Q1. Which of the following statements about the variance of a normally distributed population is least accurate?

A)     The Chi-squared distribution is a symmetric distribution.

B)     The test of whether the population variance equals σ₀² requires the use of a Chi-squared distributed test statistic, [(n − 1)s²] / σ₀².

C)     A test of whether the variance of a normally distributed population is equal to some value σ₀², the hypotheses are: H₀: σ² = σ₀², versus H_a: σ² ≠ σ₀².

Q2. A test of the population variance is equal to a hypothesized value requires the use of a test statistic that is:

A)     F-distributed.

B)     Chi-squared distributed.

C)     t-distributed.

Q3. A munitions manufacturer claims that the standard deviation of the powder packed in its shotgun shells is 0.1% of the stated nominal amount of powder. A sport clay association has reviewed a sample of 51 shotgun shells and found a standard deviation of 0.12%. What is the Chi-squared value, and what are the critical values at a 95% confidence level, respectively?

A)   72; 32.357 and 71.420.

B)   72; 34.764 and 67.505.

C)   70; 34.764 and 79.490.

答案和详解如下：

Q1. Which of the following statements about the variance of a normally distributed population is least accurate?

A)     The Chi-squared distribution is a symmetric distribution.

B)     The test of whether the population variance equals σ₀² requires the use of a Chi-squared distributed test statistic, [(n − 1)s²] / σ₀².

C)     A test of whether the variance of a normally distributed population is equal to some value σ₀², the hypotheses are: H₀: σ² = σ₀², versus H_a: σ² ≠ σ₀².

Correct answer is A)

The Chi-squared distribution is not symmetrical, which means that the critical values will not be numerically equidistant from the center of the distribution, though the probability on either side of the critical values will be equal (that is, if there is a 5% level of significance and a two-sided test, 2.5% will lie outside each of the two critical values).

Q2. A test of the population variance is equal to a hypothesized value requires the use of a test statistic that is:

A)     F-distributed.

B)     Chi-squared distributed.

C)     t-distributed.

Correct answer is B)

In tests of whether the variance of a population equals a particular value, the chi-squared test statistic is appropriate.

Q3. A munitions manufacturer claims that the standard deviation of the powder packed in its shotgun shells is 0.1% of the stated nominal amount of powder. A sport clay association has reviewed a sample of 51 shotgun shells and found a standard deviation of 0.12%. What is the Chi-squared value, and what are the critical values at a 95% confidence level, respectively?

A)   72; 32.357 and 71.420.

B)   72; 34.764 and 67.505.

C)   70; 34.764 and 79.490.

Correct answer is A)

To compare standard deviations we use a Chi-square statistic. X2 = (n – 1)s2 / σ02 = 50(0.0144) / 0.01 = 72. With 50 df, the critical values at the 95% confidence level are 32.357 and 71.420. Since the Chi-squared value is outside this range, we can reject the hypothesis that the standard deviations are the same.

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.