Name: ___________________________________
Grade: ___________
Math 223 – Exam 2
Note: There are 104 points on this test. However, the test is worth 100 points.
Multiple Choice:
Choose the best response to each of the following questions and circle the correct letter.
These questions (1-21) are worth 3 points each.
1. A researcher claims that 60% of voters favor gun control. Determine which type of hypothesis test should
be used to test this claim.
A) a left-tailed test
B) a right-tailed test
C) a two-tailed test
2. Given the same sample statistics, which level of confidence will produce the narrowest confidence interval?
A) 75%
B) 85%
C) 92%
D) 98%
3. Suppose a result is statistically significant at the l% level. Which of the following is true at the 5% level of
significance?
A) the result is not statistically significant at the 5% level
B) the result is also statistically significant at the 5% level
C) there is not enough information to draw a conclusion about the statistical significance at the 5% level
4. Consider the following claim: The mean RDA of sodium is 2400 mg. It is also known that the population
is approximately normal with a standard deviation of 550 mg. Suppose that a sample of size 15 resulted in
x =2600 mg. Determine which is the appropriate approach to use to conduct a hypothesis test.
A) Use the normal distribution.
B) Use the Student t distribution.
C) Use the Chi-square distribution
D) Cannot determine from the information provided.
5. Medical researchers are studying the improvement of individuals with Down Syndrome on a language and
memory test after taking a certain drug. They report, “With 99% confidence, the mean test score
improvement is between 13.96 and 16.04 points. The phrase “99% confidence” means
A) 99% of all observed patients will show an improvement between 13.96 and 16.04 points.
B) the method used to get the interval produces intervals which include the true mean 99% of the time.
C) there is a 99% chance that an individual with Down Syndrome will show improvement on this test after
taking the drug.
D) there is a 99% chance that the sample mean will be between 13.96 and 16.04 points.
6. Consider the following claim: The standard deviation of the diameter at breast height, or DBH, of the slash
pine tree is less than one inch. Which of the following statements represents a Type I error?
A) Fail to support the claim σ < 1 when σ < 1 is true.
B) Support the claim μ < 1 when μ = 1 is true.
C) Support the claim σ < 1 when σ = 1 is true.
D) Fail to support the claim μ < 1 when μ < 1 is true.
7. You plan to construct a confidence interval for the mean µ of a population with a standard deviation σ.
Which of the following would result in a larger margin of error?
A) Using a lower level of confidence.
B) Decreasing the sample size.
C) Reducing σ.
D) All of the above.
Use the following to answer question 8:
In order to track the percentage of misdiagnoses by Emergency Room doctors, a sample of 900 patients
revealed that 45 patients were misdiagnosed with indigestion instead of a heart attack. Suppose the hospital
uses this information to test the claim that physicians misdiagnose less than 6% of heart attacks. The hypothesis
test results in a p-value of 0.1033.
8. What is the risk that the p-value describes in the above situation?
A) If we decide to say “the physicians misdiagnose less than 6% of heart attacks”, then the probability we’ll
be right is 1%.
B) If we decide to say “the physicians misdiagnose less than 6% of heart attacks”, then the probability we’ll
be wrong is 10%.
C) If we decide to say “the physicians misdiagnose less than 6% of heart attacks”, then the probability we’ll
be right is 10%.
D) If we decide to say “the physicians misdiagnose less than 6% of heart attacks”, then the probability we’ll
be wrong is less than 6%.
9. An advantage of a 90% confidence interval over a 95% confidence interval is
A) the method used to construct the interval, when used over and over, produces intervals which include the
true population parameter more often.
B) the margin of error is larger.
C) the margin of error is smaller.
D) the width of the interval is larger.
10. Suppose you wish to disprove the claim that μ = 3.5. Given a sample size of n = 50 and a level of
significance of α = 0.01, when should you reject the null hypothesis?
A) If the standardized test statistic is greater than 1.645 or less than -1.645.
B) If the standardized test statistic is greater than 2.576 or less than -2.576.
C) If the standardized test statistic is greater than 2.576.
D) If the standardized test statistic is less than -1.645.
11. In a recent study of 100 eighth graders, the mean number of hours per week that they watched television
was 19.6. The population standard deviation is 5.8. A confidence interval results in (18.25, 20.95).
Determine the confidence level.
A) 90%
B) 92%
C) 98%
D) 99%
12. If you want to require stronger evidence against the null hypothesis before you are willing to reject it, you
should
A) lower the p-value.
B) increase the p-value.
C) lower the level of significance.
D) increase the level of significance.
13. In a hypothesis test, the p-value tells us
A) the probability the null hypothesis is true.
B) the probability the null hypothesis is false.
C) the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as
extreme as the one observed in the sample.
D) the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as
extreme as the one observed in the sample.
14. The financial aid office of a university asks a sample of students about their employment and earnings. The
report says, “For academic year earnings, a significant difference ( P-value = .038) was found between the
sexes, with men earning more on the average.” This means
A) the chance of obtaining a difference in academic year earnings for male and female students as large as
that observed in the sample, or one more extreme, if there is really no difference is 0.038.
B) the chance of obtaining a difference in academic year earnings for male and female students as large as
that observed in the sample, or one more extreme, is 0.962.
