Practice Test for Stat 515
Second Examination
Part A: For questions 1-3 please refer to the following situation. The pulse rate per minute of the
adult male population between 18 and 25 years of age in the United States is known to have a
normal distribution with mean of 72 beats per minute and a standard deviation of 9.7.
1. Suppose that a male between the age of 18 to 25 years is randomly selected. What is the
probability that his pulse rate will exceed 100 beats per minute? [Such an event will
disqualify him from getting into the Marine Corps (not “cool enough”)!]
2. What is the probability that his pulse rate will be between 70 and 82 beats per minute?
3. What is the lowest pulse rate that a male between 18 and 25 years of age could have but still
belong to the highest 5% of all pulse rates?
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Part B: For questions 4-6, refer to the information provided in Part A regarding the distribution
of pulse rates of American males between 18 and 25 years old, and consider the situation where
36 of these males are randomly sampled. Let X denote the sample mean of the pulse rates of
these 36 males.
4. Describe the sampling distribution of X . That is, describe its shape and also state its mean
and its standard error.
5. Find the probability that X will either be between 70 and 74 beats per minute.
6. Find the value of A such that the probability that X will be at most A is 0.05.
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Part C: For questions 7-8 please refer to the following situation. Assume that when a person is
afflicted with melanoma (a form of skin cancer), the probability that it will kill him/her is 0.45.
Suppose that we have 500 patients who are afflicted by melanoma. Let p̂ denote the proportion
of these 500 patients who will be killed by the melanoma.
7. What is the approximate sampling distribution of p̂ , and what are the mean and standard
error of p̂ ?
8. Find the approximate probability that p̂ is between 0.44 and 0.47?
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Part D: For questions 9-11 please refer to the following situation. A researcher wants to estimate
, the mean academic year salary of female faculty members in US universities. She takes a
random sample of n = 100 female faculty members from US universities and determines their
academic year salaries. From this sample she observes a sample mean academic year salary of
X  $51,000 with a sample standard deviation of S = $10,000.
9. Based on this information, construct a 95% confidence interval for the mean academic year
salary of female faculty members in US universities.
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10. What will happen to the length of your confidence interval if you change the confidence level
from 95% to 99%?
11. Personally, which interval would you prefer in this situation: the 95% confidence interval or
the 99% confidence interval. Explain your reasoning for your choice.
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Part E: For questions 12-13, refer to the following situation. A researcher would like to
determine the proportion, p, of South Carolinians who play the state lottery. Suppose that this
researcher desires to have a 95% confidence interval for p whose margin of error is at most 0.03.
12. How many SC voters should this researcher sample [Assume that you don't have any prior
information about the value of p.]?
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12. Discuss what will happen to the desired sample size if
(a) the confidence coefficient 100(1 - )% is changed from 95% to 99%;
(b) the maximum tolerable margin of error (denoted by B) is changed from 0.03 to 0.02.
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Part F: For questions 14-15 refer to the following information. Suppose that the researcher in
Part E decided to take a sample of size n = 1500 South Carolinians, and from this sample, the
researcher finds that 400 play the state lottery.
14. Based on this information, construct a 99% confidence interval for the population proportion
p?
15. Based on the confidence interval in question 14, could the researcher conclude that the
proportion (in percent) of all South Carolinians who play the state lottery is at least 30%?
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Part G: Consider a population whose probability function is described by:
x
p(x)
1
.5
2
.2
3
.3
16. Determine the population mean, , and the population standard deviation, .
17. Enumerate all possible samples of size n=2 together with their respective probabilities. Also,
for each of these samples, compute the sample mean.
18. Determine the sampling distribution of the sample mean. Present this in tabular form.
19. Based on the sampling distribution you constructed in (18), determine the mean of the sample
mean. Is this equal to the population mean?
20. Determine the standard error of the sample mean. How does this relate to the population
standard deviation?
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