Exam 5: Understanding and Comparing Distributions

A state's Department of Education reports that 12% of the high school students in that state attend private high schools. The State University wonders if the percentage is the same in their applicant pool. Admissions officers plan to check a random sample of the over 10,000 applications on file to estimate the percentage of students applying for admission who attend private schools. -Interpret the confidence interval in this context.

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Question 1

Correct Answer:

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We are 90% confident that between 7.9% and 12.6% of the applicants attend private high schools.

In 2000, the United Nations claimed that there was a higher rate of illiteracy in men than in women from the country of Qatar. A humanitarian organization went to Qatar to conduct a random sample. The results revealed that 45 out of 234 men and 42 out of 251 women were classified as illiterate on the same measurement test. Do these results indicate that the United Nations findings were correct? -Find a 95% confidence interval for the difference in the proportions of illiteracy in men and women from Qatar. Interpret your interval.

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Question 2

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With the conditions satisfied (from Problem 1), the sampling distribution of the difference in proportions is approximately Normal with a mean of pm - pw, the true difference between the population proportions.
We can find a two-proportion z-interval.
We know: $n_{m}=234, \hat{p}_{m}=0.192, n_{w}=251, \hat{p}_{w}=0.167$

We estimate $S D\left(\hat{p_{m}}-\hat{p_{w}}\right)$ as

$S E\left(\hat{p}_{m}-\hat{p}_{w}\right)=\sqrt{\frac{\hat{p}_{m} \hat{q}_{m}}{n_{m}}+\frac{\hat{p}_{w} \hat{q}_{w}}{n_{w}}}=\sqrt{\frac{(0.192)(0.808)}{234}+\frac{(0.167)(0.833)}{201}}=0.0349$

$\begin{array}{l}M E=z^{*} \times S E\left(\hat{p_{m}}-\hat{p}_{w}\right)=1.96(U .0349)=0.0684 \\\end{array}$

The observed difference in sample proportions = pm - pw = 0.192 - 0.167 = 0.025, so the 95% confidence interval is 0.025 ± 0.0684, or -4% to 9%. We 95% confident that the proportion of illiterate men in Qatar is between
4-percentage points
lower and 9-percentage points higher than the proportion of illiterate women.
According to the 2010 census, 20.3% of the population of the United States (ages 5 and up) live in a home in which a language other than English is spoken. Advocates for providing government programs to assist non-English speakers are convinced that, with the increasing non-white population in the United States, this proportion has probably increased. They plan to conduct a survey, and if they find the proportion of people who live in such homes has increased, they will organize a campaign to increase government investment in these assistance programs.

Researchers conduct a study to test a potential side effect of a new allergy medication. A random sample of 160 subjects with allergies was selected for the study. The new "improved" Brand I medication was randomly assigned to 80 subjects, and the current Brand C medication was randomly assigned to the other 80 subjects. 14 of the 80 patients with Brand I reported drowsiness, and 22 of the 80 patients with Brand C reported drowsiness. -Does the interval in question 1 provide evidence that the side effect of drowsiness is different with the new medication?

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Question 3

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There is not sufficient evidence because 0 is contained in the interval. There may be no difference in the proportion of drowsiness reported.

A company manufacturing computer chips finds that 8% of all chips manufactured are defective. Management is concerned that employee inattention is partially responsible for the high defect rate. In an effort to decrease the percentage of defective chips, management decides to offer incentives to employees who have lower defect rates on their shifts. The incentive program is instituted for one month. If successful, the company will continue with the incentive program. -Write the company's null and alternative hypotheses.

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Question 4

A pharmaceutical company investigating whether drug stores are less likely than food stores to remove over-the-counter drugs from the shelves when the drugs are past the expiration date found a P-value of 2.8%. This means that:

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Question 5

Not wanting to risk poor sales for a new soda flavor, a company decides to run one more taste test on potential customers, this time requiring a higher approval rating than they had for earlier tests. This higher standard of proof will increase I. the risk of Type I error II. the risk of Type II error III. power

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Question 6

According to the 2010 census, 20.3% of the population of the United States (ages 5 and up) live in a home in which a language other than English is spoken. Advocates for providing government programs to assist non-English speakers are convinced that, with the increasing non-white population in the United States, this proportion has probably increased. They plan to conduct a survey, and if they find the proportion of people who live in such homes has increased, they will organize a campaign to increase government investment in these assistance programs. -In this context describe a Type I error and the impact such an error would have on this group's efforts.

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Question 7

We test the hypothesis that p = 35% versus p < 35%. We don't know it but actually p = 26%. With which sample size and significance level will our test have the greatest power?

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Question 8

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels. -Describe to the board of directors an advantage and a disadvantage of using a 5% alpha level of significance instead.

