Deck 8: Hypothesis Testing for Population Proportions

Question 1

From the TI- 84 graphing calculator screenshots below, choose the screenshot whose shaded area correctly depicts the following hypothesis test results: H₀: p = 0.15, H_a: p × 0.15, a = 0.05, z = - 1.82, p- value = 0.0688

A)
B)

C)

Answer 1

Question 2

Which of the following is not a condition that must be checked before proceeding with a two- sample test?

A)The samples must be independent of each other.
B)The observations within each sample must be independent of one another.
C)Each sample must be a random sample.
D)All of these are conditions that must be checked to proceed with a two- sample test.

Answer 2

All of these are conditions that must be checked to proceed with a two- sample test.

Question 3

From the TI- 84 graphing calculator screenshots below, choose the screenshot whose shaded area depicts a p- value for a left- tailed test.

A)
B)

C)

Answer 3

Question 4

A polling agency is interested in testing whether the proportion of women who support a female candidate for office is less than the proportion of men. The null hypothesis is that there is no difference in the proportions of men and women who support the female candidate. The alternative hypothesis is that the proportion of women who support the female candidate is less than the proportion of men. The test results in a p- value of 0.041. Which of the following is the best interpretation of the p- value?

A)The p- value is the probability that the majority of women will support the female candidate.
B)The p- value is the probability of getting a result that is as extreme as or more extreme than the one obtained, assuming that there is no difference in the proportions.
C)The p- value is the probability that the majority of men will support the female candidate.
D)The p- value is the probability of getting a result that is as extreme as or more extreme than the one obtained, assuming that the proportion of women who support the female candidate is less than the proportion of men.

Question 5

Which statement best describes the power of a hypothesis test?

A)The probability of rejecting the null hypothesis when the null hypothesis is true.
B)The probability of failing to reject the null hypothesis when the null hypothesis is not true.
C)The probability of rejecting the null hypothesis when the null hypothesis is not true.
D)None of these

Question 6

Use the following information to answer the question. A janitor at a large office building believes that his supply of light bulbs has too many defective bulbs. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.07 (the light bulb manufacturer's stated defect rate). Suppose he does a hypothesis test with a significance level of 0.05.
Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.07 and H_a: p > 0.07.
Suppose that the janitor tests 300 randomly selected light bulbs and finds that 27 bulbs are defective. What value of the test statistic should he report? Round to the nearest hundredth.

A)z = - 1.96
B)z = 1.96
C)z = 1.36
D)z = - 1.36

Question 7

From the TI- 84 graphing calculator screenshots below, choose the screenshot whose shaded area depicts a p- value for a two- tailed test.

A)
B)

C)

Question 8

Which of the following is not one of the four steps of the hypothesis test?

A)State the level of significance.
B)Make a decision to reject or not reject the null hypothesis.
C)State the null and alternative hypothesis about the population parameter.
D)All of these are steps of the hypothesis test.

Question 9

Which statement best describes the significance level of a hypothesis test?

A)The probability of failing to reject the null hypothesis when the null hypothesis is not true.
B)The probability of rejecting the null hypothesis when the null hypothesis is true.
C)The probability of rejecting the null hypothesis when the null hypothesis is not true.
D)None of these

Question 10

Use the following information to answer the question. A researcher is wondering whether the drinking habits of adults in a certain region of the country are in the same proportion as the general population of adults. Suppose a recent study stated that the proportion of adults who reported drinking once a week or less in the last month was 0.26. The null hypothesis for this test is H₀: p = 0.26 and the alternative hypothesis is H_a: p < 0.26. The researcher collected data from 150 surveys he handed out at a busy park located in the region.
Check that the conditions hold so that the sampling distribution of the z- statistic will approximately follow the standard normal distribution. Are the conditions satisfied? If not, choose the condition that is not satisfied.

A)No, the researcher did not collect a large enough sample.
B)No, the population of interest is not large enough to assume independence.
C)Yes, all the conditions are satisfied.
D)No, the researcher did not collect a random sample.

Question 11

Read the following then choose the appropriate test and name the population(s). A researcher asks random samples of men and women whether they had purchased organically grown food in the last three months. He wants to determine whether the proportion of women who purchase organically grown food is greater than the proportion of men who purchase organically grown food.

