Download Test 2 from Fall - Marietta College

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Math 223 – Exam 2
Note: There are 113 points on this test. However, the test is worth 110 points.
Multiple Choice:
Choose the best response to each of the following questions and circle the correct letter.
These questions (1-18) are worth 3 points each.
1. 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) there is a 99% chance that the sample mean will be between 13.96 and 16.04 points.
B) 99% of all observed patients will show an improvement between 13.96 and 16.04 points.
C) the method used to get the interval produces intervals which include the true mean 99% of the time.
D) there is a 99% chance that an individual with Down Syndrome will show improvement on this test after
taking the drug.
2. According to Gordon, Churchill, et al., men have weights with a mean of 172 lbs and a standard deviation of
28.7 lb. The weights of men are normally distributed. Suppose you wish to show that men’s weights vary
less than reported. Determine which of the following 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.
3. In a hypothesis test, the p-value tells us
A) the probability the null hypothesis is false.
B) the probability the null hypothesis is true.
C) 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.
D) 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.
4. Given the same sample statistics, which level of confidence will produce the second widest confidence
interval?
A) 75%
B) 85%
C) 92%
D) 98%
5. If we are testing the null hypothesis H0: p = .60 versus HA: p < .60 and we compute our test statistic to be
z = 2.3, then:
A) we have strong evidence that p is less than .60.
B) we fail to have strong evidence that p is less than .6.
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)
6. A disadvantage of a 95% confidence interval over a 90% 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 smaller.
7. If you want to require stronger evidence against the null hypothesis before you are willing to reject it, you
should
A) increase the level of significance.
B) lower the level of significance.
C) increase the p-value.
D) lower the p-value.
8. Which of the following would result in a larger margin of error?
A) Using a higher level of confidence.
B) Decreasing the sample size.
C) Having a sample with a larger σ.
D) All of the above.
9. A researcher claims that more than 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
10. Consider the following claim: The EPA claims that fluoride in children’s drinking water should be at a
mean level of less than 1.2 ppm, or parts per million, to reduce the number of dental cavities. Which of the
following statements represents a Type II error?
A) Fail to support the claim σ < 1.2 when σ < 1.2 is true.
B) Support the claim μ < 1.2 when μ = 1.2 is true.
C) Support the claim σ < 1.2 when σ = 1.2 is true.
D) Fail to support the claim μ < 1.2 when μ < 1.2 is true.
11. A university administrator obtains a sample of the academic records of past and present scholarship athletes
at the university. The administrator reports that no significant difference was found in the mean GPA for
male and female scholarship athletes (P-value = 0.287). This means
A) the GPAs for male and female scholarship athletes are identical except for 28.7% of the athletes.
B) the maximum difference in GPAs between male and female scholarship athletes is 0.287.
C) the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as
that observed in the sample, or one more extreme, if there is no difference in mean GPAs is 0.287.
D) the chance that a pair of randomly chosen male and female scholarship athletes would have a significant
difference in GPAs is 0.287.
12. 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) There would not be much difference in the confidence interval’s width because the original population is
already normal.
B) The confidence interval would be narrower than the true confidence interval.
C) The confidence interval would be wider than the true confidence interval.
D) There is not enough information given to determine how the confidence intervals would compare.
Use the following to answer question 13:
Suppose we know that 61% of all calls made by surveyors on weekday mornings are answered. We hope to
show that the proportion of answered calls is higher during the weekday evenings. We make 2,450 calls to
randomly chosen numbers during the evenings, of which 1,540 are answered. The resulting P-value is 0.03.
13. What is the risk that this P-value describes?
A) If we decide to say “more than 61% of all survey calls are answered during the weekday evenings”, then
the probability we’ll be wrong is .97.
B) If we decide to say “more than 61% of all survey calls are answered during the weekday evenings”, then
the probability we’ll be right is .03.
C) If we decide to say “more than 61% of all survey calls are answered during the weekday evenings”, then
the probability we’ll be wrong is 1/61.
D) If we decide to say “more than 61% of all survey calls are answered during the weekday evenings”, then
the probability we’ll be wrong is .03.
14. A study used X-ray computed tomography to collect data on brain volumes for a group of patients with
obsessive-compulsive disorders and a control group of healthy persons. Suppose you wish to test the claim
that the populations of total brain volumes for obsessive-compulsive patients and the control group vary
differently. Which type of hypothesis test should you use on the data?
A) comparing two independent proportions
B) comparing two independent means
C) comparing two dependent means
D) comparing two variances
15. When doing a hypothesis test, which of the following would be strong 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
16. Suppose a result is statistically significant at the 5% level. Which of the following is true at the 1% level of
significance?
A) the result is not statistically significant at the 1% level
B) the result is also statistically significant at the 1% level
C) there is not enough information to draw a conclusion about the statistical significance at the 1% level
17. Suppose that you wish to compare the amounts of carbon monoxide from samples of filtered and nonfiltered
king-size cigarettes. Which type of confidence interval should you construct?
A) a two independent proportions confidence interval
B) a two independent means confidence interval
C) a two dependent means confidence interval
D) a two variances confidence interval
18. If we reject the null hypothesis when, in fact, it is true, we have
A) made a correct decision.
B) made a Type I error.
C) made a Type II error.
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.
19. Label each of the following samples as independent samples or dependent samples. (2 pts each)
A) Savannah moths monitored for survival at plot 1 of Iowa prairie; Savannah moths monitored for survival
at plot 2 of Iowa prairie.
B) Patients given the drug Requip for Restless Leg Syndrome; Patients given a placebo for Restless Leg
Syndrome
C) Mothers who gave birth before week 35 of pregnancy; Their babies who developed autism
D) Survival rate of reestablishing saguaro cacti; Survival rate of reestablishing sagebrush
E) Number of in-channel pools within a certain river; Number of pool-inhabiting juvenile coho salmon in
the river
20. When evaluating a loan applicant, a financial officer is faced with the task of granting loans to people who
are good risks and denying loans to people who appear to be poor risks. In effect, the financial officer is
carrying out a hypothesis test where the hypotheses are as follows:
Ho: The applicant is a good risk.
HA: The applicant is a poor risk.
For each of the following situations, indicate whether a correct or incorrect decision has been made. If the
decision is incorrect, specify whether it is a Type I or a Type II error. (2 pts each)
a) The financial officer does not give a loan to an applicant who is a good risk.
b) The financial officer gives a loan to an applicant who is a good risk.
c) The financial officer does not give a loan to an applicant who is a bad risk.
d) The financial officer gives a loan to an applicant who is a bad risk.
Use the following situation to answer the next two questions.
When people smoke, the nicotine they absorb is converted to cotinine, which can be measured. Suppose
you wish to test the claim that the mean cotinine level of all smokers is equal to 200.0.
21. What hypotheses should be used? (3 pts)
22. Suppose after you perform the above hypothesis test, the test shows that you should fail to reject the null
hypothesis. How should you interpret this decision? (i.e. what final conclusion should you draw) (4 pts)
23. An economist wants to estimate the mean income for the first year of work for college graduates who
majored in biology. How many such incomes must be found if we want to be 95% confident that the
sample mean is within $400 of the true population mean? Assume that a previous study has revealed that
for such incomes  = $5370. (6 pts)
24. Assume that a simple random sample has been taken and the population is normally distributed. Engineers
assessing the variation in wait time at Starbucks, Dunkin Donuts, and Caribou Coffee determined that the
standard deviation of the wait time for a sample of twelve customers was 25.6 seconds. Assuming a 99%
confidence level, do the following. (2.5 pts each)
a) Find  L2 .
b) Find  R2
25. In a random sample of 15 days at a small West Virginia coal mine, the mean daily output was x = 36 tons of
coal with a standard deviation of 4.3 tons. It appears that the population has a fairly normal distribution.
Construct a 95% confidence interval for the true average daily output. (6 pts)
26. A campus coffee machine is supposed to put, on average, 8 ounces of coffee into a cup. You think that the
machine always underfills the cups. To test this theory, you ran a hypothesis test using a sample of 50 cups
with a sample mean of 7.83 ounces. The resulting p-value was 0.027. Clearly explain the meaning of this
p-value. (4 pts)
27. Someone proposes that on average, women get their first driver’s license later than men do. To test this
theory, a random sample of 96 women yielded a mean of 19.6 years and a standard deviation of 5.1 years.
A random sample of 75 men was taken which yielded a mean of 17.3 years and a standard deviation of 4.7
years. The 99% confidence interval for this data is (0.3629, 4.2371), where the women represent
population 1. What conclusion should you draw from this interval? Be sure to justify your answer. (5 pts)
28. Clearly explain (using complete sentences) why a result that is statistically significant at the 1% level must
always be statistically significant at the 5% level. If a result is statistically significant at the 5% level, what,
if anything, can you say about its significance at the 1% level? (8 pts)