AP Statistics Tests of Significance

A significance test was performed to test the null hypothesis H₀ : p = 0.5 versus the alternative H_a: p > 0.5. The test statistic is z = 1.40. Which of the following is closest to the P-value for this test?

• 0.0808

• 0.1492

• 0.1616

• 0.2984

• 0.9192

The mean time it takes for a person to experience pain relief from aspirin is 25 minutes. A new ingredient is added to help speed up relief. Let µ denote the mean time to obtain pain relief with the new product. An experiment is conducted to verify if the new product works more quickly. What are the null and alternative hypotheses for the appropriate test of significance?

• H0 : µ = 25 vs. Ha : µ ≠ 25

• H0 : µ = 25 vs. Ha : µ < 25

• H0 : µ < 25 vs. Ha : µ = 25

• H0 : µ < 25 vs. Ha : µ > 25

• H0 : µ = 25 vs. Ha : µ > 25

A test of H₀: µ=60 versus H_a: µ≠60 produces a sample mean of x-bar = 58 and a P-value of 0.04. At an α = 0.05 level, which of the following is an appropriate conclusion?

• There is sufficient evidence to conclude that µ < 60.

• There is sufficient evidence to conclude that µ = 60.

• There is insufficient evidence to conclude that µ = 60.

• There is insufficient evidence to conclude that µ ≠ 60

• There is sufficient evidence to conclude that µ ≠ 60.

Because t procedures are robust, the most important condition for their use is

• the population standard deviation is known.

• the population distribution is approximately Normal.

• the data can be regarded as a random sample from the population.

• np and n(1 – p) are both at least 10.

• all values in the sample are within two standard deviations of the mean.

We want to test H₀: µ = 1.5 vs. H_a : µ ≠ 1.5 at α = 0.05 . A 95% confidence interval for µ calculated from a given random sample is (1.4, 3.6). Based on this finding we

• fail to reject H₀

• reject H₀

• cannot make any decision at all because the value of the test statistic is not available

• cannot make any decision at all because the distribution of the population is unknown

• cannot make any decision at all because (1.4, 3.6) is only a 95% confidence interval for µ

Which of the following statements is/are correct?

• The power of a significance test depends on the effect size.

• The probability of a Type II error is equal to the significance level of the test.

• Error probabilities can be expressed only when a significance level has been specified.

The water diet requires you to drink two cups of water every half hour from the time you get up until you go to bed, but otherwise allows you to eat whatever you like. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are (in photo). Which of the following conditions must be met in order to use a t-procedure on these paired data?

• Only the distribution of pre-diet weights must be approximately Normal.

• Only the distribution of differences (after 6 weeks – before) must be approximately Normal.

• The distribution of both pre-diet weights and six-week weights must be approximately Normal.

• The distribution of pre-diet weights and the distribution of differences (after 6 weeks – before)

  must be approximately Normal.

• All three distributions—before diet, after 6 weeks, and the difference—must be approximately Normal.

The water diet requires you to drink two cups of water every half hour from the time you get up until you go to bed, but otherwise allows you to eat whatever you like. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are (see photo). What would a Type II error be for this test of the water diet?

• Concluding that the diet leads to weight loss when it doesn’t.

• Concluding that the diet leads to weight loss when it really does.

• Not concluding that the diet leads to weight loss when it does.

• Not concluding that the diet leads to weight loss when it really doesn’t.

• Drawing a conclusion from this test when the Normality condition has not been satisfied.

A researcher wishes to determine if people are able to complete a certain pencil and paper maze more quickly while listening to classical music. Suppose previous research has established that the mean time needed for people to complete a certain maze (without music) is 40 seconds. The researcher, therefore, decides to test the hypotheses H₀: µ = 40 versus H_a: µ < 40 , where µ is the time in seconds needed to complete the maze while listening to classical music. To do so, the researcher has 10,000 people complete the maze with classical music playing. The mean time for these people is x-bar = 39.92 seconds, and the P-value of his significance test is 0.0002. Which statement below best describes the appropriate conclusion to draw from this study?

• The researcher has proved that listening to classical music substantially improves the time it takes to complete the maze.

• The researcher has strong evidence that listening to classical music substantially improves the time it takes to complete the maze.

• The researcher has moderate evidence that listening to classical music substantially improves the time it takes to complete the maze.

• Although the researcher has obtained a statistically significant result, it appears to have little practical significance.

• Since the P-value is greater than the reciprocal of the sample size, this is not a significant result.

The recommended daily Calcium intake for women over 21 (and under 50) is 1000 mg per day. The health services at a college are concerned that women at the college get less Calcium than that, so they take a random sample of female students in order to test the hypotheses H₀: μ =1000 versus H_a: μ<1000. Prior to the study they estimate that the power of their test against the alternative H_a:μ=900 is 0.85. Which of the following is the best interpretation of this value?

• The probability of making a Type II error.

• The probability of rejecting the null hypothesis when the parameter value is 1000.

• The probability of rejecting the null hypothesis when the parameter value is 900.

• The probability of failing to reject the null hypothesis when the parameter value is 1000.

• The probability of failing to reject the null hypothesis when the parameter value is 900.