Research carried out to investigate the relationship between smoking status of workers and short-term absenteeism rate (hr/mo) yielded the accompanying summary information (Work-Related Consequences of Smoking Cessation, Academy of Management Journal [1989]: 606621). In addition, F
2.56. Construct an ANOVA table, and then state and test the appropriate hypotheses using a .01 significance level. Sample Sample Status Size Mean Continuous smoker 96 2.15 Recent ex-smoker 34 2.21 Long-term ex-smoker 86 1.47 Never smoked 206 1.69

An investigation carried out to study purchasers of luxury automobiles reported data on a number of different attributes that might affect purchase decisions, including comfort, safety, styling, durability, and reliability (Measuring Values Can Sharpen Segmentation in the Luxury Car Market, Journal of Advertising Research [1995]: 9 22). Here is summary information on the level of importance of speed, rated on a seven-point scale: Type of Car American German Japanese Sample size 58 38 59 Sample mean rating 3.05 2.87 2.67 In addition, SSE
459.04. Carry out a hypothesis test to determine if there is sufficient evidence to conclude that the mean importance rating of speed is not the same for owners of these three types of cars.

The Gunning Fog index is a measure of reading difficulty based on the average number of words per sentence and the percentage of words with three or more syllables. High values of the Gunning Fog index are associated with difficult reading levels. Independent random samples of six advertisements were taken from three different magazines, and Gunning Fog indices were computed to obtain the data given in the accompanying table (Readability Levels of Magazine Advertisements, Journal of Advertising Research [1981]: 4550). Construct an ANOVA table, and then use a significance level of .01 to test the null hypothesis of no difference between the mean Gunning Fog index levels for advertisements appearing in the three magazines. Scientific American 15.75 11.55 11.16 9.92 9.23 8.20 Fortune 12.63 11.46 10.77 9.93 9.87 9.42 New Yorker 9.27 8.28 8.15 6.37 6.37 5.66

Some investigators think that the concentration (mg/mL) of a particular antigen in supernatant fluids could be related to onset of meningitis in infants. The accompanying data are typical of that given in plots in the article Type-Specific Capsular Antigen Is Associated with Virulence in Late-Onset Group B Streptococcal Type III Disease (Infection and Immunity [1984]: 124129). Construct an ANOVA table, and use it to test the null hypothesis of no difference in mean antigen concentrations for the three groups. Asymptomatic infants 1.56 1.06 0.87 1.39 0.71 0.87 Infants with late onset sepsis 1.51 1.78 1.45 1.13 1.87 1.89 1.07 1.72 Infants with late onset meningitis 1.21 1.34 1.95 2.27 0.88 1.67 2.57

Parents are frequently concerned when their child seems slow to begin walking (although when the child finally walks, the resulting havoc sometimes has the parents wishing they could turn back the clock!). The article Walking in the Newborn (Science, 176 [1972]: 314315) reported on an experiment in which the effects of several different treatments on the age at which a child first walks were compared. Children in the first group were given special walking exercises for 12 min per day beginning at age 1 week and lasting 7 weeks. The second group of children received daily exercises but not the walking exercises administered to the first group. The third and fourth groups were control groups: They received no special treatment and differed only in that the third groups progress was checked weekly, whereas the fourth groups progress was checked just once at the end of the study. Observations on age (in months) when the children first walked are shown in the accompanying table. Also given is the ANOVA table, obtained from the SPSS computer package. Age n Total Treatment 1 9.00 9.50 9.75 6 60.75 10.00 13.00 9.50 Treatment 2 11.00 10.00 10.00 6 68.25 11.75 10.50 15.00 Treatment 3 11.50 12.00 9.00 6 70.25 11.50 13.25 13.00 Treatment 4 13.25 11.50 12.00 5 61.75 13.50 11.50 Analysis of Variance Source df Sum of sq. Mean Sq. F Ratio F Prob Between Groups 3 14.778 4.926 2.142 .129 Within Groups 19 43.690 2.299 Total 22 58.467 a. Verify the entries in the ANOVA table. b. State and test the relevant hypotheses using a signifi- cance level of .05.

