STATS 4

01 Aug STATS 4

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. There were three treatments: a control with no jumping, a low-jump condition (the jump height was 30 centimeters), and a high-jump condition (60 centimeters). After 8 weeks of 10 jumps per day, 5 days per week, the bone density of the rats (expressed in mg/cm3 ) was measured. Here are the data. data449.dat

(a) Make a table giving the sample size, mean, and standard deviation for each group of rats. Consider whether or not it is reasonable to pool the variances. (Round your answers for x, s, and  to one decimal place.)

Group	n		s
Control
Low jump
High jump

(b) Run the analysis of variance. Report the F statistic with its degrees of freedom and P-value. What do you conclude? (Round your test statistic to two decimal places and your P-value to three decimal places.)

F	=
P	=

Conclusion: There is    statistically significant difference between the three treatment means at the α = .05 level.

2. A study of reading comprehension in children compared three methods of instruction. The three methods of instruction are called Basal, DRTA, and Strategies. As is common in such studies, several pretest variables were measured before any instruction was given. One purpose of the pretest was to see if the three groups of children were similar in their comprehension skills. The READING data set described in the Data Appendix gives two pretest measures that were used in this study. Use one-way ANOVA to analyze these data and write a summary of your results.

3. A study of reading comprehension in children compared three methods of instruction. The three methods of instruction are called Basal, DRTA, and Strategies. Basal is the traditional method of teaching, while DRTA and Strategies are two innovative methods based on similar theoretical considerations. The READING data set includes three response variables that the new methods were designed to improve. Analyze these variables using ANOVA methods. Be sure to include multiple comparisons or contrasts as needed. Write a report summarizing your findings.

4. The One-Way ANOVA applet lets you see how the F statistic and the P-value depend on the variability of the data within groups and the differences among the means.

(a) The black dots are at the means of the three groups. Move these up and down until you get a configuration that gives a P-value of about 0.01. What is the value of the F statistic?

This answer has not been graded yet.

(b) Now increase the variation within the groups by dragging the mark on the pooled standard error scale to the right. Describe what happens to the F statistic and the P-value. Explain why this happens.

5. Corporate advertising tries to enhance the image of the corporation. A study compared two ads from two sources, the Wall Street Journal and the National Enquirer. Subjects were asked to pretend that their company was considering a major investment in Performax, the fictitious sportswear firm in the ads. Each subject was asked to respond to the question “How trustworthy was the source in the sportswear company ad for Performax?” on a 7-point scale. Higher values indicated more trustworthiness. Here is a summary of the results.

Ad source	n	x	s
Wall Street Journal	66	4.77	1.50
National Enquirer	61	2.43	1.64

Find the two-sample pooled t statistic. Then formulate the problem as an ANOVA and report the results of this analysis. Verify that F = t 2.

6. A survey looked at the value of recreational sports on college campuses. One of the questions asked each student to rate the importance of recreational sports to college satisfaction and success. Responses were on a 10-point scale with 1 indicating total lack of importance and 10 indicating very high importance. The following table summarizes these results:

Class	n	Mean score
Freshman Sophomore Junior Senior	720 535 595 439	7.6 7.6 7.5 7.3

(a) To compare the mean scores across classes, what are the degrees of freedom for the ANOVA F statistic?

numerator df
denominator df

(b) The MSG = 11.606. If sp = 2.15, what is the F statistic? (Round your answer to two decimal places.) F =   (c) Give an approximate (from Table E) or exact (from software) P-value. (Enter your answer to four decimal places.)

7. Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Over time, the lengths of the flowers and the form of the hummingbirds’ beaks have evolved to match each other. Here are data on the lengths in millimeters of three varieties of these flowers on the island of Dominica. data466.dat Do a complete analysis that includes description of the data and a significance test to compare the mean lengths of the flowers for the three species. (Round your answers for x to four decimal places, s to three decimal places, and  to three decimal places. Round your test statistic to two decimal places. Round your P-value to three decimal places.)

flower type	n		s
H. bihai
H. caribaea red
H. caribaea yellow

F =   P =

8. Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. One of the foods was yesiga wet, beef cut into small pieces and prepared with several Ethiopian spices. The iron content of four samples of yesiga wet cooked in each of the three types of pots is given below. The units are milligrams of iron per 100 grams of cooked food. data378.dat

(a) Make a table giving the sample size, mean, and standard deviation for each type of pot. Is it reasonable to pool the variances? Note that with the small sample sizes in this experiment, we expect a large amount of variability in the sample standard deviations.

Type of pot	n		s
Aluminum
Clay
Iron

(b) Run the analysis of variance. Report the F statistic with its degrees of freedom and P-value. What do you conclude? (Round your test statistic to two decimal places. Round your P-value to three decimal places.) F =   P =   Conclusion: There is    statistically significant difference between the three treatment means at the α = .05 level.

9. The National Crime Victimization Survey estimates that there were over 400,000 violent crimes committed against women by their intimate partner that resulted in physical injury. An intervention study designed to increase safety behaviors of abused women compared the effectiveness of six telephone intervention sessions with a control group of abused women who received standard care. Fifteen different safety behaviors were examined. One of the variables analyzed was the total number of behaviors (out of 15) that each woman performed. Here is a summary of the means of this variable at baseline (just before the first telephone call) and at follow-up 3 and 6 months later.

	Time
Group	Baseline	3 months	6 months
Intervention	10.4	12.5	11.9
Control	9.6	9.9	10.4

(a) Find the marginal means. (Round your answers to three decimal places.)

	Time
Group	Baseline	3 months	6 months	Mean
Intervention	10.4	12.5	11.9
Control	9.6	9.9	10.4
Mean

Are they useful for understanding the results of this study?

The row means suggest that the control group showed more improvement than the 3 months group.The row means suggest that the intervention group showed more improvement than the control group.    The row means suggest that the control group showed more improvement than the baseline group.The row means suggest that the intervention group showed more improvement than the 6 months group.The row means suggest that the intervention group showed less improvement than the control group.

(b) Plot the means. Do you think there is an interaction? Describe the meaning of an interaction for this study.

Yes there is an interaction, it means that the mean number of actions does not change over time for the two groups.Yes there is an interaction, it means that the mean number of actions changes consistently over time for the two groups.    Yes there is an interaction, it means that the mean number of actions changes differently over time for the two groups.No there is not an interaction, meaning that the mean number of actions does not change over time for the two groups.

(Note: This exercise is from a repeated-measures design, and the data are not particularly normal because they are counts with values from 1 to 10. Although we cannot use the methods in this chapter for statistical inference in this setting, the example does illustrate ideas about interactions.)

10. The amount of chromium in the diet has an effect on the way the body processes insulin. In an experiment designed to study this phenomenon, four diets were fed to male rats. There were two factors. Chromium had two levels: low (L) and normal (N). The rats were allowed to eat as much as they wanted (M) or the total amount that they could eat was restricted (R). We call the second factor Eat. One of the variables measured was the amount of an enzyme called GITH. The means for this response variable appear in the following table.

	Eat
Chromium	M	R
L	4.545	5.175
N	4.425	5.317

A plot of the mean GITH for these diets, with the factor Chromium on the x axis and GITH on the y axis is given below. For each Eat group the points for the two Chromium means are connected.

(a) Describe the patterns you see. Does the amount of chromium in the diet appear to affect the GITH mean? Does restricting the diet rather than letting the rats eat as much as they want appear to have an effect? Is there an interaction?

This answer has not been graded yet.

(b) Compute the marginal means. Compute the differences between the M and R diets for each level of Chromium. Use this information to summarize numerically the patterns in the plot.

11. Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. Foods considered were yesiga wet’, beef cut into small pieces and prepared with several Ethiopian spices; shiro wet’, a legume-based mixture of chickpea flour and Ethiopian spiced pepper; and ye-atkilt allych’a, a lightly spiced vegetable casserole. Four samples of each food were cooked in each type of pot. The iron in the food is measured in milligrams of iron per 100 grams of cooked food. The data are shown in the table below.

		Iron Content (mg/100 g) of Food Cooked in Different Pots
Type of pot		Meat		Legumes		Vegetables
Aluminum		1.77	2.36	1.96	2.14		2.40	2.17	2.41	2.34		1.03	1.53	1.07	1.30
Clay		2.27	1.28	2.48	2.68		2.41	2.43	2.57	2.48		1.55	0.79	1.68	1.82
Iron		5.27	5.17	4.06	4.22		3.69	3.43	3.84	3.72		2.45	2.99	2.80	2.92

(a) Make a table giving the sample size, mean, and standard deviation for each type of pot. Is it reasonable to pool the variances? Although the standard deviations vary more than we would like, this is partially due to the small sample sizes, and we will proceed with the analysis of variance.

This answer has not been graded yet.

(b) Plot the means. Give a short summary of how the iron content of foods depends upon the cooking pot.

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(c) Run the analysis of variance. Give the ANOVA table, the F statistics with degrees of freedom and P-values, and your conclusions regarding the hypotheses about main effects and interactions.

12. Do left-handed people live shorter lives than right-handed people? A study of this question examined a sample of 949 death records and contacted next of kin to determine handedness. Note that there are many possible definitions of “left-handed.” The researchers examined the effects of different definitions on the results of their analysis and found that their conclusions were not sensitive to the exact definition used. For the results presented here, people were defined to be right-handed if they wrote, drew, and threw a ball with the right hand. All others were defined to be left-handed. People were classified by gender (female or male) and handedness (left or right), and a 2 × 2 ANOVA was run with the age at death as the response variable. The F statistics were 22.36 (handedness), 37.44 (gender), and 2.10 (interaction). The following marginal mean ages at death (in years) were reported: 77.39 (females), 71.32 (males), 75.00 (right-handed), and 66.03 (left-handed).

(a) For each of the F statistics given, find the degrees of freedom and an approximate P-value. Summarize the results of these tests.

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(b) Using the information given, write a short summary of the results of the study.

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15.–/0.64 pointsMIntroStat8 13.E.515.XP.

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My Notes

A study of undergraduate computer science students examined changes in major after the first year. The study examined the fates of 256 students who enrolled as first-year students in the same fall semester. The students were classified according to gender and their declared major at the beginning of the second year. For convenience we use the labels CS for computer science majors, EO for engineering and other science majors, and O for other majors. The explanatory variables included several high school grade summaries coded as 10 = A, 9 = A-, etc. Here are the mean high school mathematics grades for these students.

	Major
Gender	CS	EO	O
Males	8.68	8.35	7.65
Females	9.11	9.36	8.04

Describe the main effects and interaction using appropriate graphs and calculations.

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16.–/0.64 pointsMIntroStat8 13.E.516.XP.

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A study of undergraduate computer science students examined changes in major after the first year. The study examined the fates of 256 students who enrolled as first-year students in the same fall semester. The students were classified according to gender and their declared major at the beginning of the second year. For convenience we use the labels CS for computer science majors, EO for engineering and other science majors, and O for other majors. The mean SAT mathematics scores for the students are summarized in the following table.

	Major
Gender	CS	EO	O
Males	628	618	589
Females	582	631	543

Summarize the results of this study using appropriate plots and calculations to describe the main effects and interaction.

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17.–/0.8 pointsMIntroStat8 13.E.517.XP.

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My Notes

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A study of undergraduate computer science students examined changes in major after the first year. The study examined the fates of 256 students who enrolled as first-year students in the same fall semester. The students were classified according to gender and their declared major at the beginning of the second year. The students studied were enrolled at a large Midwestern university several years ago. Discuss how you would conduct a similar study at a college or university of your choice today. Include a description of all variables that you would collect for your study.

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