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?

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?

A marketing agency would like to determine if there is a relationship between union membership and type of vehicle owned (domestic or foreign brand). The goal is to develop targeted advertising campaigns for particular vehicle brands likely to appeal to specific groups of customers. A survey of $500$ potential customers revealed the following results.

$$\begin{array}{|l|c|c|} \hline & \text { Union Member } & \begin{array}{c} \text { Not Union } \\ \text { Member } \end{array} \\ \hline \text { Domestic brand } & 133 & 147 \\ \hline \text { Foreign brand } & 67 & 153 \\ \hline \end{array}$$

a. Specify the competing hypotheses to determine whether vehicle brand (domestic, foreign) is associated with union membership.

b. Use the critical value approach to conduct the test at the 10% significance level. What is your conclusion?

c. Is the conclusion reached in part (b) sensitive to the choice of significance level?

He also calculates the following summary statistics over this time period:

$$\begin{array}{|c|c|c|c|c|} \hline \text { Mean } & \text { Median } & \text { Standard Deviation } & \text { Skewness } & \text { Kurtosis } \\ \hline 1.50 \% & 1.31 \% & 6.98 \% & 0.11 & -0.33 \\ \hline \end{array}$$

Perform the Jarque-Bera test in order to determine whether the monthly stock returns are not normally distributed at the 5% significance level.

$$\begin{array}{|l|c|c|} \hline \text { Importance of Religion } & \text { U.S. Results } & \begin{array}{c} \text { Responses of } \\ \text { Massachusetts Residents } \end{array} \\ \hline \text { Very important } & 0.58 & 160 \\ \hline \text { Somewhat important } & 0.25 & 140 \\ \hline \text { Not too important } & 0.15 & 96 \\ \hline \text { Don't know } & 0.02 & 4 \\ \hline \end{array}$$

Discuss whether you would expect to find the same conclusions if you conducted a similar test for the state of Utah or states in the Southern Belt of the United States.

Consider a multinomial experiment with $n=400$ and $k=3$. The null hypothesis is $H_0: p_1=0.60, p_2=0.25$, and $p_3=0.15$. The observed frequencies resulting from the experiment are

$$\begin{array}{|l|r|r|r|} \hline \text { Category } & 1 & 2 & 3 \\ \hline \text { Frequency } & 250 & 94 & 56 \\ \hline \end{array}$$

a. Define the alternative hypothesis.

b. Calculate the value of the test statistic and approximate the $p$-value for the test.

c. At the 5% significance level, what is the conclusion to the hypothesis test?