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Question 9

The countries of Europe report that 46% of the labor force is female. The United Nations wonders if the percentage of females in the labor force is the same in the United States. Representatives from the United States Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate a percentage of females in the United States labor force. -They actually select a random sample of 525 employment records, and find that 229 of the people are females. Create the confidence interval.

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Question 10

Pew Research found that, in 2013, 50% of American adults favored allowing same-sex couples to marry legally. This is up from 48% in 2012. The 2013 estimate was based on a random sample of 1,501 adults. Assume the same sample size was used in 2012. ["Changing Attitudes on Gay Marriage," Pew Internet and American Life Project, June 2013 -Because it is known that support for allowing same-sex couples to marry has been rising, it would be reasonable to perform a one-sided hypothesis test with the alternative hypothesis that the proportion of Americans who favor allowing such marriages is greater in 2013 than 2012. Would such a test cause you to reach the same conclusion you reached in question 3?

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Question 11

A report on health care in the US said that 28% of Americans have experienced times when they haven't been able to afford medical care. A news organization randomly sampled 801 black Americans, of whom 38% reported that there had been times in the last year when they had not been able to afford medical care. Does this indicate that this problem is more severe among black Americans? -Was your test one-tail upper tail, one-tail lower tail, or two-tail? Explain why you chose that kind of test in this situation.

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Question 12

Employment program A city council must decide whether to fund a new "welfare-to-work" program to assist long-time unemployed people in finding jobs. This program would help clients fill out job applications and give them advice about dealing with job interviews. A six-month trial has just ended. At the start of this trial a number of unemployed residents were randomly divided into two groups; one group went through the help program and the other group did not. Data about employment at the end of this trial are shown in the table. Should the city council fund this program? Test an appropriate hypothesis and state your conclusion. Current job status Employed Unemployed Group 1 (Help program) 20 34 Group 2 (No help) 13 33

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Question 13

A researcher investigating whether joggers are less likely to get colds than people who do not jog found a P-value of 3%. This means that:

(Multiple Choice)

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Question 14

A company manufacturing computer chips finds that 8% of all chips manufactured are defective. Management is concerned that employee inattention is partially responsible for the high defect rate. In an effort to decrease the percentage of defective chips, management decides to offer incentives to employees who have lower defect rates on their shifts. The incentive program is instituted for one month. If successful, the company will continue with the incentive program. -Management decided to extend the incentive program so that the decision can be made on three months of data instead. Will the power increase, decrease, or remain the same?

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Question 15

A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote. -What is the probability that the true proportion of the professor's students who were registered to vote is in your confidence interval?

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Question 16

Wildlife scientists studying a certain species of frogs know that past records indicate the adults should weigh an average of 118 grams with a standard deviation of 14 grams. The researchers collect a random sample of 50 adult frogs and weigh them. In their sample the mean weight was only 110 grams. One of the scientists is alarmed, fearing that environmental changes may be adversely affecting the frogs. Do you think this sample result is unusually low? Explain.

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Question 17

The International Olympic Committee states that the female participation in the 2004 Summer Olympic Games was 42%, even with new sports such as weight lifting, hammer throw, and modern pentathlon being added to the Games. Broadcasting and clothing companies want to change their advertising and marketing strategies if the female participation increases at the next games. An independent sports expert arranged for a random sample of pre-Olympic exhibitions. The sports expert reported that 202 of 454 athletes in the random sample were women. Is this strong evidence that the participation rate may increase? -Test an appropriate hypothesis and state your conclusion.

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Question 18

A pharmaceutical company has a drug they think is better at lowering blood pressure than the current medication, but it costs more money for consumers. They conduct a significance test on their clinical study (H₀: The new drug works the same as the old drug, Ha: The new drug works better than the old drug.) Which statement about errors is correct?

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Question 19

A company manufacturing computer chips finds that 8% of all chips manufactured are defective. Management is concerned that employee inattention is partially responsible for the high defect rate. In an effort to decrease the percentage of defective chips, management decides to offer incentives to employees who have lower defect rates on their shifts. The incentive program is instituted for one month. If successful, the company will continue with the incentive program. -In this context describe a Type I error and the impact such an error would have on the company.

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Question 20

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A pharmaceutical company investigating whether drug stores are less likely than food stores to remove over-the-counter drugs from the shelves when the drugs are past the expiration date found a P-value of 2.8%. This means that:

We test the hypothesis that p = 35% versus p < 35%. We don't know it but actually p = 26%. With which sample size and significance level will our test have the greatest power?

A researcher investigating whether joggers are less likely to get colds than people who do not jog found a P-value of 3%. This means that:

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