A)Two- proportion z- test; one population is all adults who buy organically grown food and the other population is all adults who do not buy organically grown food.
B)Two- proportion z- test; one population is all men and the other population is all women.
C)One- proportion z- test; the population is all men.
D)One- proportion z- test; the population is all women.

Question 12

Use the following information to answer the question. A janitor at a large office building believes that his supply of light bulbs has too many defective bulbs. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.07 (the light bulb manufacturer's stated defect rate). Suppose he does a hypothesis test with a significance level of 0.05.
Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.07 and H_a: p > 0.07.
Choose the statement that best describes the significance level in the context of the hypothesis test.

A)The significance level of 0.05 is the defect rate we believe is the true defect rate.
B)The significance level of 0.05 is the probability of concluding that the defect rate is equal to 0.07 when in fact it is greater than 0.07.
C)The significance level of 0.05 is the test statistic that we will use to compare the observed outcome to the null hypothesis.
D)The significance level of 0.05 is the probability of concluding that the defect rate is higher than 0.07 when in fact the defect rate is equal to 0.07.

Question 13

A researcher conducts a hypothesis test on a population proportion. Her null and alternative hypothesis are H0: p = 0.6 and Ha: p < 0.6. The test statistic and p- value for the test are z = - 1.51 and p- value = 0.0655. For a significance level of a = 0.05, choose the correct conclusion regarding the null hypothesis.

A)There is sufficient evidence to conclude that the population proportion is significantly different from 0.6.
B)There is insufficient evidence to reject the null hypothesis that the population proportion is equal to 0.6.
C)None of these.
D)There is sufficient evidence to accept the null hypothesis that the population proportion is equal to 0.6.

Question 14

Use the following information to answer the question. A researcher is wondering whether the drinking habits of adults in a certain region of the country are in the same proportion as the general population of adults. Suppose a recent study stated that the proportion of adults who reported drinking once a week or less in the last month was 0.26. The null hypothesis for this test is H₀: p = 0.26 and the alternative hypothesis is H_a: p < 0.26. The researcher collected data from 150 surveys he handed out at a busy park located in the region.
To continue the study into the drinking habits of adults, the researcher decides to collect data from adults working in "white collar" jobs to see whether their drinking habits are in the same proportion as the general public. The null hypothesis for this test is H0: p = 0.26 and the alternative hypothesis is Ha: p < 0.26. The researcher collected data from a random sample of 120 adults with "white collar" jobs of which 25 stated that they drank once a week or less in the last month. Assume that the conditions that must be met in order for us to use the N(0, 1)distribution as the sampling distribution are satisfied. Find the values of the sample proportion, p^, the test statistic, and the p- value associated with the test statistic. Round all values to the nearest thousandth.

A)p^{^} = 0.30, z = 0.803, p- value = 0.041
B)p^{^} = 0.208, z = - 1.290, p- value = 0.098
C)p^{^} = 0.75, z = - 1.32, p- value = 0.599
D)p^{^} = 0.208, z = - 0.250, p- value = 0.401

Question 15

Suppose a city official conducts a hypothesis test to test the claim that the majority of voters support a proposed tax to build sidewalks. Assume that all the conditions for proceeding with a one- sample test on proportions have been met. The calculated test statistic is approximately 1.40 with an associated p- value of approximately 0.081. Choose the conclusion that provides the best interpretation for the p- value at a significance level of a = 0.05.

A)The p- value should be considered extreme; therefore the hypothesis test proves that the null hypothesis is true.
B)If the null hypothesis is true, then the probability of getting a test statistic that is as extreme or more extreme than the calculated test statistic of 1.40 is 0.081. This result is surprising and could not easily happen by chance.
C)If the null hypothesis is true, then the probability of getting a test statistic that is as extreme or more extreme than the calculated test statistic of 1.40 is 0.081. This result is not surprising and could easily happen by chance.
D)None of these.

