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Introduction to Statistics: Chapter 8 Homework (Hypothesis Testing for Population Proportions)
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Hypothesis Testing for Population Proportions
Key Concepts:
Terms in this set (43)
8.1
The null hypothesis is always a statement about what?
The null hypothesis is always a statement about a population parameter.
In statistical inference, measurements are made on a
______ and _______________ are made to a population.
In statistical inference, measurements are made on a
sample and generalizations are made to a population.
A recent poll estimated that 2.4% of American adults are vegetarian. A nutritionist thinks this rate has increased and will take a random sample of American adults and record whether or not they are vegetarian. State the null and alternative hypotheses in words and in symbols.
State the null hypothesis in words.
The proportion of American adults who are vegetarian is equal to 2.4%.
State the null hypothesis in symbols.
H0: p=.0024
State the alternative hypothesis in words.
The proportion of American adults who are vegetarian is greater than 2.4%.
State the alternative hypothesis in symbols.
Ha: p > 0.024
A researcher is testing someone who claims to have ESP by having that person predict whether a coin will come up heads or tails. The null hypothesis is that the person is guessing and does not have ESP, and the population proportion of success is 0.50. The researcher tests the claim with a hypothesis test, using a significance level of 0.05. Fill in the blanks below with an accurate statement about the potential conclusion of this test.
The probability of concluding that the person has ESP when in fact she or he does not have ESP is 0.05.
According to a reputable magazine, the dropout rate for all college students with loans is 30%. Suppose that 132 out of 400 random college students with loans drop out. Complete parts (a) and (b) below.
a. Give the null and alternative hypotheses to test that the dropout rate is not 30%.
b. Report the test statistic (z) from the output given.
a. H0: p = .30; Ha: p ≠ .30
b. z = 1.3093
According to a reputable magazine, 33% of all cars sold in a certain country are SUVs. Suppose a random sample of
500 recently sold cars shows that 146 are SUVs. Complete parts (a) and (b) below.
a. Write the null and alternative hypotheses to test that fewer than 33% of cars sold are SUVs.
b. Report the value of the test statistic (z) from the figure.
a. H0: p = .33; Ha: p < .33
b. z = -1.8071
Suppose a researcher is testing someone to see whether she or he can tell Soda X from Soda Y, and the researcher is using 30 trials, half with Soda X and half with Soda Y. The null hypothesis is that the person is guessing.
a. About how many should the researcher expect the person to get right under the null hypothesis that the person is guessing?
b. Suppose person A gets 17 right out of 30, and person B gets 24 right out of 30. Which will have a smaller p-value, and why?
a. The person should get 15 right.
b. Person B will have a smaller p-value because that person's number of successes is further from the hypothesized number of successes.
A certain county is 25% African American. Suppose a researcher is looking at jury pools, each with 200 members, in this county. The null hypothesis is that the probability of an African American being selected into the jury pool is 25%.
a. How many African Americans would the researcher expect on a jury pool of 200 people if the null hypothesis is true?
b. Suppose pool A contains 37 African American people out of 200, and pool B contains 27 African American people out of 200. Which will have a smaller p-value and why?
a. 50 African Americans
b. Pool B will have a smaller p-value because that pool's number of AA people is further from the hypothesized number of AA people.
8.2
A 2003 study of dreaming found that out of a random sample of
123 people, 80 reported dreaming in color. However, the rate of reported dreaming in color that was established in the 1940s was 0.27. Check to see whether the conditions for using a one-proportion z-test are met assuming the researcher wanted to test to see if the proportion dreaming in color had changed since the 1940s.
Are the conditions met?
Ans: Yes, all the conditions are met.
About 31% of the population in a large region are between the ages of 40 and 65, according to the country's census. However, only 4% of the 2500 employees at a company in the region are between the ages of 40 and 65. Lawyers might argue that if the company hired people regardless of their age, the distribution of ages would be the same as though they had hired people at random from the surrounding population. Check whether the conditions for using the one-proportion z-test are met.
Are all the conditions satisfied? Select all that apply.
Ans: Yes, all conditions are satisfied or can be reasonably assumed.
When asked whether marriage is becoming obsolete, 763
out of 2008 randomly selected adults answering a particular survey said yes. We are testing the hypothesis that the population proportion that believes marriage is becoming obsolete is more than 36% using a significance level of 0.05. One of the following figures is correct. Indicate which graph matches the alternative hypothesis, p>0.36. Report and interpret the correct p-value.
