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consisted of 10 participants in an e\c. program. A control group (C) of 5 subjects volunteered for the program buł. were unable to attend for various reasons. Subjects in the other two groups were selected to be similar to those in the first two groups in age and other characteristics. These were 11 joggers (J) and 10 sedentary people (S) who did not regularly exercise.¹³ One of the variables measured at the end of the program was a physical fitness score. Part of the ANOVA table used to analyze these data is given below:

	Degrees
Source	of freedom	Sum of sąuares Mean sąuare F
Groups	3	104,855.87
Error	32	70,500.59
Total

(a)    Fili in the missing entries in the ANOVA table.

(b)    State Ho and H_a for this experiment.

(c)    What is the distribution of the F statistic under the assumption that Ho is true? Using Table E, give an approximate P-value for the ANOVA test. Write a brief conclusion.

(d)    What is Sp, the estimate of the within-group variance? What is s_p ?

12.33 Another variable measured in the experiment described in the previous

exercise was a depression score. Higher values of this score indicate morę depression. Part of the ANOVA table for these data appears below:

	Degrees
Source	of freedom Sum of sąuares	Mean sąuare F
Groups	3	158.96
Error	32	62.81
Total

(a)    Fili in the missing entries in the ANOVA table.

(b)    State H o and H_a for this experiment.

(c)    What is the distribution of the F statistic under the assumption that H o is true? Using Table E, give an approximate P-value for the ANOVA test. What do you conclude?

(d)    What is s_p, the estimate of the within-group variance? What is s_p ?

12.34 The weight gain of women during pregnancy has an important effect on the birth weight of their children. If the weight gain is not adequate, the infant is morę likely to be smali and will tend to be less healthy. In a study

conducted in three countries, weight gains (in kilograms) of women during the third trimester of pregnancy were measured.¹⁴ The results are summarized in the following table:

Country	n	X	5
Egypt	46	3.7	2.5
Kenya	111	3.1	1.8
Mexico	52	2.9	1.8

(a)    Find the pooled estimate of the within-country variance s². What entry in the ANOVA table gives this ąuantity?

(b)    The sum of squares for countries (groups) is 17.22. Use this information and that given above to complete all the entries in an ANOVA table.

(c)    State Ho and H_a for this study.

(d)    What is the distribution of the F statistic under the assumption that Ho is true? Use Table E to find an approximate P-value for the significance test. Report your conclusion.

(e)    Calculate R², the coefficient of determination.

12.35

12.36

In another part of the study described in the previous exercise, measurements of food intake in kilocalories were taken on many individuals several times during the period of a year. From these data, average daily food intake values were computed for each individual. The results for toddlers aged 18 to 30 months are summarized in the following table:

Country	n	X	5
Egypt	88	1217	327
Kenya	91	844	184
Mexico	54	1119	285

(a)    Find the pooled estimate of the within-country variance s². What entry in the ANOVA table gives this ąuantity?

(b)    The sum of sąuares for countries (groups) is 6,572,551. Use this information and that given above to complete all the entries in an ANOVA table.

(c)    State Ho and H_a for this study.

<d) What is the distribution of the F statistic under the assumption that Ho is true? Use Table E to find an approximate P-value for the significance test. Report your conclusion.

(e) Calculate R², the coefficient of determination.

Refer to the change-of-majors study described in Example 12.3 (page 750).

Let fi i, /*2> and /*3 represent the population mean SAT mathematics scores

for the three groups.