The article Heavy Drinking and Problems Among Wine Drinkers (Journal of Studies on Alcohol [1999]: 467471) analyzed drinking problems among Canadians. For each of several different groups of drinkers, the mean and standard deviation of highest number of drinks consumed were calculated: s n Beer only 7.52 6.41 1256 Wine only 2.69 2.66 1107 Spirits only 5.51 6.44 759 Assume that each of the seven samples studied can be viewed as a random sample for the respective group. Is there sufficient evidence to conclude that the mean value of highest number of drinks consumed is not the same for all seven groups?

Leaf surface area is an important variable in plant gas-exchange rates. The article Fluidized Bed Coating of Conifer Needles with Glass Beads for Determination of Leaf Surface Area (Forest Science [1980]: 2932) included an analysis of dry matter per unit surface area (mg/cm3 ) for trees raised under three different growing conditions. Let m1, m2, and m3 represent the true mean dry matter per unit surface area for the growing conditions 1, 2, and 3, respectively. The given 95% simultaneous con- fidence intervals are based on summary quantities that appear in the article: Difference m1  m2 m1  m3 m2  m3 Interval (3.11, 1.11) (4.06, 2.06) (1.95, .05) Which of the following four statements do you think describes the relationship between m1, m2, and m3? Explain your choice. a. m1
m2, and m3 differs from m1 and m2. b. m1
m3, and m2 differs from m1 and m3. c. m2
m3, and m1 differs from m2 and m3. d. All three ms are different from one another.

The degree of success at mastering a skill often depends on the method used to learn the skill. The article Effects of Occluded Vision and Imagery on Putting Golf Balls (Perceptual and Motor Skills [1995]: 179186) reported on a study involving the following four learning methods: (1) visual contact and imagery, (2) nonvisual contact and imagery, (3) visual contact, and (4) control. There were 20 subjects randomly assigned to each method. The following summary information on putting performance score was reported: Method 1 2 3 4 16.30 15.25 12.05 9.30
13.23 s 2.03 3.23 2.91 2.85 a. Is there sufficient evidence to conclude the mean putting performance score is not the same for the four methods? b. Calculate the 95% TK intervals, and then use the underscoring procedure described in this section to identify significant differences among the learning methods.

The article referenced in Exercise 15.24 reported on a study involving the following four learning methods: (1) visual contact and imagery, (2) nonvisual contact and imagery, (3) visual contact, and (4) control. There were 20 subjects assigned to each method. Calculate the 99% TK intervals, indicate which methods differ significantly from one another, and summarize the results by underscoring. How do the 99% TK intervals compare to the 95% TK intervals of Exercise 15.24?

The accompanying data resulted from a flammability study in which specimens of five different fabrics were tested to determine burn times. 1 17.8 16.2 15.9 15.5 2 13.2 10.4 11.3 Fabric 3 11.8 11.0 9.2 10.0 4 16.5 15.3 14.1 15.0 13.9 5 13.9 10.8 12.8 11.7 MSTr
23.67 MSE
1.39 F
17.08 P-value
.000 The accompanying output gives the TK intervals as calculated by MINITAB. Identify significant differences and give the underscoring pattern. Individual error rate = 0.00750 Critical value = 4.37 Intervals for (column level mean) (row level mean) 1234 1.938 2 7.495 3.278 1.645 3 8.422 3.912 1.050 5.983 6.900 4 3.830 0.670 2.020 1.478 3.445 4.372 0.220 5 6.622 2.112 0.772 5.100

Sample mean chlorophyll concentrations for the four Jerusalem artichoke varieties introduced in Exercise 15.9 were .30, .24, .41, and .33, with corresponding sample sizes of 5, 5, 4, and 6, respectively. In addition, MSE
.0130. Calculate the 95% TK intervals, and then use the underscoring procedure described in this section to identify significant differences among the varieties.