The graph right-shaded. (z =1.8653
p = 0.0311); Because the p-value is less than the significance level, reject the null hypothesis. There is sufficient evidence to conclude that the population proportion is greater than 0.36.
According to a recent survey, 17% of an adult population suffer from an anxiety disorder. Suppose a test of 300 random college students showed that 47 suffered from an anxiety disorder. Complete parts (a) and (b) below.
Graph: Shades left and right, not full.
Interpret the p-value. Select the correct choice below and fill in the answer box to complete your choice.
If 17% of the population suffer from an anxiety disorder, then there is a 53.87% chance we will observe a z-score less than −0.6148 or greater than 0.6148.
According to a reputable magazine, 32% of people in a certain large country have sleepwalked at least once in their lives. Suppose a random sample of 200 people showed that 48
reported sleepwalking. Carry out the first two steps of a hypothesis test that will test whether the proportion of people who have sleepwalked is 0.32. Use a significance level of 0.05. Explain how someone would fill in the required entries for p0, x, n, and prop shown in the accompanying figure, where p0 is the hypothesized proportion, x is the number of successes, n is the number of observations, and prop is the alternative hypothesis.
P0=.32; x=48; n=200; prop:≠p0
(NOTE: Use the technology input figure.)
According to a reputable magazine, 33% of Americans planned to take a family vacation (a vacation more than 50 miles from home involving two or more immediate family members.) Suppose a recent survey of 250 Americans found that 60 planned on taking a family vacation. Carry out the first two steps of a hypothesis test to determine if the proportion of Americans planning a family vacation has changed. Explain how you would fill in the required entries in the figure for # of successes, # of observations, and the value in H0.
H0: p = .33, Ha: p ≠ .33; All conditions are satisfied or can be reasonably assumed.
# of successes: 60; # of observations: 250; Perform:
Hypothesis test for p: H0: p = .33, Ha: p ≠ .33
(NOTE: Use the technology input figure.)
According to a reputable magazine, 35% of people in a certain large country have sleepwalked at least once in their lives. Suppose a random sample of 200 people showed that 42
reported sleepwalking. The null and alternative hypotheses for a test that would test whether the proportion of people who have sleepwalked is 0.35 are H0: p=0.35 and Ha: p≠0.35, respectively. The conditions for such a test are satisfied. Use the technology output provided to determine the test statistic and p-value for the test, and provide a conclusion. Use a significance level of 0.05.
z=-4.15; p-value=0.00; Reject H0.
There is sufficient evidence to conclude that the proportion of people who have sleepwalked is not 0.35.
(NOTE: Use the technology input figure.)
According to a reputable magazine, the percentage of female CEOs in a certain group of high-revenue companies was 5%. Suppose a student did a survey of 300 randomly selected large companies (not the group of high-revenue companies), and 24
of them had female CEOs. The null and alternative hypotheses for a test that will test whether the population proportion of female CEOs is not 0.05 are H0: p=0.05 and Ha: p≠0.05, respectively. The conditions for such a test are satisfied. Determine the test statistic and p-value for the test. Use the technology output provided and a significance level of 0.05, and provide a conclusion.
z=2.38; p-value=.017; Reject H0. There is sufficient evidence to conclude that the proportion of female CEOs in large companies is not 0.05.
(NOTE: Use the technology input figure.)
A 2018 poll of 3627 randomly selected users of a social media site found that 1577 get most of their news about world events on the site. Research done in 2013 found that only 44% of all the site users reported getting their news about world events on this site.
a. Does this sample give evidence that the proportion of site users who get their world news on this site has changed since 2013? Carry out a hypothesis test and use a 0.05 significance level.
b. After conducting the hypothesis test, a further question one might ask is what proportion of all of the site users get most of their news about world events on the site in 2018. Use the sample data to construct a 95% confidence interval for the population proportion. How does your confidence interval support your hypothesis test conclusion?
H0: p=.44, Ha: p≠.44; z=-0.63; p-value=.528; Do not reject H0. The percentage is not significantly different from 44%.; The 95% confidence interval for the population proportion is (.4187,.4509).; The result of 95% confidence interval agrees with the result of the hypothesis test performed in part (a) because the value of p0 is inside the 95% confidence interval.
