Q9. Ken Wallace is interested in testing whether the average price to earnings (P/E) of firms in the retail industry is 25. Using a t-distributed test statistic and a 5% level of significance, the critical values for a sample of 40 firms is/are:

A)  -1.96 and 1.96.

B)  -2.023 and 2.023.

C)  -1.685 and 1.685.

Q10 .Simone Mak is a television network advertising executive. One of her responsibilities is selling commercial spots for a successful weekly sitcom. If the average share of viewers for this season exceeds 8.5%, she can raise the advertising rates by 50% for the next season. The population of viewer shares is normally distributed. A sample of the past 18 episodes results in a mean share of 9.6% with a standard deviation of 10.0%. If Mak is willing to make a Type 1 error with a 5% probability, which of the following statements is most accurate?

A)   With an unknown population variance and a small sample size, Mak cannot test a hypothesis based on her sample data.

B)   Mak cannot charge a higher rate next season for advertising spots based on this sample.

C)   The null hypothesis Mak needs to test is that the mean share of viewers is greater than 8.5%.

答案和详解如下：

Q9. Ken Wallace is interested in testing whether the average price to earnings (P/E) of firms in the retail industry is 25. Using a t-distributed test statistic and a 5% level of significance, the critical values for a sample of 40 firms is/are:

A)  -1.96 and 1.96.

B)  -2.023 and 2.023.

C)  -1.685 and 1.685.

Correct answer is B)

There are 40 − 1 = 39 degrees of freedom and the test is two-tailed. Therefore, the critical t-values are ± 2.023. The value 2.023 is the critical value for a one-tailed probability of 2.5%.

Q10 .Simone Mak is a television network advertising executive. One of her responsibilities is selling commercial spots for a successful weekly sitcom. If the average share of viewers for this season exceeds 8.5%, she can raise the advertising rates by 50% for the next season. The population of viewer shares is normally distributed. A sample of the past 18 episodes results in a mean share of 9.6% with a standard deviation of 10.0%. If Mak is willing to make a Type 1 error with a 5% probability, which of the following statements is most accurate?

A)   With an unknown population variance and a small sample size, Mak cannot test a hypothesis based on her sample data.

B)   Mak cannot charge a higher rate next season for advertising spots based on this sample.

C)   The null hypothesis Mak needs to test is that the mean share of viewers is greater than 8.5%.

Correct answer is B)

Mak can conclude with 95% confidence that the average share of viewers for the show this season exceeds 8.5 and thus she can charge a higher advertising rate next season. Hypothesis testing process:

Step 1:  State the hypothesis. Null hypothesis: mean ≤ 8.5%; Alternative hypothesis: mean > 8.5%

Step 2:  Select the appropriate test statistic. Use a t statistic because we have a normally distributed population with an unknown variance (we are given only the sample variance) and a small sample size (less than 30). If the population were not normally distributed, no test would be available to use with a small sample size.

Step 3:  Specify the level of significance. The significance level is the probability of a Type I error, or 0.05.

Step 4:  State the decision rule.  This is a one-tailed test. The critical value for this question will be the t-statistic that corresponds to a significance level of 0.05 and n-1 or 17 degrees of freedom. Using the t-table, we determine that we will reject the null hypothesis if the calculated test statistic is greater than the critical value of 1.74.

Step 5:  Calculate the sample (test) statistic.  The test statistic = t = (9.6 – 8.5) / (10.0 / √ 18) = 0.479 (Note: Remember to use standard error in the denominator because we are testing a hypothesis about the population mean based on the mean of 18 observations.)

Step 6: Make a decision. The calculated statistic is less than the critical value. Mak cannot conclude with 95% confidence that the mean share of viewers exceeds 8.5% and thus she cannot charge higher rates.

Note: By eliminating the two incorrect choices, you can select the correct response to this question without performing the calculations.

d

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