Significance Tests and Decision Making

A test of H0: µ=60 versus Ha: 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.

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

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.

Which of the following describes a Type II error in this setting?

• Concluding that the students are less fit (on average) than the general population when in fact they have equal fitness on average.

• Not concluding that the students are less fit (on average) as the general population when in fact they are less fit (on average).

• Not concluding that the students are less fit (on average) as the general population when in fact they have the same fitness (on average).

• Concluding that the students are less fit (on average) than the general population, when, in fact, they are less fit (on average).

• Concluding that the students have the same fitness (on average) when in fact they are more fit (on average).

A significance test was performed to test the null hypothesis H₀: µ = 2 versus the alternative H_a: µ ≠ 2. A sample of size 28 produced a test statistic is t = 2.051. Assuming all conditions for inference were met, which of the following intervals contains the P-value for this test?

• P < 0.01

• 0.01≤ P < 0.02

• 0.02 ≤ P < 0.025

• 0.025 ≤ P < 0.05

• 0.05 ≤ P < 0.10

You collect test scores on four members of a population which you can safely assume is approximately Normally distributed and test the hypotheses H₀: μ=100 versus H_a: μ > 100. You obtain a P-value of 0.052. Which of the following statements is true?

• At the 5% significance level, you have proved that

  H₀ is true.

• You have failed to obtain any evidence for H_a.

• There is some evidence against H₀ , and a study using a larger sample size may be worthwhile

• You can accept H_a at the 5% significance level

• You can accept H₀ at the 10% significance level.

Some people say that more babies are born in September than in any other month. To test this claim, you take a simple random sample of 150 students at your school and find that 21 of them were born in September. You are interested in whether the proportion born in September is higher than 1/12—what you would expect if September was no different from any other month. Thus your null hypothesis is H₀: p= 1/12. The P-value for your test is 0.0056. Which of the following statements best describes what the P-value measures?

• The probability that September birthdays are no more common that any other month is 0.0056.

• The probability that September birthdays are more common is 0.0056

• The probability that the proportion of September birthdays in the population is not equal to 1/12 is 0.0056

• 0.0056 is the probability of getting a sample with a proportion of September birthdays this far or farther above 1/12 if the true proportion is 1/12

• 0.0056 is the probability of getting a sample with a proportion of September birthdays this close to 1/12 if the true proportion is not 1/12

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