Download Review Session 3

Review Session 3
Normal Distributions
1. Assume that a distribution of 150 scores is normally distributed with a mean of 92 and a
standard deviation of 11.5.
a. Find the z scores corresponding to observed scores of 87 and 96.
b. Determine the number of scores between observed scores 80 and 100.
c. Find the 80th percentile, P80.
2. Assume that a set of 200 scores is normally distributed with a mean of 60 and a standard deviation of
12.
a. What are the z scores corresponding to the raw scores of 76, 38, and 50?
b. How many scores lie between the values of 48 and 80? 65 and 75? 34 and 52?
c. How many scores exceed the values of 80, 60, and 40?
d. How many scores are less than the values of 35, 50, and 75?
e. Find P35, P80, P55, and P70.
3. The norms for a standard mathematics test, assumed to be normally distributed, are as follows:
National norms:
mean = 75
s = 12
Large-city norms:
mean = 68
s = 15
John has a score of 80, and Mary has a score of 65. What are their percentile ranks in terms of the
national norms? In terms of large-city norms?
4. A statistics instructor tells the class that grading will be based on the normal distribution. He plans to
give 10% As, 20% Bs, 40%Cs, 20%Ds, and 10%Fs. If the final examination scores have a mean of 75
and a standard deviation of 9.6, what is the range of scores for each letter grade?
5. The dean of a college wishes to know the percentage of students who finish a bachelor’s degree
program in 3.5 years or less. He knows the average number of years to earn the degree is 4.3 years,
with a standard deviation of 0.3 years. Assume a normal distribution in calculating the percentage.
6. The results of an English exam show a mean of 55, with a standard deviation of 6. What percentage of
students scored below 46? Above 68? Assume a normal distribution.
7. In an ancient culture, the average male life span was 37.6 years, with a standard deviation of 4.8 years.
The average female life span was 41.2 years, with a standard deviation of 7.7 years. Use the properties of
a normal distribution to find:
a. What percentage of men died before age 30?
b. What percentage of women lived to an age of at least 50?
c. At what age is a female death at the same relative position in the distribution as a male death at
age 35?
(More problems on next
page)
Review Session 3
(continued)
Sampling Distributions
1. Given a distribution of 3000 numbers that are assigned to be normally distributed with  = 120 and  =
25, use the properties of the normal distribution to determine the following probabilities:
a. P(selecting a number less than 100).
b. P(selecting a number greater than 150).
c. P(selecting a number between 110 and 125).
d. P(selecting a number less than 75 or greater than 160).
2. Over the past ten years, a high school Spanish teacher has been giving the same comprehensive final
exam to all students in first-year Spanish. The mean of this exam is 78.4, and the standard deviation is
14.8. To pass the course, a student must score at least 55. To be placed in the honors section, a student
must score at least 99. If a student is selected at random, use the properties of the normal distribution to
determine the following properties:
a. P(student will pass the course).
b. P(student will be placed in the honors section).
c. P(student will score higher than 80).
3. Consider the distribution of scores given in Problem 1. Use the central limit theorem to describe the
sampling distribution of the mean for samples of size 144 and 400.
4. Using the sampling distribution of the mean for the sample size 144 in Problem 3 and the properties of a
the normal distribution, determine the following probabilities:
a. P(X>121.4)
c. P(X<118.2)
d. P(X<120.8)
5. Using the sampling distribution of the mean for the sample size 400 in Problem 3 and the properties of
the normal distribution, determine the following probabilities:
a. P(X>121.4)
c. P(X<118.2)
d. P(X<120.8)
6. Suppose that, in the latest census for a large region, the average income for a family unit was  =
$35,285 and  = $7200. Suppose a sample of size 400 is randomly selected. Develop and describe the
sampling distribution of the mean, and then use the properties of the normal distribution to determine
the following probabilities:
a. P(X>$35000)
c. P(X<$35700)
d. P($34900 < X < $35500)
7. For the census information in Problem 6, what proportion of the families in this region have incomes
exceeding $40000? Incomes below $25000?
Review Session 4
Confidence Intervals and Hypothesis Testing
1. The physical education teachers in a large school district administer a physical performance test to a
random sample of seventh grade boys, n = 64. The sample mean is 84.20 and the standard
deviation is 8.48. Construct the 95-percent confidence interval for the mean of the population of
seventh grade boys in this district.
2. A social scientist believes that working people today want to retire at a later age than workers did
previously. She reviews the available literature and concludes that the desired retirement age five years
ago was 65. To investigate this conjecture, she randomly selects 400 workers between the ages of 58
and 62 and interviews them. Based on these interviews, she finds the mean desired retirement age to
be 65.6 and the standard deviation of 4.0.
a. Test: H0:  = 65 against Ha:  > 65. Use  = .05.
b. Construct CI95. (Hint: The critical value for the confidence interval will not be the same
as for the test statistic.)
3. Given that Mean = 22.6 and s = 2.34, compute and compare the 95-percent confidence
intervals for the sample sizes n = 150 and n = 10.
4. A population mean is being estimated based on a sample size of 49. The sample mean is 95, and the
sample standard deviation is 21.
a. Construct CI95.
b. Give the confidence statement for the interval in Part A.
c. Suppose CI99 is constructed. How does this interval compare to the interval in Part A.
5. Describe what is meant by statistical precision when constructing confidence intervals. Why is
statistical precision enhanced as sample size is increased?
6. A 95-percent confidence interval for the mean, based on a supple of size 169, goes from 80.57 to
87.43.
a. What are the mean and standard deviation of the sample?
b. Assume that the standard deviation of the sample remains the same, how large a sample
would be necessary to obtain a 95-percent confidence interval one-half the length of the one
above?