A 2018 poll of 2220 randomly selected U.S. adults found that 41.58% planned to watch at least a "fair amount" of a particular sporting event in 2018. In 2014, 49% of U.S. adults reported planning to watch at least a "fair amount."
a. Does this sample give evidence that the proportion of U.S. adults who planned to watch the 2018 sporting event was less than the proportion who planned to do so in 2014? Use a 0.05 significance level.
b. After conducting the hypothesis test, a further question one might ask is what proportion of all U.S. adults planned to watch at least a "fair amount" of the sporting event in 2018. Use the sample data to construct a 90% confidence interval for the population proportion. How does your confidence interval support your hypothesis test conclusion?
a. State the null and alternative hypotheses. Let p be the proportion of U.S. adults that planned to watch at least a "fair amount" of the sporting event.
H0: p=.49 and Ha: p<.49; z=-6.99; p-value=0.00; Reject H0. There is sufficient evidence to suggest that the percentage in 2018 is significantly less than 49%.
b. The 90% confidence interval for the population proportion is (0.3986,0.4330).; The result of the 90% confidence interval agrees with the result of the hypothesis test performed in part (a) because the 90% confidence interval only includes p0.
8.3
A researcher carried out a hypothesis test using a two-tailed alternative hypothesis. Which of the following z-scores is associated with the smallest p-value? Explain.
i. z=−0.68 ii. z=1.18 iii. z=2.49 iv. z=3.35
z=3.35 because The z-score farthest from 0 has the smallest tail area and thus has the smallest p-value.
A test is conducted in which a coin is flipped 30 times to test whether the coin is unbiased. The null hypothesis is that the coin is fair. The alternative is that the coin is not fair. One of the accompanying figures represents the p-value after getting 16 heads out of 30 flips, and the other represents the p-value after getting 18 heads out of 30 flips. Which is which, and how do you know?
Figure (A; z=.3651 and p=.715) represents getting 16 heads out of 30, and Figure (B; z=1.0954 and p=.2733) represents getting 18 heads out of 30. Since 18 heads out of 30 is further from half (50% heads), it results in a z-value further from 0 and a smaller p-value.
Suppose you are spinning pennies to test whether you get biased results. When you reject the null hypothesis when it is actually true, that is often called the first kind of error. The second kind of error is when the null is false and you fail to reject. Report the first kind of error and the second kind of error.
The null hypothesis is that the penny is not biased. The first kind of error is saying the penny is biased when in fact the penny is not biased. The second kind of error is saying the penny is not biased when in fact the penny is biased.
Suppose you are testing someone to see whether he or she can tell butter from margarine when it is spread on toast. You use many bite-sized pieces selected randomly, half from buttered toast and half from toast with margarine. The taster is blindfolded. The null hypothesis is that the taster is just guessing and should get about half right. When you reject the null hypothesis when it is actually true, that is often called the first kind of error. The second kind of error is when the null is false and you fail to reject. Report the first kind of error and the second kind of error.
The first kind of error is saying the person can tell butter from margarine when in fact he or she
cannot. The second kind of error is saying the person
cannot tell butter from margarine when in fact he or she can.
What superpower do people from a certain region want most? In past years, 10% of people from this region chose invisibility as the most desired superpower. Assume that this is an accurate representation of the people from this region. A group of futurists examines a more recent 2018 poll that found that 12% of those sampled picked invisibility as their desired superpower. The futurists carry out a hypothesis test using a significance level of 0.10, and the result is shown in the StatCrunch output. Based on this, can they conclude that the percentage of all people from this region who would pick invisibility as their superpower is still 10%? If not, what conclusion would be appropriate based on these sample data?
No, they cannot conclude that the percentage of all people from this region who would pick invisibility as their superpower is still 10%. Given thep-value and significance level, we would reject H0. This means that there is enough evidence to conclude the proportion of people from the region who would pick invisibility as their superpower has increased. This does not prove the null hypothesis to be true.
(NOTE: view the StatCrunch output.)
The null hypothesis on true/false tests is that the student is guessing, and the proportion of right answers is 0.50. A student taking a five-question true/false quiz gets 4 right out of 5. She says that this shows that she knows the material, because the one-tailed p-value from the one-proportion z-test is 0.090, and she is using a significance level of 0.10. What is wrong with her approach?
The sample size is not large enough to use the one-proportion z-test. To use that test, the tester must expect at least 10 successes and at least 10 failures.
A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. The proposition will pass if it gets more than 50% of the votes. Suppose a poll is taken, and 557 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. Use a 2.5% significance level. If a confidence interval is appropriate, find the approximate 95% confidence interval. In both cases, assume that the necessary conditions have been met.
