Significance Tests for Proportions

The average yield of a certain crop is 10.1 bushels per plant. A biologist claims that a new fertilizer will result in a greater yield. The appropriate null and alternative hypotheses would be:

• H₀ : µ = 10.8; H_a: µ > 10.8

• H₀ : µ = 10.8; H_a: µ ≠ 10.8

• H₀ : µ = 10.1; H_a: µ > 10.1

• H₀ : µ = 10.1; H_a: µ < 10.1

• H₀ : µ = 10.1; H_a: µ ≠ 10.1

An opinion poll asks a SRS of 200 adults how they feel about voting for an amendment. What hypothesis would test if the proportion is greater than 60%?

• H₀ : p = 0.6; H_a: p > 0.6

• H₀ : p = 0.6; H_a: p ≠ 0.6

• H₀ : p = 0.6; H_a: p < 0.6

• H₀ : p = 0.6; H_a: p = 0.75

• H₀ : p = 0.75; H_a: p < 0.6

A test of significance produces a P-value of 0.035. Which of the following conclusions is appropriate?

• Accept H_a at the α = 0.05 level

• Reject H_a at the α = 0.01 level

• Fail to reject H₀ at the α = 0.05 level

• Reject H₀ at the α = 0.05 level

• Accept H₀ at the α = 0.05 level

Type II error is

• rejecting the null hypothesis when it is true

• failing to reject the null hypothesis when it is false

• rejecting the null hypothesis when it is false

• failing to reject the null hypothesis when it is true

• more serious than a Type I error

A researcher plans to conduct a significance test at the significance level of 0.05. She designs her study to have a power of 0.85 for a particular alternative value of the parameter. The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is

• 0.05

• 0.15

• 0.80

• 0.95

• equal to the 1 - (P-value) and cannot be determined until the data have been collected

In hypothesis testing, β is the probability of committing a Type II error in a test with significance level α. The probability of committing a Type I error is

• 1 - β

• 1 - α

• β - α

• α

• cannot be determined

James, a claimed psychic, was presented with 200 cards face down and was asked to determine if the card was one of 5 symbols. He was correct in 50 cases. When testing the hypotheses H₀: p= 0.20 and H_a: p>0.20 and assuming conditions are met, the P-value of this test is

• between 0.10 and 0.05

• between 0.05 and 0.025

• between 0.025 and 0.01

• between 0.01 and 0.001

• below 0.001

The most important condition for drawing sound conclusions from statistical inference is usually

• that the population standard deviation is known

• that at least 30 people are included in the study

• that the data come from a random sample or a randomized experiment

• that the population distribution is exactly Normal

• that no calculation errors are made in the confidence interval or test statistic

It is believed that a mean weight of 160 pounds would be normal for a type of athletes. To see if there is evidence that the mean weight of the population of all athletes is significantly higher than 160 pounds you test the hypotheses H₀: μ = 160; H_a: μ > 160 and obtain a P-value of 0.0742. Which of the following is true?

• At the 5% significance level, you have proved H₀ is true.

• You have failed to obtain sufficient evidence against H₀.

• At the 5% significance level, you have failed to prove that H₀ is true, and a larger sample size is needed to do so.

• Only 7.42% of the athletes weigh less than 160 pounds.

• None of the above. A significance test is inappropriate in this setting.

A rubber band man claims that p, the proportion of bands that snap when stretched beyond 8 inches, is no more than 0.03. Some customers have complained that this happens more frequently. Which of the following sets of hypotheses would the customers used to conduct an experiment to test the rubber band man's claim?

• H₀: p ≠ 0.03, H_A: p = 0.03

• H₀: p = 0.03, H_A: p > 0.03

• H₀: p = 0.03, H_A: p ≠ 0.03

• H₀: p = 0.03, H_A: p < 0.03

• H₀: p > 0.03, H_A: p = 0.03