Question 16

Use the following information to answer the question. A researcher is wondering whether the drinking habits of adults in a certain region of the country are in the same proportion as the general population of adults. Suppose a recent study stated that the proportion of adults who reported drinking once a week or less in the last month was 0.26. The researcher's null hypothesis for this test is H₀: p = 0.26 and the alternative hypothesis is H_a: p > 0.26. The researcher collected data from a random sample of 75 adults in the region of interest.
To continue the study into the drinking habits of adults, the researcher decides to collect data from adults working in "blue collar" jobs to see whether their drinking habits are in the same proportion as the general public. The null hypothesis for this test is H0: p = 0.26 and the alternative hypothesis is Ha: p > 0.26. The researcher collected data from a random sample of 90 adults with "blue collar" jobs of which 30 stated that they drank once a week or less in the last month. Assume that the conditions that must be met in order for us to use the N(0, 1)distribution as the sampling distribution are satisfied. Find the values of the sample proportion, p^, the test statistic, and the p- value associated with the test statistic. Round all values to the nearest thousandth.

A)p^{^} = 0.333, z = 1.586, p- value = 0.056
B)p^{^} = 0.289, z = - 0.829, p- value = 0.407
C)p^{^} = 0.667, z = 8.795, p- value = 0.000⁺ D)p^{^} = 0.333, z = 0.067, p- value = 0.946

Question 17

A research firm carried out a hypothesis test on a population proportion using a right- tailed alternative hypothesis. Which of the following z- scores would be associated with a p- value of 0.04? Round to the nearest hundredth.

A)z = - 1.75
B)z = 1.75
C)z = - 2.50
D)z = 2.50

Question 18

Suppose that the following is to be tested: H0: p = 0.35 and Ha: p > 0.35. Calculate the observed z- statistic for the following sample data: Forty out of eighty test subjects have the characteristic of interest. Round to the nearest hundredth.

A)z = - 1.87
B)z = 1.88
C)z = - 0.94
D)z = 2.81

Question 19

Use the following information to answer the question. A janitor at a large office building believes that his supply of light bulbs has too many defective bulbs. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.07 (the light bulb manufacturer's stated defect rate). Suppose he does a hypothesis test with a significance level of 0.05.
Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.07 and H_a: p > 0.07.
The janitor calculates a p- value for the hypothesis test of approximately 0.087. Choose the correct interpretation for the p- value.

A)The p- value tells us that the probability of concluding that the defect rate is equal to 0.07, when in fact it is greater than 0.07, is approximately 0.087.
B)The p- value tells us that the true population rate of defective light bulbs is approximately 0.087.
C)The p- value tells us that if the defect rate is 0.07, then the probability that the janitor will have 27 defective light bulbs out of 300 is approximately 0.087. At a significance level of 0.05, this would not be an unusual outcome.
D)None of these

Question 20

A researcher believes that the proportion of women that exercise with a friend is greater than the proportion men. He takes a random sample from each population and records the response to the question, "Have you exercised with a friend at least once in the last seven days?" The null hypothesis is H0: pwomen = pmen. Choose the correct alternative hypothesis.

A)Ha: p_women < p_men
B)Ha: p = 0
C)H_a: p_women > p_men
D)Ha: p_women × p_men

Question 21

Which of the following is not a condition that must be checked before proceeding with a two- sample test?

A)The samples must be independent of each other.
B)Both samples must be large enough so that the product of each sample size (n1 and n2)and the pooled estimate, p^, is greater than or equal to 10.
C)Each sample must be a random sample.
D)All of these are conditions that must be checked to proceed with a two- sample test.

Question 22

A researcher conducts a hypothesis test on a population proportion. Her null and alternative hypothesis are H0: p = 0.4 and Ha: p < 0.4. The test statistic and p- value for the test are z = - 3.01 and p- value = 0.0013. For a significance level of a = 0.05, choose the correct conclusion regarding the null hypothesis.

A)There is not sufficient evidence to conclude that the population proportion is significantly different from 0.04.
B)There is sufficient evidence to conclude that the population proportion is significantly different from 0.04.
C)There is not sufficient evidence to reject the null hypothesis that the population proportion is equal to 0.04.
D)There is sufficient evidence to accept the null hypothesis that the population proportion is equal to 0.04.

Question 23

Use the following information to answer the question. A janitor at a large office building believes that his supply of light bulbs has a defect rate that is different than the defect rate stated by the manufacturer. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.09 (the light bulb manufacturer's stated defect rate). Suppose we do a hypothesis test with a significance level of 0.01. Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.09 and H_a: p > 0.09.
The janitor calculates a p- value for the hypothesis test of approximately 0.113. Choose the correct interpretation for the p- value.

