Q1. Ryan McKeeler and Howard Hu, two junior statisticians, were discussing the relation between confidence intervals and hypothesis tests. During their discussion each of them made the following statement:

McKeeler: A confidence interval for a two-tailed hypothesis test is calculated as adding and subtracting the product of the standard error and the critical value from the sample statistic. For example, for a level of confidence of 68%, there is a 32% probability that the true population parameter is contained in the interval.

Hu: A 99% confidence interval uses a critical value associated with a given distribution at the 1% level of significance. A hypothesis test would compare a calculated test statistic to that critical value. As such, the confidence interval is the range for the test statistic within which a researcher would not reject the null hypothesis for a two-tailed hypothesis test about the value of the population mean of the random variable.

With respect to the statements made by McKeeler and Hu:

A)   both are correct.

B)   only one is correct.

C)   both are incorrect.

Q2. Given a mean of 10% and a standard deviation of 14%, what is a 95% confidence interval for the return next year?

A)   -17.44% to 37.44%.

B)   -4.00% to 24.00%.

C)   -17.00% to 38.00%.

答案和详解如下：

Q1. Ryan McKeeler and Howard Hu, two junior statisticians, were discussing the relation between confidence intervals and hypothesis tests. During their discussion each of them made the following statement:

McKeeler: A confidence interval for a two-tailed hypothesis test is calculated as adding and subtracting the product of the standard error and the critical value from the sample statistic. For example, for a level of confidence of 68%, there is a 32% probability that the true population parameter is contained in the interval.

Hu: A 99% confidence interval uses a critical value associated with a given distribution at the 1% level of significance. A hypothesis test would compare a calculated test statistic to that critical value. As such, the confidence interval is the range for the test statistic within which a researcher would not reject the null hypothesis for a two-tailed hypothesis test about the value of the population mean of the random variable.

With respect to the statements made by McKeeler and Hu:

A)   both are correct.

B)   only one is correct.

C)   both are incorrect.

Correct answer is B)

McKeeler’s statement is incorrect. Specifically, for a level of confidence of say, 68%, there is a 68% probability that the true population parameter is contained in the interval. Therefore, there is a 32% probability that the true population parameter is not contained in the interval. Hu’s statement is correct.

Q2. Given a mean of 10% and a standard deviation of 14%, what is a 95% confidence interval for the return next year?

A)   -17.44% to 37.44%.

B)   -4.00% to 24.00%.

C)   -17.00% to 38.00%.

Correct answer is A)

10% +/- 14(1.96) = -17.44% to 37.44%.

Q6. If the null hypothesis is H₀: ρ ≤ 0, what is the appropriate alternative hypothesis?

A)     H_a: ρ ≠ 0.

B)     H_a: ρ > 0.

C)     H_a: ρ < 0.

Correct answer is B)

The alternative hypothesis must include the possible outcomes the null does not.

Q7. In order to test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 1.2. If you choose a 5% significance level you should:

A)   fail to reject the null hypothesis and conclude that the population mean is not greater than 100.

B)   reject the null hypothesis and conclude that the population mean is greater than 100.

C)   fail to reject the null hypothesis and conclude that the population mean is greater than 100.

Correct answer is A)

At a 5% significance level, the critical t-statistic using the Student’s t distribution table for a one-tailed test and 29 degrees of freedom (sample size of 30 less 1) is 1.699 (with a large sample size the critical z-statistic of 1.645 may be used). Because the critical t-statistic is greater than the calculated t-statistic, meaning that the calculated t-statistic is not in the rejection range, we fail to reject the null hypothesis and we conclude that the population mean is not significantly greater than 100.

Q8. In order to test whether the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 3.4. The null and alternative hypotheses are:

A)H₀: μ ≤ 100; H_a: μ > 100.

B)H₀: μ = 100; H_a: μ ≠ 100.

C)H₀: X ≤ 100; H_a: X > 100.

Correct answer is A)

The null hypothesis is that the theoretical mean is not significantly different from zero. The alternative hypothesis is that the theoretical mean is greater than zero.

Q9. In a two-tailed test of a hypothesis concerning whether a population mean is zero, Jack Olson computes a t-statistic of 2.7 based on a sample of 20 observations where the distribution is normal. If a 5% significance level is chosen, Olson should:

A)   reject the null hypothesis and conclude that the population mean is not significantly different from zero.

B)   fail to reject the null hypothesis that the population mean is not significantly different from zero.

C)   reject the null hypothesis and conclude that the population mean is significantly different from zero.

Correct answer is C)

At a 5% significance level, the critical t-statistic using the Student’s t-distribution table for a two-tailed test and 19 degrees of freedom (sample size of 20 less 1) is ± 2.093 (with a large sample size the critical z-statistic of 1.960 may be used). Because the critical t-statistic of 2.093 is to the left of the calculated t-statistic of 2.7, meaning that the calculated t-statistic is in the rejection range, we reject the null hypothesis and we conclude that the population mean is significantly different from zero.

Q10. What kind of test is being used for the following hypothesis and what would a z-statistic of 1.68 tell us about a hypothesis with the appropriate test and a level of significance of 5%, respectively?

H₀: B ≤ 0

H_A: B > 0

A)   One-tailed test; fail to reject the null.

B)   Two-tailed test; fail to reject the null.

C)   One-tailed test; reject the null.

Correct answer is C)

The way the alternative hypothesis is written you are only looking at the right side of the distribution. You are only interested in showing that B is greater than 0. You don't care if it is less than zero. For a one-tailed test at the 5% level of significance, the critical z value is 1.645. Since the test statistic of 1.68 is greater than the critical value we would reject the null hypothesis.