The data that accompanies this exercise show miles per gallon (MPG) for a sample of 25 cars.

a. Using the Jarque-Bera test, state the competing hypotheses in order to determine whether or not MPG follow the normal distribution.

b. Calculate the value of the Jarque-Bera test statistic. Use Excel to calculate the $p$-value.

c. At $\alpha=0.05$, can you conclude that MPG are not normally distributed?

Total 2005 CE0 compensation for the largest U.S. companies by revenue is reported in the following frequency distribution, along with some summary statistics. Total compensation includes salary, bonuses, stock and incentives, the potential value of stock options, and gains from stock options exercised.

$$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Total Compensation } \\ \text { (in millions of \$) } \end{array} & \text { Frequency } \\ \hline \text { Less than } 5 & 43 \\ \hline 5 \text { up to } 10 & 65 \\ \hline 10 \text { up to } 15 & 32 \\ \hline 15 \text { up to } 20 & 38 \\ \hline 20 \text { or more } & 60 \\ \hline & n=238 \\ \hline \end{array}$$

Other summary statistics for CEO compensation (in millions of $) are as follows:$

$$\begin{array}{|c|c|c|c|c|} \hline \text { Mean } & \text { Median } & \begin{array}{c} \text { Standard } \\ \text { Deviation } \end{array} & \text { Skewness } & \text { Kurtosis } \\ \hline 19.03 & 11.02 & 27.61 & 5.26 & 35.53 \\ \hline \end{array}$$

$a. Conduct a goodness-of-fit test for normality of CEO compensation at the 1% significance level.

b. Conduct the Jarque-Bera test at the 1$ significance level.

c. Does total compensation of CEOs for the largest U.S. companies not follow the normal distribution?

Fabco, a precision machining shop, uses statistical process control (SPC) techniques to ensure quality and consistency of their steel shafts. The control limits used in their SPC charts are based on the assumption that shaft diameters are normally distributed. To verify this assumption, a quality engineer has measurec the diameters for a sample of 50 of its popular 1/2-inch shafts.

a. Using the Jarque-Bera test, state the competing hypotheses in order to determine whether or not the data follow the normal distribution.

b. Calculate the value of the Jarque-Bera test statistic Use Excel to calculate the $p$-value.

c. At $\alpha=0.10$, can you conclude that the shaft diameters are not normally distributed?

d. Would your conclusion change at the 5% significance level?

The quality department at an electronics component manufacturer must ensure that their components will operate at prespecified levels. The accompanying table shows a frequency distribution with measured resistance values (in ohms) for a sample of $520$ resistors. The sample mean and the sample standard deviation are $4790 \mathrm{ohms}$ and 40 ohms, respectively.

$$\begin{array}{|l|c|} \hline \text { Resistance (ohms) } & \text { Frequency } \\ \hline \text { Under } 4740 & 44 \\ \hline 4740 \text { up to } 4780 & 145 \\ \hline 4780 \text { up to } 4820 & 197 \\ \hline 4820 \text { up to } 4860 & 107 \\ \hline 4860 \text { or more } & 27 \\ \hline & \text { Total }=520 \\ \hline \end{array}$$

a. Using the goodness-of-fit test for normality, state the competing hypotheses to test if the sample data suggest that resistance does not follow the normal distribution.

b. At $\alpha=0.05$, what is the critical value?

c. Calculate the value of the test statistic.

d. What is the conclusion to the test? Would your conclusion change at the 10% significance level?

A multinomial experiment produced the following results:

$$\begin{array}{|l|r|r|r|r|r|} \hline \text { Category } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Frequency } & 57 & 63 & 70 & 55 & 55 \\ \hline \end{array}$$

Can we conclude at the 1% significance level that not all population proportions are equal to $0.20$ ?

Fifty cities provided information on vacancy rates (in percent) in local apartments in the following frequency distribution. The sample mean and the sample standard deviation are 9% and 3.6%, respectively.

$$\begin{array}{|l|c|} \hline \text { Vacancy Rate (in percent) } & \text { Frequency } \\ \hline \text { Less than } 6 & 10 \\ \hline 6 \text { up to } 9 & 10 \\ \hline 9 \text { up to } 12 & 20 \\ \hline 12 \text { or more } & 10 \\ \hline \end{array}$$

Apply the goodness-of-fit test for normality at the 5% significance level. Do the sample data suggest that vacancy rates do not follow the normal distribution?