A)The p- value tells us that the true population rate of defective light bulbs is approximately 0.113.
B)The p- value tells us that the probability of concluding that the defect rate is equal to 0.09, when in fact it is greater than 0.09, is approximately 0.113.
C)The p- value tells us that if the defect rate is 0.113, then the probability that the janitor will have 33 defective light bulbs out of 300 is approximately 0.113. At a significance level of 0.01, this would not be an unusual outcome.
D)None of these

Question 24

A researcher believes that children who attend elementary school in a rural setting are more physically active then children who attend elementary school in an urban setting. The researcher collects a random sample from each population and records the proportion of children in each sample who reported participating in at least one hour of rigorous activity a day. The data is summarized in the table below. Assume the all conditions for proceeding with a two- sample test have been met. Find the z- statistic (rounded to the nearest hundredth)and p- value (rounded to the nearest thousandth)for this hypothesis test. Using a 5% significance level, state the correct conclusion regarding the null hypothesis, H₀: p_rural = p_urban.

A)z = 0.82, p = 0.073. There is sufficient evidence to accept the null hypothesis.
B)z = - 1.79, p = 0.037. There is insufficient evidence to reject the null hypothesis.
C)z = 1.79, p = 0.037. There is sufficient evidence to reject the null hypothesis.
D)z = 0.71, p = 0.073. There is sufficient evidence to reject the null hypothesis.

Question 25

Choose the statement that best describes the significance level in the context of the hypothesis test.

A)The significance level of 0.01 is the defect rate we believe is the true defect rate.
B)The significance level of 0.01 is the probability of concluding that the defect rate is different than 0.09 when in fact the defect rate is equal to 0.09.
C)The significance level of 0.01 is the probability of concluding that the defect rate is equal to 0.09 when in fact it is greater than 0.09.
D)The significance level of 0.01 is the z- statistic that we will use to compare the observed outcome to the null hypothesis.

Question 26

Read the following then choose the appropriate test and name the population(s). A researcher asks a random sample of 200 men whether they had made an online purchase in the last three months. He wants to determine whether the proportion of men who make online purchases is less than 0.18.

A)One- proportion z- test; the population is all adults who make online purchases.
B)Two- proportion z- test; the population is the 200 men surveyed.
C)Two- proportion z- test; one population is all men who make online purchases and the other population is all men who do not make online purchases.
D)One- proportion z- test; the population is all men.

Question 27

Complete the statement by filling in the blanks. The null hypothesis is to be and is only rejected when the observed outcome is shown to be .

A)Likely; false; extremely likely
B)Known; true; the population parameter
C)Proven; true; impossible
D)Assumed; true; extremely unlikely

Question 28

Use the following information to answer the question. A researcher is wondering whether the drinking habits of adults in a certain region of the country are in the same proportion as the general population of adults. Suppose a recent study stated that the proportion of adults who reported drinking once a week or less in the last month was 0.26. The researcher's null hypothesis for this test is H₀: p = 0.26 and the alternative hypothesis is H_a: p > 0.26. The researcher collected data from a random sample of 75 adults in the region of interest.
Check that the conditions hold so that the sampling distribution of the z- statistic will approximately follow the standard normal distribution. Are the conditions satisfied? If not, choose the condition that is not satisfied.

A)No, the researcher did not collect a large enough sample.
B)No, the researcher did not collect a random sample.
C)Yes, all the conditions are satisfied.
D)No, the population of interest is not large enough to assume independence.

Question 29

Suppose that the following is to be tested: H0: p = 0.72 and Ha: p × 0.72. Calculate the observed z- statistic for the following sample data: Sixty- eight out of ninety test subjects have the characteristic of interest. Round to the nearest thousandth.

A)z = 0.751
B)z = 0.756
C)z = - 0.751
D)z = 0.453

Question 30

Use the following information to answer the question. A janitor at a large office building believes that his supply of light bulbs has a defect rate that is different than the defect rate stated by the manufacturer. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.09 (the light bulb manufacturer's stated defect rate). Suppose we do a hypothesis test with a significance level of 0.01. Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.09 and H_a: p > 0.09.
Suppose the janitor tests 300 light bulbs and finds that 33 bulbs are defective. What value of the test statistic should he report? Round to the nearest hundredth.

