150
Spring 2013
Name:
Problems 1 – 6 are each worth 5 points.
1. Which is true about a 98% confidence interval for a population proportion based on a given sample?
I. We are 98% confident that the sample proportion is in our interval.
II. We are 98% confident that the population proportion is in our interval.
III. The interval is wider than a 95% confidence interval would be.
(a) I only (b) II only (c) III only (d) I and III (e) II and III
2. Suppose that for a certain population,  = 7500 and  = 85. Suppose also that samples of size 100 will be selected from this population. Which of the following is true of the distribution of the sample means (i.e. the distribution of x )?
(a) m ( ) =
7500,  s ( ) =
85, and we cannot assume the distribution is ≈ normal
(b) m ( ) =
7500,  s ( ) =
85, and we can assume the distribution is ≈ normal
(c) m ( ) =
7500,  s ( ) =
8.5, and we cannot assume the distribution is ≈ normal
(d) m ( ) =
7500,  s ( ) =
8.5, and we can assume the distribution is ≈ normal
(e) None of the above
3. Suppose that for a certain population, the proportion of success is 8.2%. Suppose also that samples of size 100 will be selected from this population. Which of the following is true of the distribution of the sample proportions of success (i.e. the distribution of p
)?
(a) m ( ) =
0.082,  s ( ) =
0.0274, and we cannot assume the distribution is ≈ normal
(b) m ( ) =
0.082,  s ( ) =
0.0274, and we can assume the distribution is ≈ normal
(c) m ( ) =
0.082,  s ( ) =
0.0027,
and we cannot assume the distribution is ≈ normal
(d) m ( ) =
0.082,  s ( ) =
0.0027,
and we can assume the distribution is ≈ normal
(e) None of the above

4.
Suppose that in a hypothesis test of H
0
: m =
1250
vs. H a
: m >
1250
, the P-value is 0.0155.
Which of the following is true?
(a) We fail to reject the null hypothesis, potentially making a Type I error.
(b) We fail to reject the null hypothesis, potentially making a Type II error.
(c) We reject the null hypothesis, potentially making a Type I error.
(d) We reject the null hypothesis, potentially making a Type II error.
(e) None of the above
5. In a certain hypothesis test, a P-value of 0.104 is calculated. Which of the following is true?
(a) There is a 10.4% chance that the null hypothesis is true.
(b) There is a 10.4% chance that the alternative hypothesis is true.
(c) There is a 10.4% chance of the observed statistic given the null hypothesis is true.
(d) Both (a) and (c)
(e) None of the above
6. A confidence interval for a population mean was constructed based on a sample of size 160.
Another confidence is to be constructed with the same confidence level, but is to have a margin of error ¼ the size of the first. What size sample is required?
(a) 40 (b) 10 (c) 640 (d) 2560 (e) any size of 30 or more
7. Find the P-value for each of the following scenarios.
(6 points each)
(a) (b) (c)
H
0
: p
=
0.68
H
0
:
H a
: p
<
0.68
H a
: m =
1500
H
0
: p
1
= p
2
= p
3
= p
4 m ¹
1500
H a
: p ’s not all equal z = -1.93 t = -2.65 c
2
=
5.0594
n = 15

8. A new battery is about to be released by the BatCo company. For advertising purposes, the president of the company wants his research team to construct a 95% confidence interval for the mean lifespan of this battery with a margin of error of no more than 5 hours. If the standard deviation of lifespans of similar batteries is 26.4 hours, how many batteries must be tested to construct the desired interval?
(8 points)
9. The human resources department of a very large company plans to construct a 90% confidence interval for the proportion of the employees that are pleased with the health care benefits the company offers them with a margin of error of no more than 4 percentage points.
How many employees should be randomly polled?
(8 points)
10. A company has developed a new light bulb that they claim has a life span 4200 hours with a standard deviation of 72 hours. What is the probability that in a sample of 100 such bulbs, the average life span is less than 4185 hours?
(9 points)
11. A company that manufactures computer chips believes that only 2% of their chips are defective. Suppose that this is true. What is the probability that in a sample of 1000 of this company’s computer chips, more than 3% are defective?
(9 points)