Do lizards play a role in spreading plant seeds? Some research carried out in South Africa would suggest so (Dispersal of Namaqua Fig (Ficus cordata cordata) Seeds by the Augrabies Flat Lizard (Platysaurus broadleyi), Journal of Herpetology [1999]: 328330). The researchers collected 400 seeds of this particular type of fig, 100 of which were from each treatment: lizard dung, bird dung, rock hyrax dung, and uneaten figs. They planted these seeds in batches of 5, and for each group of 5 they recorded how many of the seeds germinated. This resulted in 20 observations for each treatment. The treatment means and standard deviations are given in the accompanying table. Treatment n s Uneaten figs 20 2.40 .30 Lizard dung 20 2.35 .33 Bird dung 20 1.70 .34 Hyrax dung 20 1.45 .28 a. Construct the appropriate ANOVA table, and test the hypothesis that there is no difference between mean number of seeds germinating for the four treatments. b. Is there evidence that seeds eaten and then excreted by lizards germinate at a higher rate than those eaten and then excreted by birds? Give statistical evidence to support your answer.

The article Growth Response in Radish to Sequential and Simultaneous Exposures of NO2 and SO2 (Environmental Pollution [1984]: 303325) compared a control group (no exposure), a sequential exposure group (plants exposed to one pollutant followed by exposure to the second four weeks later), and a simultaneous-exposure group (plants exposed to both pollutants at the same time). The article states, Sequential exposure to the two pollutants had no effect on growth compared to the control. Simultaneous exposure to the gases significantly reduced plant growth. Let and represent the sample means for the control, sequential, and simultaneous groups, respectively. Suppose that Use the given information to construct a table where the sample means are listed in increasing order, with those that are judged not to be significantly different underscored.

The nutritional quality of shrubs commonly used for feed by rabbits was the focus of a study summarized in the article Estimation of Browse by Size Classes for Snowshoe Hare (Journal of Wildlife Management [1980]: 3440). The energy content (cal/g) of three sizes (4 mm or less, 57 mm, and 810 mm) of serviceberries was studied. Let m1, m2, and m3 denote the true energy content for the three size classes. Suppose that 95% simultaneous con- fidence intervals for m1  m2, m1  m3, and m2  m3 are (10, 290), (150, 450), and (10, 310), respectively. How would you interpret these intervals?

The accompanying underscoring pattern appeared in the article Effect of SO2 on Transpiration, Chlorophyll Content, Growth, and Injury in Young Seedlings of Woody Angiosperms (Canadian Journal of Forest Research [1980]: 7881). Water loss of plants (Acer saccharinum) exposed to 0, 2, 4, 8, and 16 hours of fumigation was recorded, and a multiple comparison procedure was used to detect differences among the mean water losses for the different fumigation durations. How would you interpret this pattern? Duration of fumigation 16 0 8 2 4 Sample mean water loss 27.57 28.23 30.21 31.16 36.21

Samples of six different brands of diet or imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids (PAPUFA, in percent), resulting in the data shown in the accompanying table. (The data are fictitious, but the sample means agree with data reported in Consumer Reports.) Imperial 14.1 13.6 14.4 14.3 Parkay 12.8 12.5 13.4 13.0 12.3 Blue Bonnet 13.5 13.4 14.1 14.3 Chiffon 13.2 12.7 12.6 13.9 Mazola 16.8 17.2 16.4 17.3 18.0 Fleischmanns 18.1 17.2 18.7 18.4 a. Test for differences among the true average PAPUFA percentages for the different brands. Use a
.05. b. Use the TK procedure to compute 95% simultaneous confidence intervals for all differences between means and interpret the resulting intervals.

formation on satisfaction levels for employees on three different work schedules: n1
24 n2
24 n3
20
6.60
5.37
5.20 MSE was computed to be 2.028. Calculate the 95% TK intervals, and identify any significant differences.

Consider the accompanying data on plant growth after the application of different types of growth hormone. 1 13 17 7 14 2 21 13 20 17 Hormone 3 18 14 17 21 4 7 11 18 10 5 6 11 15 8 a. Carry out the F test at level a
.05. b. What happens when the TK procedure is applied? (Note: This contradiction can occur when is barely rejected. It happens because the test and the multiple comparison method are based on different distributions. Consult your friendly neighborhood statistician for more information.)