A)z = 1.21
B)z = - 2.17
C)z = 2.17
D)z = - 1.21

Question 31

A polling agency is interested in testing whether the proportion of women who support a female candidate for office is greater than the proportion of men. The null hypothesis is that there is no difference in the proportion of men and women who support the female candidate. The alternative hypothesis is that the proportion of women who support the female candidate is greater than the proportion of men. The test results in a p- value of 0.112. Which of the following is the best interpretation of the p- value?

A)The p- value is the probability of getting a result that is as extreme as or more extreme than the one obtained, assuming that the proportion of women who support the female candidate is greater than the proportion of men.
B)The p- value is the probability that men will support the female candidate.
C)The p- value is the probability that women will support the female candidate.
D)The p- value is the probability of getting a result that is as extreme as or more extreme than the one obtained, assuming that there is no difference in the proportions.

Question 32

Suppose a city official conducts a hypothesis test to test the claim that the majority of voters oppose a proposed school tax. Assume that all the conditions for proceeding with a one- sample test on proportions have been met. The calculated test statistic is approximately 1.46 with an associated a- value of approximately 0.072. Choose the conclusion that provides the best interpretation for the p- value at a significance level of a = 0.05.

A)If the null hypothesis is true, then the probability of getting a test statistic that is as extreme or more extreme than the calculated test statistic of 1.46 is 0.072. This result is surprising and could not easily happen by chance.
B)The p- value should be considered extreme; therefore the hypothesis test proves that the null hypothesis is true.
C)If the null hypothesis is true, then the probability of getting a test statistic that is as extreme or more extreme than the calculated test statistic of 1.46 is 0.072. This result is not surprising and could easily happen by chance.
D)None of these.

Question 33

A research firm carried out a hypothesis test on a population proportion using a left- tailed alternative hypothesis. Which of the following z- scores would be associated with a p- value of 0.04? Round to the nearest hundredth.

A)z = 2.50
B)z = - 2.50
C)z = - 1.75
D)z = 1.75

Question 34

A researcher believes that children who attend elementary school in a rural setting have lower obesity rates then children who attend elementary school in an urban setting. The researcher collects a random sample from each population and records the proportion of children in each sample who were clinically obese. The data is summarized in the table below. Assume the all conditions for proceeding with a two- sample test have been met. Find the z- statistic (rounded to the nearest hundredth)and p- value (rounded to the nearest thousandth)for this hypothesis test. Using a 5% significance level, state the correct conclusion regarding the null hypothesis H₀: p_rural = p_urban.

A)z = - 1.95, p = 0.026. There is not sufficient evidence to reject the null hypothesis.
B)z = - 1.85, p = 0.032. There is sufficient evidence to accept the null hypothesis.
C)z = - 1.95, p = 0.026. There is sufficient evidence to reject the null hypothesis.
D)z =1.95, p = 0.026. There is sufficient evidence to prove that the population proportions are the same.

Question 35

From the TI- 84 graphing calculator screenshots below, choose the screenshot whose shaded area correctly depicts the following hypothesis test results: H₀: p = 0.25, H_a: p > 0.25, a = 0.05, z = 2.01, p- value = 0.022

A)
B)

C)

Question 36

A researcher believes that the reading habits of men and women are different. He takes a random sample from each population and records the response to the question, "Did you read at least one book last month?" The null hypothesis is H0: pwomen = pmen. Choose the correct alternative hypothesis.

A)Ha: pwomen < pmen
B)Ha: p = 0

C)Ha: p_women × p_men

D)Ha: p_women > p _men

Question 37

Read the following problem description then choose the correct null and alternative hypothesis. A new drug is being tested to see whether it can reduce the rate of asthma attacks in children ages 5 to 14 with asthma ages. The rate of asthma attacks in the population of concern is 0.0744.

A)H₀: p > 0.0744; H_a: p × 0.0744
B)H₀: p = 0.0744; H_a: p < 0.0744
C)H₀: p < 0.0744; H_a: p < 0.0744
D)H₀: p = 0.0744; H_a: p > 0.0744

Question 38

Which of the following is not one of the components of a hypothesis test?

A)State the level of significance.
B)Check the conditions that are required for future computations.
C)Make a decision to either accept the null hypothesis or accept the alternative hypothesis.
D)State the null and alternative hypothesis about the population parameter.

Question 39

Which of the following is not true about the alternative hypothesis?