12. The data below gives the number of calories in one serving of 8 randomly selected bran cereals.
70 120 70 50 110 110 130 90
(a) Show that the data is consistent with having come from a normal population.
(4 points)
(b) Using this data, construct a 96.5% confidence interval for the true mean caloric content of all bran cereals.
(5 points)
(c) If we were going to test the hypotheses H
0
:  = 100 vs. H a
:  < 100, what test statistic would be generated by this data? Show your calculation.
(6 points)
Note, you will first need to compute the sample mean and sample standard deviation for this sample. You may use your calculator for this.
13. Some people fear the use of nuclear energy because of potential accidents. However, a
Gallup Poll in 2009 indicated that 74% of a random sample of 504 men supported the use of nuclear energy. Construct a 94% confidence interval for the true proportion of all men that would be in favor of the use of nuclear energy.
(8 points)

14. The numbers below give summary data for the average ACT scores for incoming freshman at a certain university for two groups: those receiving football scholarships and those not receiving football scholarships.
(5 points each)
(a) Construct a 98% confidence x s
Scholarship 21.86 2.84 n
44 interval for the difference in average ACT scores for football
No Scholarship 24.75 3.29 scholarship students and non-football scholarship students.
31
(b) Based on your answer to (a), would you conclude that average ACT scores are the same for these two populations or not. Support your answer.
15. In a recent study, the subjects were asked, “How many of your friends do you consider close enough to that you would discuss important personal matters with them?” The researcher collected the data, and its summary is below. Does the data support that, on average, females have more “close friends” than men. Test the appropriate hypotheses at the  = 0.05 level.
(10 points) n x s
Male 645 1.861 1.777
Female 813 2.089 1.760
H
0
: Validity:
H a
:
Test Statistic:
P -value:
Conclusion:

16. A study was conducted to investigate how effective a new nicotine lozenge is at helping people quit smoking. Subjects were monitored for a year. Of the 459 subjects who had been taking the nicotine lozenge, 82 successfully abstained from smoking for the year. Of the 458 subjects who had been taking a placebo lozenge, 66 successfully abstained from smoking.
Does this data support that taking the nicotine lozenge increases the chance of quitting smoking (at least for a year)? Test the relevant hypotheses at the  = 0.05 level.
(10 points)
H
0
: Validity:
H a
:
Test Statistic:
P -value:
Conclusion:

17. Capstone Question
Two candidates, Morgan and Roebling, are each running for the same political office in a large city. Morgan’s staff conducts a poll of 600 randomly selected likely voters in the upcoming election. They find that 342 or 57% intend to vote for Morgan and provide for him a 95% confidence interval for the true proportion, p
M
, namely, p
M
is
in
( )
or
53% < p
M
<
61%
Based on these data, Morgan calls a press conference and proudly announces, “I have this election in the bag; 57% of the voters are solidly behind me!”
On the same day, Roebling’s staff conducts a poll of 500 randomly selected individuals who had voted in the last presidential election and find that 275 or 55% say they intend to vote for him. His staff likewise provides a 95% confidence interval for the true proportion, p
R
, namely, p
R
is
in
( )
or
50.6% < p
R
<
59.4%
Not to be outdone, Roebling calls a press conference and also announces, “ I have this election sewn up. Clearly the majority of voters favor me.”
The local media has been following this contest closely. It reports, “We must conclude that this election belongs to Morgan. After all, the difference in proportions computed from the confidence intervals show that the lead of Morgan over Roebling lies between 1.6% and
2.4%. Clearly, Morgan has the lead.”
A friend of yours is puzzled by all this and hopes that you can answer his questions. Please respond in carefully worded complete sentences. His questions are:
(3 points each)
(a) “So, does this mean that there is a 95% probability that p
M
lies between 53% and
61%?” Explain what the confidence interval means.
(b) “Must it not be the case that p
M
+ p
R
≤ 1? Then why is 57% + 55% > 100%?”
(c) “Morgan and Roebling each claim the majority of voters. How can you resolve that?
(d) "Since Morgan has a larger sample size, does this mean that his claim is more likely to be true and that his confidence interval will contain p
M
?"
(e) “I am really confused by the media claim that Morgan leads Roebling by a percentage between 1.6% and 2.4%. Can you clarify their reasoning? Is it correct?”
Please write answers to these on the separate page provided. Make sure to use complete sentences and be as detailed as possible.

Answers for #17 Name: