Chapter 7 Exercises
Prerequisites
All material presented in chapter 7
Selected answers
You may want to use the "r to z' calculator" and the "Calculate Area for a given X" applet for some of these exercises.
1. A population has a mean of 50 and a standard deviation of 6. (a) What are the mean and standard deviation of the sampling distribution of the mean for N = 16? (b) What are the mean and standard deviation of the sampling distribution of the mean for N = 20? (Ch. 7.E)
2. Given a test that is normally distributed with a mean of 100 and a standard
deviation of 12, find:
(a) the probability that a single score drawn at random will be greater than 110 (Ch. 6.C)
(b) the probability that a sample of 25 scores will have a mean greater than 105 (Ch. 7.E)
(c) the probability that a sample of 64 scores will have a mean greater than 105 (Ch. 7.E)
(d) the probability that the mean of a sample of 16 scores will be either less than 95 or greater than 105 (Ch. 7.E)
3. What term refers to the standard deviation of the sampling distribution? (Ch. 7.A)
4. (a) If the standard error of the mean is 10 for N = 12, what is the standard error
of the mean for N = 22?
(b) If the standard error of the mean is 50 for N = 25, what is it for N = 64? (Ch. 7.E)
5. A questionnaire is developed to assess women's and men's attitudes toward using
animals in research. One question asks whether animal research is wrong and is
answered on a 7point scale. Assume that in the population, the mean for women is 5,
the mean for men is 4, and the standard deviation for both groups is
1.5. Assume the scores are normally distributed. If 12 women and 12 men are
selected randomly, what is the probability that the mean of the women will be
more than 1.5 points higher than the mean of the men? (Ch. 7.F)
6. If the correlation between reading achievement and math achievement in the
population of fifth graders were 0.60, what would be the probability that in a
sample of 28 students, the sample correlation coefficient would be greater than
0.65? (Ch. 7.G)
7. If numerous samples of N = 15 are taken from a uniform distribution and a
relative frequency distribution of the means is drawn, what would be the shape of
the frequency distribution? (Ch. 7.E & Ch. 7.D)
8. A normal distribution has a mean of 20 and a standard deviation of 10. Two
scores are sampled randomly from the distribution and the second score is
subtracted from the first. What is the probability that the difference score
will be greater than 5? Hint: Read the Variance Sum Law section of Chapter 3. (Ch. 7.A & Ch. 3.E)
9. What is the shape of the sampling distribution of r? In what way does the shape
depend on the size of the population correlation? (Ch. 7.G)
10. If you sample one number from a standard normal distribution, what is the
probability it will be 0.5? (Ch. 7.A & Ch. 6.F)
11. A variable is normally distributed with a mean of 120 and a standard deviation
of 5. Four scores are randomly sampled. What is the probability that the mean of the
four scores is above 127? (Ch. 7.E)
12. The correlation between self esteem and extraversion is .30. A sample of 84
is taken. (a) What is the probability that the correlation will be less than 0.10? (b) What is the probability that the correlation will be greater than 0.25? (Ch. 7.G)
13. The mean GPA for students in School A is 3.0; the mean GPA for students in School B is 2.8. The standard deviation in both schools is 0.25. The GPAs of both schools are normally distributed. If 9 students are randomly sampled from each school, what is the probability that:
(a) the sample mean for School A will exceed that of School B by 0.5 or more? (Ch. 7.F)
(b) the sample mean for School B will be greater than the sample mean for School A? (Ch. 7.F)
14. In a city, 70% of the people prefer Candidate A. Suppose 30 people from this city were sampled. (a) What is the mean of the sampling distribution of p? (b) What is the standard error of p? (c) What is the probability that 80% or more of this sample will prefer Candidate A? (d) What is the probability that 45% or more of this sample will prefer some other candidate? (Ch. 7.H)
15. When solving problems where you need the sampling distribution of r, what is the reason for converting from r to z'? (Ch. 7.G)
16. In the population, the mean SAT score is 1000. Would you be more likely (or equally likely) to get a sample mean of 1200 if you randomly sampled 10 students or if you randomly sampled 30 students? Explain. (Ch. 7.C & Ch. 7.E)
17. True/false: The standard error of the mean is smaller when N = 20 than when N = 10. (Ch. 7.E)
18. True/false: The sampling distribution of r = .8 becomes normal as N increases. (Ch. 7.G)
19. True/false: You choose 20 students from the population and calculate the mean of their test scores. You repeat this process 100 times and plot the distribution of the means. In this case, the sample size is 100. (Ch. 7.E & Ch. 7.B)
20. True/false: In your school, 40% of students watch TV at night. You randomly ask 5 students every day if they watch TV at night. Every day, you would find that 2 of the 5 do watch TV at night. (Ch. 7.C & Ch. 7.H)
21. True/false: The median has a sampling distribution. (Ch. 7.A)
22. True/false: Refer to the figure below. The population distribution is shown in black, and its corresponding sampling distribution of the mean for N = 10 is labeled "A." (Ch. 7.D & Ch. 7.E)
Questions from Case Studies:
The following questions use data from the Angry Moods (AM) case study.
23. (a) How many men were sampled? (b) How many women were sampled?
24. What is the mean difference between men and women on the AngerOut scores?
25. Suppose in the population, the AngerOut score for men is two points higher than it is for women. The population variances for men and women are both 20. Assume the AngerOut scores for both genders are normally distributed. Given this information about the population parameters:
(a) What is the mean of the sampling distribution of the difference between means? (Ch. 7.F)
(b) What is the standard error of the difference between means? (Ch. 7.F)
(c) What is the probability that you would have gotten this mean difference (see #24) or less in your sample? (Ch. 7.F)
The following questions use data from the Animal Research (AR) case study.
26. How many people were sampled to give their opinions on animal research?
27. (AR#11) What is the correlation in this sample between the belief that animal research is wrong and belief that animal research is necessary? (Ch. 4.E)
28. Suppose the correlation between the belief that animal research is wrong and the belief that animal research is necessary is .68 in the population.
(a) Convert .68 to z'. (Ch. 7.G)
(b) Find the standard error of this sampling distribution. (Ch. 7.G)
(c) Assuming the data used in this study was randomly sampled, what is the probability that you would get this correlation or stronger (closer to 1)? (Ch. 7.G)
Answers:
1) (a) Mean = 50, SD = 1.5
2) (b) .019
4) (a) 7.39
11) .0026
12) (b) .690
13) (a) .0055
14) (c) .116
23) (a) 30
25) (a) 2
28) (c) .603