An economics professor states on her syllabus that final grades will be distributed using the normal distribution. The final averages of $300$ students are calculated, and she groups the data into a frequency distribution as shown in the accompanying table. The mean and the standard deviation of the final are $\bar{x}=72$ and $s=10$.

$$\begin{array}{|l|c|} \hline \text { Final Averages } & \text { Frequency } \\ \hline \text { F: Less than } 50 & 5 \\ \hline \text { D: } 50 \text { up to } 70 & 135 \\ \hline \text { C: } 70 \text { up to } 80 & 105 \\ \hline \text { B: } 80 \text { up to } 90 & 45 \\ \hline \text { A: } 90 \text { or above } & 10 \\ \hline & \text { Total }=300 \\ \hline \end{array}$$

a. Using the goodness-of-fit test for normality, state the competing hypotheses in order to determine if we can reject the professor's normality claim.

b. At a 5% significance level, what is the critical value?

c. Calculate the value of the test statistic.

d. What is the conclusion to the test?

You are given the following summary statistics from a sample of 50 observations:

$$\begin{array}{|l|r|} \hline \text { Mean } & 77.25 \\ \hline \text { Standard Deviation } & 11.36 \\ \hline \text { Skewness } & 1.12 \\ \hline \text { Kurtosis } & 1.63 \\ \hline \end{array}$$

a. Using the Jarque-Bera test, specify the null and alternative hypotheses to determine whether or not the data are normally distributed.

b. At the 5% significance level, what is the critical value?

c. Calculate the value of the test statistic.

d. What is the conclusion? Can you conclude that the data do not follow the normal distribution? Explain.

The following frequency distribution has a sample mean of $-3.5$ and a sample standard deviation of 9.7.

$$\begin{array}{|l|c|} \hline \text { Class } & \text { Frequency } \\ \hline \text { Less than }-10 & 70 \\ \hline-10 \text { up to } 0 & 40 \\ \hline 0 \text { up to } 10 & 80 \\ \hline 10 \text { or more } & 10 \\ \hline \end{array}$$

At the 1%$ significance level, use the goodness-of-fit test for normality to determine whether or not the data are normally distributed.

Consider the following sample data with mean and standard deviation of $20.5$ and $5.4$, respectively.

$$\begin{array}{|l|c|} \hline \text { Class } & \text { Frequency } \\ \hline \text { Less than } 10 & 25 \\ \hline 10 \text { up to } 20 & 95 \\ \hline 20 \text { up to } 30 & 65 \\ \hline 30 \text { or more } & 15 \\ \hline & n=200 \\ \hline \end{array}$$

a. Using the goodness-of-fit test for normality, specify the competing hypotheses in order to determine whether or not the data are normally distributed.

b. At the 5% significance level, what is the critical value? What is the decision rule?

c. Calculate the value of the test statistic.

d. What is the conclusion?

(Use Excel) According to a 2008 survey by the Pew Research Center, people in China are highly satisfied with their roaring economy and the direction of their nation (USA Today, July 22, 2008). Eighty-six percent of those who were surveyed expressed positive views of the way China is progressing and described the economic situation as good. A political analyst wants to know if this optimism among the Chinese depends on age. In an independent survey of 280 Chinese residents, the respondents are asked how happy they are with the direction that their country is taking. Their responses are tabulated in the following table.

$$\begin{array}{|l|c|c|c|} \hline \text { Age } & \begin{array}{c} \text { Very } \\ \text { Happy } \end{array} & \begin{array}{c} \text { Somewhat } \\ \text { Happy } \end{array} & \begin{array}{c} \text { Not } \\ \text { Happy } \end{array} \\ \hline 20 \text { up to } 40 & 23 & 50 & 18 \\ \hline 40 \text { up to } 60 & 51 & 38 & 16 \\ \hline 60 \text { and above } & 19 & 45 & 20 \\ \hline \end{array}$$

a. Set up the appropriate hypotheses to test the claim that optimism regarding China's direction depends on the age of the respondent.

b. Calculate the value of the test statistic.

c. Use Excel's CHISQ.DIST.RT function to calculate the $p$-value.

d. At a 1% level of significance, can we infer that optimism among the Chinese is dependent on age?