The proponent should use a hypothesis test because the proponent wants to know whether or not the proposition will pass. However, the proponent could also use a confidence interval.; H0: p=.50 and Ha: p > .50; z=3.60; p-value=0.00; Reject H0. There is enough evidence to conclude that the proposition will pass.; A confidence interval is not the most appropriate approach. The proponent should use a hypothesis test.
A proponent of a new proposition on a ballot wants to know the population percentage of people who support the bill. Suppose a poll is taken, and 566 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. Use a 55% significance level. If a confidence interval is appropriate, find the approximate 90% confidence interval. In both cases, assume that the necessary conditions have been met.
The proponent should use a confidence interval because the proponent wants to know the proportion of the population who will vote for the proposition. A hypothesis test would be impossible.; Hypotheses are not appropriate. The proponent should use a confidence interval.; A test statistic is not appropriate. The proponent should use a confidence interval.; A p-value is not appropriate. The proponent should use a confidence interval.; A test conclusion is not appropriate. The proponent should use a confidence interval.; (0.540, 0.592)
Is it acceptable practice to look at your research results, note the direction of the difference, and then make the alternative hypothesis one-sided in order to achieve a significant difference? Explain.
No. Changing the alternative hypothesis is considered cheating.
A magazine advertisement claims that wearing a magnetized bracelet will reduce arthritis pain in those who suffer from arthritis. A medical researcher tests this claim with 233 arthritis sufferers randomly assigned to wear either a magnetized bracelet or a placebo bracelet. The researcher records the proportion of each group who report relief from arthritis pain after 6 weeks. After analyzing the data, he fails to reject the null hypothesis. What are valid interpretations of his findings?
There's insufficient evidence that the magnetized bracelets are effective at reducing arthritis pain.; There were no statistically significant differences between the magnetized bracelets and the placebos in reducing arthritis pain.
8.4
When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.
The p-value will be smaller if the sample proportions are far apart because a larger difference results in a larger absolute value of the numerator of the test statistic.
When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.
The p-value will be smaller with a larger sample size because a larger sample results in a smaller standard error, which is the denominator of the test statistic.
A study for the treatment of patients with HIV-1 was a randomized, controlled, double-blind study that compared the effectiveness of ritonavir-boosted darunavir (rbd), the drug currently used to treat HIV-1, with dorovirine, a newly developed drug. Of the 376 subjects taking ritonavir-boosted darunavir, 307 achieved a positive result. Of the 376 subjects taking dorovirine, 328
achieved a positive result. Complete parts (a) and (b).
a. The sample percentage of subjects who achieve a positive outcome with ritonavir-boosted darunavir is 81.65%. The sample percentage of subjects who achieve a positive outcome with dorovirine is 87.23%.
b. H0: p1=p2 and Ha: p1≠p2; z=-2.11; p-value=0.035; Since the p-value is greater than the significance level of α=0.01, fail to reject the null hypothesis. There is insufficient evidence to support the claim that the percentage from the ritonavir-boosted darunavir group is different from the percentage from the dorovirine group.; No, since the decision of the test was to fail to reject the null hypothesis of no difference.
A study was conducted on the efficacy and safety of varenicline for smoking cessation in people living with HIV. The study was a randomized, double-blind, placebo-controlled trial. Of the 116 subjects treated with varenicline, 19 abstained from smoking for the entire 48-week study period. Of the 115 subjects assigned to the placebo group, 99 abstained from smoking for the entire study period. Complete parts (a) and (b).
a. The sample percentage of subjects treated with varenicline who abstained from smoking for the entire 48-week study period is 16.38%. The sample percentage of subjects in the placebo group who abstained from smoking for the entire 48-week study period is 7.83%.
b. H0:p1=p2 and Ha: p1>p2; z=1.99; p-value=0.023; Since the p-value is greater than the significance level of α=0.01, fail to reject the null hypothesis. There is insufficient evidence to support the claim that the percentage from the varenicline group is greater than the percentage from the placebo group.
A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of the press was secure or threatened in the country today. In 2016, 2484 of 3091 students surveyed said that freedom of the press was secure or very secure. In 2017, 1797 of 2037 students surveyed felt this way. Complete parts (a) and (b).
a. H0:p1=p2 and Ha: p1≠p2; z=-7.41; p-value=.00; Since the p-value is less than the significance level of α=0.05, reject the null hypothesis. There is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
b. The 90% confidence interval is (−0.095,−0.062).; Because the confidence interval does not contain 0, it appears that the two proportions are not equal. This conclusion supports the hypothesis test conclusion.
