Reading 11: Hypothesis Testing-LOS b 习题精选 involving, interpret, hypothesis, 2011

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Session 3: Quantitative Methods: Application
Reading 11: Hypothesis Testing

LOS b: Define and interpret a test statistic, a Type I and a Type II error, and a significance level, and explain how significance levels are used in hypothesis testing.

 

 

For a two-tailed test of hypothesis involving a z-distributed test statistic and a 5% level of significance, a calculated z-statistic of 1.5 indicates that:

A) the null hypothesis cannot be rejected.

B) the null hypothesis is rejected.

C) the test is inconclusive.

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For a two-tailed test at a 5% level of significance the calculated z-statistic would have to be greater than the critical z value of 1.96 for the null hypothesis to be rejected.

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Which of the following statements about hypothesis testing is most accurate? A Type II error is the probability of:

A) rejecting a true alternative hypothesis.

B) failing to reject a false null hypothesis.

C) rejecting a true null hypothesis.

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The Type II error is the error of failing to reject a null hypothesis that is not true.

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If the probability of a Type I error decreases, then the probability of:

A) a Type II error increases.

B) incorrectly accepting the null decreases.

C) incorrectly rejecting the null increases.

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If P(Type I error) decreases, then P(Type II error) increases. A null hypothesis is never accepted. We can only fail to reject the null.

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Which of the following statements about hypothesis testing is most accurate?

A) A Type I error is rejecting the null hypothesis when it is true, and a Type II error is accepting the alternative hypothesis when it is false.

B) A hypothesized mean of 3, a sample mean of 6, and a standard error of the sampling means of 2 give a sample Z-statistic of 1.5.

C) When the critical Z-statistic is greater than the sample Z-statistic in a two-tailed test, reject the null hypothesis and accept the alternative hypothesis.

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Z = (6 - 3)/2 = 1.5. A Type II error is wrongly accepting the null hypothesis. The null hypothesis should be rejected when the sample Z-statistic is greater than the critical Z-statistic.

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A researcher is testing whether the average age of employees in a large firm is statistically different from 35 years (either above or below). A sample is drawn of 250 employees and the researcher determines that the appropriate critical value for the test statistic is 1.96. The value of the computed test statistic is 4.35. Given this information, which of the following statements is least accurate? The test:

A) has a significance level of 95%.

B) indicates that the researcher will reject the null hypothesis.

C) indicates that the researcher is 95% confident that the average employee age is different than 35 years.

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This test has a significance level of 5%. The relationship between confidence and significance is: significance level = 1 ? confidence level. We know that the significance level is 5% because the sample size is large and the critical value of the test statistic is 1.96 (2.5% of probability is in both the upper and lower tails).

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Identify the error type associated with the level of significance and the meaning of a 5 percent significance level.

Error type

α = 0.05 means there is a 5 percent probability of

A) Type I error   failing to reject a true null hypothesis

B) Type II error    rejecting a true null hypothesis

C) Type I error    rejecting a true null hypothesis

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The significance level is the risk of making a Type 1 error and rejecting the null hypothesis when it is true.

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A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $57,000 per year. Assuming a normal distribution, what is the test statistic given a sample of 115 newly acquired CFA charterholders with a mean starting salary of $65,000 and a standard deviation of $4,500?

A) 19.06.

B) -19.06.

C) 1.78.

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With a large sample size (115) the z-statistic is used. The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean – hypothesized mean) / (population standard deviation / (sample size)^1/2 = (X ? μ) / (σ / n^1/2) = (65,000 – 57,000) / (4,500 / 115^1/2) = (8,000) / (4,500 / 10.72) = 19.06.

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If a two-tailed hypothesis test has a 5% probability of rejecting the null hypothesis when the null is true, it is most likely that the:

A) probability of a Type I error is 2.5%.

B) power of the test is 95%.

C) significance level of the test is 5%.

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Rejecting the null hypothesis when it is true is a Type I error. The probability of a Type I error is the significance level of the test. The power of a test is one minus the probability of a Type II error, which cannot be calculated from the information given.

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A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $59,000 per year. What is the test statistic given a sample of 135 newly acquired CFA charterholders with a mean starting salary of $64,000 and a standard deviation of $5,500?

A) 10.56.

B) -10.56.

C) 0.91.

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With a large sample size (135) the z-statistic is used. The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean – hypothesized mean) / (population standard deviation / (sample size)^1/2) = (X ? μ) / (σ / n^1/2) = (64,000 – 59,000) / (5,500 / 135^1/2) = (5,000) / (5,500 / 11.62) = 10.56.

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A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $58,500 per year. What is the test statistic given a sample of 175 newly acquired CFA charterholders with a mean starting salary of $67,000 and a standard deviation of $5,200?

A) -1.63.

B) 21.62.

C) 1.63.

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With a large sample size (175) the z-statistic is used. The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean – hypothesized mean) / (population standard deviation / (sample size)^1/2 = (X ? μ) / (σ / n^1/2) = (67,000 – 58,500) / (5,200 / 175^1/2) = (8,500) / (5,200 / 13.22) = 21.62.