C) the chance that a pair of randomly chosen male and female students would have a significant difference
in academic year earnings is 0.962.
D) the academic year earnings for male and female students are different except in 3.8% of all cases.
15. If we reject the null hypothesis when, in fact, it is false, we have
A) made a correct decision.
B) made a Type I error.
C) made a Type II error.
16. If we are testing the null hypothesis H0: p = .40 versus HA: p > .40 and we compute our test statistic to be
z = 1.5, then:
A) we have strong evidence that p is greater than .40.
B) we fail to have strong evidence that p is greater than .4.
C) we have “significant” results, with a p-value that is less than 0.05.
D) we fail to have “significant” results, with a p-value that is greater than 0.05.
E) both (A) and (C)
F) both (B) and (D)
17. When doing a hypothesis test, which of the following would be weak evidence against the null hypothesis?
A) using a small level of significance
B) using a large level of significance
C) obtaining data with a small p-value
D) obtaining data with a large p-value
18. The mean age of bus drivers in Chicago is 48.9 years. If a hypothesis test is performed, how should you
interpret a decision that fails to reject the null hypothesis?
A) There is sufficient evidence to support the claim that μ = 48.9.
B) There is not sufficient evidence to support the claim that μ = 48.9.
C) There is sufficient evidence to reject the claim that μ = 48.9.
D) There is not sufficient evidence to reject the claim that μ = 48.9.
19. Assume that a small simple random sample is selected from a normally distributed population for which σ is
unknown. Construction of a confidence interval should use the t distribution, but suppose you used the
normal distribution instead. How would this mistake affect the confidence interval limits?
A) The confidence interval would be wider than the true confidence interval.
B) The confidence interval would be narrower than the true confidence interval.
C) There would not be much difference in the confidence interval’s width because the original population is
already normal.
D) There is not enough information given to determine how the confidence intervals would compare.
20. How many of the below statements about correlation are true?
A) 1
B) 2
C) 3
D) 4
E) 5
i) The value of r is heavily influenced by outliers.
ii) r can be any value between -1 and 1.
iii) r describes how tightly the points on a scatterplot cluster about a straight line.
iv) It makes sense to talk about the correlation between a student’s height and his/her gender.
v) The value of r does not change as you change the units of the two variables.
21. Children in the same family exhibit a very strong tendency to have similar heights at similar ages. Hence,
a plausible value for the correlation between the heights of two children of the same parents, at similar ages,
is
A) r = -0.96.
B) r = -0.23.
C) r = 0.96.
D) r = 0.23.
Short Answer:
Answer the following questions in the space provided after each question. Remember to show all of your work
to receive full credit. The point value for each part/question is noted after each part/question.
22. An airline claims that the no-show rate for passengers is less than 5%. In a sample of 420 randomly
selected reservations, 19 were no-shows. Assuming that you wish to test the airline’s claim, state the correct
hypotheses and find the test statistic. (Note: You do NOT need to complete the hypothesis test.) (5 pts)
23. A company is often interested in determining if there is a relationship between advertising expenditures and
number of items sold. A certain pharmaceutical company gathered the following data from a randomly
selected sample. The paired data consist of the cost (x) of regionally advertising (in thousands of dollars) a
certain pharmaceutical drug and the number of new prescriptions (y) written (in thousands).
Advertising Cost (x)
Number written (y)
9
85
2
52
3
55
4
68
2
67
5
86
9
83
10
73
The resulting linear regression equation is ŷ = 55.788 + 2.7885 x and the correlation is r = 0.708.
a) Find the predicted value of the number of new prescriptions written if $6000 is spent in regional
advertising. (2.5 pts)
b) Find the residual when the company spends $3000 in regional advertising. (3 pts)
c) Find r 2 and clearly explain what it means. (3.5 pts)
24. A random sample of 56 fluorescent light bulbs has a mean life of 645 hours with a standard deviation of 31
hours.
a) Construct a 95% confidence interval for the true mean life of fluorescent light bulbs. (5 pts)
b) Based on your answer from part a), do you have reason to believe that the average fluorescent light bulb
has a lifetime of more than 640 hours? Why or why not? (4 pts)
25. In order to fairly set flat rates for auto mechanics, a shop foreman needs to estimate the average time it takes
to replace a fuel pump in a car. How large a sample must he select if he wants to be 99% confident that the
true average time is within 5 minutes of the sample average? Assume the standard deviation of all times is
26 minutes. (6 pts)
26. Biologists are investigating their efforts to prevent erosion on the bank of a stream. For this stream, a
narrow channel width is a good indicator that erosion is not occurring. Therefore, the biologists know that
their efforts are failing if the mean width is more than 3.7 meters. In order to test if this is the case, they
measured the channel width at 12 locations. This sample resulted in a mean of 4.52 meters and a standard
deviation of 1.347 meters.
a) Assuming the widths follow a normal distribution, find the corresponding p-value for their test. (7 pts)
(Be sure to show all of the steps required to get to the p-value.)
b) If the biologists use a 5% level of significance, what conclusion should they make? (2.5 pts)
c) How, if any, would their conclusion change if the level of significance was changed to 1%? (2.5 pts)