A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of religion was secure of threatened in the country today. In 2016, 2077 of 3105 students surveyed said that freedom of religion was secure or very secure. In 2017, 1930 of 2951 students surveyed felt this way. Complete parts (a) and (b).
a. H0:p1=p2 and Ha: p1≠p2; z=1.23; p-value=0.220; Since the p-value is greater than the significance level of α=0.05, fail to reject the null hypothesis. There is insufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
b. The 90% confidence interval is (−0.005,0.035). Because the confidence interval contains 0, it appears that the two proportions are equal. This conclusion supports the hypothesis test conclusion.
Chapter 8 CR
For each of the following, state whether a one-proportion z-test or a two-proportion z-test would be appropriate, and name the populations.
a. Upper A student watches a random sample of men and women leaving a supermarket with carts to see whether the proportion of men who put the carts back in the designated area is greater than the proportion of women who do so.
b. Upper A polling agency takes a random sample to determine the proportion of people in a state who support a certain proposition to determine if it will pass.
a. A two-proportion z-test would be appropriate.; Both men and women who shop at the market
b. A one-proportion z-test would be appropriate.; people from the state
For each test in parts a and b below, state whether a one-proportion z-test or a two-proportion z-test would be appropriate and give the null and alternative hypotheses for the appropriate test.
a. One-proportion z-test; H0:p=0.5 and Ha: p>0.5
b. two-proportion z-test; H0: pa=pn and Ha: pa≠pn
A community college used enrollment records of all students and reported that that the percentage of the student population identifying as female in 2011 was 56% whereas the proportion identifying as female in 2017 was 54%. Would it be appropriate to use this information for a hypothesis test to determine if the proportion of students identifying as female at this college had declined? Explain.
No, since these are measures of all students at the college, inference is not needed or appropriate.
A friend claims he can predict the suit of a card drawn from a standard deck of 52 cards. There are four suits and equal numbers of cards in each suit. The parameter, p, is the probability of success, and the null hypothesis is that the friend is just guessing.
a. Identify the null hypothesis.
b. What hypothesis best fits the friend's claim? (This is the alternative hypothesis.)
a. p=1/4
b. p>1/4
Judging on the basis of experience, a politician claims that 53% of voters in a certain area have voted for an independent candidate in past elections. Suppose you surveyed 2020 randomly selected people in that area, and 13 of them reported having voted for an independent candidate. The null hypothesis is that the overall proportion of voters in the area that have voted for an independent candidate is 53%. What value of the test statistic should you report?
The test statistic is z=1.08.
The mother of a teenager has heard a claim that 28% of teenagers who drive and use a cell phone reported texting while driving. She thinks that this rate is too high and wants to test the hypothesis that fewer than 28% of these drivers have texted while driving. Her alternative hypothesis is that the percentage of teenagers who have texted when driving is less than 28%.
H0: p=0.28
Ha: p<0.28
She polls 40 randomly selected teenagers, and 44 of them report having texted while driving, a proportion of 0.1. The p-value is 0.006. Explain the meaning of the p-value in the context of this question.
The p-value says that if the true proportion of teenagers who text while driving is 0.28, then there is only a 0.006 probability that one would get a sample proportion of 0.1 or smaller with a sample size of 40.
Suppose a friend says he can predict whether a coin flip will result in heads or tails. You test him, and he gets 10 right out of 20. Do you think he can predict the coin flip (or has a way of cheating)? Or could this just be something that occurs by chance? Explain without doing any calculations.
No, the friend mostly likely cannot predict the coin flips and the result could have occurred by chance. The friend only got half of the predictions correct, which is the expected number of correct predictions from randomly guessing the results of the coin flips.
In the mid-1800s, a doctor decided to make the doctors wash their hands with a strong disinfectant between patients at a clinic with a death rate of 10.3%. The doctor wanted to test the hypothesis that the death rate would go down after the new hand-washing procedure was used. What null and alternative hypotheses should he have used? Explain, using both words and symbols. Explain the meaning of any symbols you use.
H0: The death rate has remained the same at 10.3% after starting hand-washing. Ha: The death rate has decreased to a value less than 10.3%.; H0: p=0.103, Ha:
p<0.103 (p is the proportion of deaths at theclinic)
A true/false test has 80
questions. Suppose a passing grade is 48 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 48 answers correct out of 80. Use a significance level of 0.05. Steps 1 and 2 of a hypothesis test procedure are given below. Show step 3, finding the test statistic and the p-value and step 4, interpreting the results.
z=1.79
p-value=0.037; Reject H0.
The probability of doing this well by chance alone is so small that it can be concluded that the student is not guessing.
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