A)It is assumed to be true based on the sample results.
B)Like the null hypothesis, it is always a statement about a population parameter.
C)It is sometimes called the research hypothesis.
D)It is usually a statement that the researcher hopes to demonstrate is true.

Question 40

Read the following problem description then choose the correct null and alternative hypothesis. A new drug is being tested to see whether it can reduce the rate of food- related allergic reactions in children ages 1 to 3 with food allergies. The rate of allergic reactions in the population of concern is 0.03.

A)H₀: p × 0.03; H_a: p = 0.03
B)H₀: p < 0.03; H_a: p = 0.03
C)H₀: p = 0.03; H_a: p > 0.03
D)H₀: p = 0.03; H_a: p < 0.03

Question 41

Use the following information to answer the question. A child psychologist believes that controlled physical outbursts of anger (like punching a pillow)may improve the mood of young boys with emotional impairment. He believes that the proportion of boys that would benefit from this treatment is greater than the proportion of girls. A random sample from each population receives counseling in the treatment and is asked about their mood after an episode (x is the number of test subjects that reported an improvement in mood). The results of the study are summarized in the table below.

Find the percentage of children that reported an improved mood from each group. Compare the percentages. Do the initial (untested)findings show what the psychologist expected?

Question 42

Use the following information to answer the question. A manager at a large office supply store believes that her supply of pencils has a defect rate that is higher than the defect rate stated by the manufacturer. The manager's null hypothesis is that the supply of pencils has a defect rate of p = 0.025 (the pencil manufacturer's stated defect rate). Suppose we do a test with a significance level of 0.05. Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.025 and H_a: p > 0.025.
Suppose the store manager tests 150 pencils and finds that 9 are defective. Calculate the z test statistic. Round to the nearest hundredth.

Question 43

Explain why failing to reject the null hypothesis does not prove that the null hypothesis is true.

Question 44

List and briefly summarize the essential ingredients of the hypothesis test.

Question 45

Two different students conduct a coin flipping experiment with a left- tailed alternative. The obtain the following test statistics:
Student #1: z = - 2.05
Student #2: z = - 1.28
Which of the test statistics has a smaller p- value and why?

Question 46

Use the following information to answer the question. A manager at a large office supply store believes that her supply of pencils has a defect rate that is higher than the defect rate stated by the manufacturer. The manager's null hypothesis is that the supply of pencils has a defect rate of p = 0.025 (the pencil manufacturer's stated defect rate). Suppose we do a test with a significance level of 0.05. Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.025 and H_a: p > 0.025.
The alternative hypothesis is a right- tailed with a z- score = 0.21

Question 47

Use the following information to answer the question. A child psychologist believes that controlled physical outbursts of anger (like punching a pillow)may improve the mood of young boys with emotional impairment. He believes that the proportion of boys that would benefit from this treatment is greater than the proportion of girls. A random sample from each population receives counseling in the treatment and is asked about their mood after an episode (x is the number of test subjects that reported an improvement in mood). The results of the study are summarized in the table below.

Assuming the conditions for a two proportion z- test hold, state the alternative hypothesis then find the observed test statistic and p- value. State your decision regarding the null hypothesis, H0: p1 = p2. How do the test results compare to the expectations of the psychologist? Round to the nearest hundredth.

Question 48

Write a statement describing the meaning of the level of significance in the context of the hypothesis test.

Question 49

A sociologist believes that families that eat at least one meal a day together (without the interference of any other media)will have better communication skills. The sociologist conducts a study to see if there is a difference in the proportion of meals that are eaten together as a family for families living in a rural setting compared to families living in an urban setting. She collects a random sample from each population and records the proportion of test subjects that reported that they had eaten at least 3 meals per week together as a family. The data is summarized in the table below. Assume the all conditions for proceeding with a two- sample test have been met.
Find the z test statistic (rounded to the nearest hundredth)and p- value (rounded to the nearest thousandth)for testing the hypothesis that the population proportions are different. At the 5% significance level, state the correct conclusion regarding the null hypothesis H₀: p_rural = p_urban. Round all calculations to the nearest hundredth.

