Review for Proportion Significance Test

Pew Research Center found that 39.5% of teenagers had a part-time job during a recent summer. Ari believes the percentage is different for students at his school. He will collect data and perform a significance test about p = the true proportion of all students at his school who had a part-time job during the summer. Which of the following are the appropriate hypotheses for the test?

• Ho : p-hat = 0.395, Ha : p-hat > 0.395

• Ho : p-hat = 0.395, Ha : p-hat < 0.395

• Ho : p-hat = 0.395, Ha : p-hat (not=) 0.395

• Ho : p = 0.395,

  Ha : p > 0.395

• Ho : p = 0.395,

  Ha : p (not=) 0.395

Which of the following is the best explanation for why the condition np ≥ 10 and nq ≥ 10 must be satisfied to perform a significance test for a proportion?

• So independence can be assumed even when sampling without replacement

• So the results from the sample can be generalized to a larger population

• So the sampling distribution has enough data to be approximately normal

• So we can establish causation

• This condition is not needed

Research shows that 42% of Americans prefer to live in close-knit communities where stores and schools are within walking distance. A construction company believes this percentage is less in a town where they plan to build. They select a random sample of 50 residents and find that 15 prefer living in close-knit communities. Which of the following is the correct test statistic for testing the company's belief?

• A) z = (0.42 - 0.30) / sqrt((0.30)(0.70)/50)

• B) z = (0.42 - 0.30) / sqrt((0.42)(0.58)/50)

• C) z = (0.15 - 0.42) / sqrt((0.42)(0.58)/50)

• D) z = (0.30 - 0.42) / sqrt((0.30)(0.70)/50)

• E) z = (0.30 - 0.42) / sqrt((0.42)(0.58)/50)

Past studies indicate that roughly 20% of adults get less than 5 hours of sleep each night. A social scientist believes the percentage is less for adults in a particular age group. The social scientist obtained a random sample of adults in this age group and conducted a test of H₀: p = 0.20 versus Hₐ: p < 0.20. The p-value of the test was 0.064. Which of the following is a correct interpretation of the p-value?

• A) If 20% of adults in this age group get less than 5 hours of sleep each night, the probability of obtaining a population proportion less than 0.20 is 0.064.

• B) If 20% of adults in this age group get less than 5 hours of sleep each night, the probability of obtaining a sample proportion as small or smaller than the one obtained by the social scientist is 0.064.

• C) If 20% of adults in this age group get less than 5 hours of sleep each night, the probability of obtaining a sample proportion as large or larger than the one obtained by the social scientist is 0.064.

• D) If less than 20% of adults in this age group get less than 5 hours of sleep each night, the probability of obtaining a population proportion less than 0.20 is 0.064.

• E) If less than 20% of adults in this age group get less than 5 hours of sleep each night, the probability of obtaining a sample proportion less than 0.20 is 0.064.

Psychologists are interested in the long-term effects of energy drink consumption among teenagers. They select a random sample of teenagers to test the claim that more than half of all teenagers consume energy drinks daily at a significance level of 0.05. The test yielded a p-value of 0.29. Assuming all conditions for inference were met, which of the following is the correct conclusion?

• We do not reject the null hypothesis. There is not convincing evidence that more than half of all teenagers consume energy drinks daily.

• We do not reject the null hypothesis. There is convincing evidence that more than half of all teenagers consume energy drinks daily.

• We do not reject the null hypothesis. There is convincing evidence that half of all teenagers consume energy drinks daily.

• We reject the null hypothesis. There is not convincing evidence that more than half of all teenagers consume energy drinks daily.

• We reject the null hypothesis. There is convincing evidence that more than half of all teenagers consume energy drinks daily.

People with insomnia have difficulty falling or staying asleep at least three times per week. Eighty people with insomnia volunteered to participate in a sleep study to see if drinking tea with lavender would reduce insomnia to once per week or less. Half of the participants were randomly assigned to drink tea with lavender each night before bedtime, and the other half were assigned to drink plain tea. Participants did not know which type of tea they were drinking. The results showed that 22 people in the lavender tea group and 14 people in the plain tea group experienced a reduction in insomnia to once per week or less. Is there convincing statistical evidence that the proportion of people who would experience reduced insomnia is greater for those who drink tea with lavender than those who drink plain tea for people similar to the volunteers in the study?

• There is convincing statistical evidence at the α=

  0.01 significance level.

• There is convincing statistical evidence at the α=

  0.05 significance level but not at the

  α=

  0.01 level.

• There is convincing statistical evidence at the α=

  0.10 significance level but not at the α=

  0.05 level.

• There is not convincing statistical evidence at the α=

  0.25 significance level.

• No conclusion can be drawn because the experiment was not blinded

A medical student recruits eighty students with athlete's foot for an experiment. Half were randomly assigned to use a Listerine-vinegar solution and the other used a hydrogen peroxide-iodine solution twice daily for two weeks. The results showed that 11 people in the Listerine group and 17 in the hydrogen peroxide group reported improvement. Is there convincing statistical evidence that the proportion of people who would experience improvement is less for those who use the Listerine solution than those who use the hydrogen peroxide solution, for people similar to the volunteers in the study?

• There is convincing statistical evidence at the α=

  0.01 significance level.

• There is convincing statistical evidence at the α=

  0.05 significance level but not at the

  α=

  0.01 level.

• There is convincing statistical evidence at the α=

  0.10 significance level but not at the α=

  0.05 level.

• There is not convincing statistical evidence at the α=

  0.25 significance level.

• No conclusion can be drawn because the experiment was not blinded

• 0.019

• 0.038

• 0.113

• 0.400

• 0.400