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Z test
Z scores
1. The height X of American young women has approximately the normal distribution with mean μ
= 65.5 inches and standard deviation σ = 2.5 inches. Find the relative frequencies of each of the
following events. In each case, sketch a normal curve and shade the area that represents the
relative frequency.
2. The Graduate Record Examinations (GRE) are widely used to help predict the performance of
applicants to graduate schools. The range of possible scores on a quantative GRE is 200 to 900.
The psychology department at a university finds that the scores of its applicants on the
quantitative GRE are approximately normal with mean μ = 544 and standard deviation σ = 103.
Find the relative frequency of applicants whose score X satisfies each of the following
conditions:
3. In a study of elite distance runners, the mean weight was reported to be 63.1 kilograms (kg),
with a standard deviation of 4.8 kg. Assuming that the distribution of weights is normal. If 90%
of elite distance runners weigh less than Peter, what is Peter's weight?
4. The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are
normally distributed with μ = 100 and σ = 15.
What percent of this population have WISC scores below 100
Below 80?
Above 140?
Between 100 and 120?
5. Steelman, Powell, and Carini (2000) explored the relationship between teacher unions and
student educational performance as measured by standardized tests, such as the American
College Testing exam (ACT). Use the ACT statistics shown here to answer the following
questions. The distribution is normally shaped. x = 22 ACT points s = 2 ACT points n = 441,574
students
a. What proportion of students scored above 26?
b. What number of students who took the ACT scored between 17 and 19?
c. Determine the score below which 90 percent of the scores fell.
6. For a large homeless population. Wong and Piliavin (2001) examined stressors, resources, and
psychological distress using the Center for Epidemiological Studies Depression Scale (CESD), a
community screening questionnaire. Among homeless persons, the mean CESD score is 23.5
with a standard deviation of 7.5 and the distribution is normal. As an intake worker at a
homeless shelter, you wish to apply their research. When new clients arrive, you administer the
CESD. Answer the following questions.
a. Any client scoring 16 or higher is to be sent to a doctor. What is the probability that your
next client will be sent to a doctor?
7. The average hobbit 3 ½ feet tall with a s = 4 inches. If the door on a hobbit house(they live in
the ground) are all 4 feet high. What percent of hobbits must duck before entering?
8. On a statistics exam, you have a score of 73. If the mean of the exam is 65 would you
prefer the standard deviation of the scores to be 8 or 16? Why?
9. A normal distribution has a mean of 120 and a standard deviation of 20. For this
distribution
a. What score separates the top 40% of the scores from the rest?
b. What score corresponds to the 90th percentile?
c. What range of scores would form the middle 60% of this distribution?
10. The population mean for a Psych test in a class that is usually taken by juniors and
seniors is 100 (out of 200), with a population standard deviation of 16. A random sample
of 16 sophomores is selected from the course, and they have a mean score of 116, with a
sample standard deviation of 10. The research question is whether or not sophomores are
different from the average student in this class. Evaluate this question, stating the null and
alternative hypotheses