Question 50

Use the following information to answer the question. A health foods shop owner is wondering if his customer's daily vitamin supplement habits are in the same proportion as the general population of adults. The shop owner heard in a news report that 60% of all adults reported that they took a daily vitamin. The shop owner believes that his customers have a greater proportion of adults who take a daily vitamin, so he decides to conduct a hypothesis test using the following null and alternative hypothesis: H₀: p = 0.6 and H_a: p > 0.6. The shop owner collected data from 50 randomly selected customers.
List and verify that the conditions hold so that the sampling distribution of the z test statistic will approximately follow the standard normal distribution.

Question 51

Use the following information to answer the question. A health foods shop owner is wondering if his customer's daily vitamin supplement habits are in the same proportion as the general population of adults. The shop owner heard in a news report that 60% of all adults reported that they took a daily vitamin. The shop owner believes that his customers have a greater proportion of adults who take a daily vitamin, so he decides to conduct a hypothesis test using the following null and alternative hypothesis: H₀: p = 0.6 and H_a: p > 0.6. The shop owner collected data from 50 randomly selected customers.
Based on a 5% significance level, write a conclusion by interpreting the p- value. Be sure to clearly state the decision regarding the null hypothesis.

Question 52

A researcher wishes to test the claim that the proportion of children with blue eyes in his region is different than one in six, the national rate of blue eyes in children. State and explain the null and alternative hypothesis that should be used to test the claim.

Question 53

Use the following information to answer the question. A manager at a large office supply store believes that her supply of pencils has a defect rate that is higher than the defect rate stated by the manufacturer. The manager's null hypothesis is that the supply of pencils has a defect rate of p = 0.025 (the pencil manufacturer's stated defect rate). Suppose we do a test with a significance level of 0.05. Symbolically, the null and alternative hypothesis are as follows: H₀: p = 0.025 and H_a: p > 0.025.
The store manager calculates a p- value for the hypothesis test of 0.0030. Write a statement explaining what the p- value means and how it should be interpreted.

Question 54

When a two- sample test of proportions is conducted, there are two conditions of independence that must be checked. State the two conditions of independence. Be sure that your statement clearly states the difference between the two conditions.

Question 55

The worker at a carnival game claims that he can communicate with a small magic rock and to prove it he tells you to hide it in your hand behind your back and he will identify the hand holding the rock. Being a wise student of statistics, you decide to stand back and observe the outcome of the next ten games before deciding whether to pay your three dollars to play the game. You have just conducted an informal hypothesis test. State the null and alternative hypothesis.

Question 56

Use the following information to answer the question. A health foods shop owner is wondering if his customer's daily vitamin supplement habits are in the same proportion as the general population of adults. The shop owner heard in a news report that 60% of all adults reported that they took a daily vitamin. The shop owner believes that his customers have a greater proportion of adults who take a daily vitamin, so he decides to conduct a hypothesis test using the following null and alternative hypothesis: H₀: p = 0.6 and H_a: p > 0.6. The shop owner collected data from 50 randomly selected customers.
To continue the study, the shop owner decides to collect data from 60 customers between the ages of 22 and 27 to see whether the proportion in this age group is different from the general population of adults. From this sample, 26 reported that they took a daily vitamin. The null hypothesis for this test is H0: p = 0.6. and the alternative hypothesis is :Ha: p × 0.6. Assume that the conditions that must be met in order for us to use the N(0, 1)distribution as the sampling distribution are satisfied. Find the values of the sample proportion, p^, the observed test statistic, and the p- value associated with this observed value. Round all values to the nearest thousandth.

Question 57

A sociologist believes that the proportion of single men who attend church on a regular basis is less than the proportion of single women. She takes a random sample from each population and records the proportion from each that reported that they attended church on a regular basis. The null hypothesis is H0: pmen = pwomen. State the correct alternative hypothesis with a sentence and symbolically.

Question 58

Suppose the following is to be tested: H0: p = 0.4 and Ha: p × 0.4. Calculate the observed z test statistic for the following sample data: n = 80 and 25 test subjects have the characteristic of interest. Round to the nearest thousandth.

Question 59

For the following description, state whether a oneproportion z- test or a two- proportion z- test would be appropriate, and name the population.
A researcher asks people who are 20- 29 years old and senior citizens (people over 65)whether they support a new tax on income. He wants to determine whether the proportions that support the tax differ for these age groups.

Question 60

The alternative hypothesis is a two- tailed with a z- score = - 1.88