Do students reduce study time in classes where they achieve a higher midterm score? In a Journal of Economic Education article (Winter $2005$), Gregory Krohn and Catherine O’Connor studied student effort and performance in a class over a semester. In an intermediate macroeconomics course, they found that “students respond to higher midterm scores by reducing the number of hours they subsequently allocate to studying for the course.” Suppose that a random sample of $n = 8$ students who performed well on the midterm exam was taken and weekly study times before and after the exam were compared. The resulting data are given in Table $10.5$. Assume that the population of all possible paired differences is normally distributed.

Table $10.5$ Weekly Study Time Data for Students Who Perform Well on the MidTerm

Students	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Before	$15$	$14$	$17$	$17$	$19$	$14$	$13$	$16$
After	$9$	$9$	$11$	$10$	$19$	$10$	$14$	$10$

Use the p-value to test the hypotheses at the $.10$, $.05$, and $.01$ levels of significance. How much evidence is there against the null hypothesis?

Paired T-Test and CI: StudyBefore, StudyAfter

$$\begin{array}{lrrrr} \text{Paired T for} & & \text{StudyBefore}&\text{- StudyAfter}\\ & \text { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \\ \text { StudyBefore } & 8 & 15.6250 & 1.9955 & 0.7055 \\ \text { StudyAfter } & 8 & 11.5000 & 3.4226 & 1.2101 \\ \text { Difference } & 8 & 4.12500 & 2.99702 & 1.05961\end{array}$$

$95 \%$ CI for mean difference: $(1.61943,6.63057)$ $T$-Test of mean difference $=0$ (vs not $=0$ ): $\mathrm{T}$-Value $=3.89 \quad \mathrm{P}$-Value $=\mathbf{0 . 0 0 6}$

Do male and female students study the same amount per week? In a recent year, 58 sophomore business students were surveyed at a large university that has more than 1,000 sophomore business students each year. The file StudyTime contains the gender and the number of hours spent studying in a typical week for the sampled students.

a. At the 0.05 level of significance, is there a difference in the variance of the study time for male students and female students?

b. Using the results of (a), which t test is appropriate for comparing the mean study time for male and female students?

c. At the 0.05 level of significance, conduct the test selected in (b).

d. Write a short summary of your findings.

In the book Business Research Methods, Donald R. Cooper and C. William Emory ($1995$) discuss a manager who wishes to compare the effectiveness of two methods for training new salespeople. The authors describe the situation as follows:

The company selects $22$ sales trainees who are randomly divided into two experimental groups-one receives type $A$ and the other type $B$ training. The salespeople are then assigned and managed without regard to the training they have received. At the year's end, the manager reviews the performances of salespeople in these groups and finds the following results:

$$\begin{array}{lcc} & \text { A Group } & \text { B Group } \\ \text { Average Weekly Sales } & \$ 1,500 & \$ 1,300 \\ \text { Standard Deviation } & 225 & 251\end{array}$$

Use the equal variances procedure to calculate a $95$ percent confidence interval for the difference between the mean weekly sales obtained when type $A$ training is used and the mean weekly sales obtained when type $B$ training is used. Interpret this interval.

An article in Fortune magazine reported on the rapid rise of fees and expenses charged by mutual funds. Assuming that stock fund expenses and municipal bond fund expenses are each approximately normally distributed, suppose a random sample of $12$ stock funds gives a mean annual expense of $1.63$ percent with a standard deviation of $.31$ percent, and an independent random sample of $12$ municipal bond funds gives a mean annual expense of $0.89$ percent with a standard deviation of $.23$ percent. Let $\mu_1$ be the mean annual expense for stock funds, and let $\mu_2$ be the mean annual expense for municipal bond funds. Do parts $a, b$, and $c$ by using the equal variances procedure. Then repeat $a, b$, and $c$ using the unequal variances procedure. Compare your results.

Set up the null and alternative hypotheses needed to attempt to establish that the mean annual expense for stock funds exceeds the mean annual expense for municipal bond funds by more than $.5$ percent. Test these hypotheses at the $.05$ level of significance. What do you conclude?

How do we decide whether to use a $z$ test or a $t$ test when testing a hypothesis about a population mean?

How to set up the null and alternative hypotheses?

Consolidated Power, a large electric power utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is $60^{\circ} \mathrm{F}$ or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of $100$ temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly (the mean temperature of waste water discharged is $60^{\circ} \mathrm{F}$ or cooler), then the plant must be shut down and appropriate actions must be taken to correct the problem.

Consolidated Power wishes to set up a hypothesis test so that the power plant will be shut down when the null hypothesis is rejected. Set up the null hypothesis $H_0$ and the alternative hypothesis $H_a$ that should be used.

National Paper Company must purchase a new machine for producing cardboard boxes. The company must choose between two machines. The machines produce boxes of equal quality, so the company will choose the machine that produces (on average) the most boxes. It is known that there are substantial differences in the abilities of the company's machine operators. Therefore, National Paper has decided to compare the machines using a paired difference experiment. Suppose that eight randomly selected machine operators produce boxes for one hour using machine 1 and for one hour using machine 2, with the following results: Ds BoxYield

Estimate the minimum and maximum differences between the mean outputs of the two machines. Justify your answer.

National Paper Company must purchase a new machine for producing cardboard boxes. The company must choose between two machines. The machines produce boxes of equal quality, so the company will choose the machine that produces (on average) the most boxes. It is known that there are substantial differences in the abilities of the company's machine operators. Therefore, National Paper has decided to compare the machines using a paired difference experiment. Suppose that eight randomly selected machine operators produce boxes for one hour using machine 1 and for one hour using machine 2, with the following results: Ds BoxYield

Assuming normality, perform a hypothesis test to determine whether there is a difference between the mean hourly outputs of the two machines. Use $\alpha=.05$

Suppose a sample of $49$ paired differences that have been randomly selected from a normally distributed population of paired differences yields a sample mean of $\bar{d}=5$ and a sample standard deviation of $s_d=7$

Test the null hypothesis $H_0: \mu_d=0$ versus the alternative hypothesis $H_a: \mu_d \neq 0$ by setting $\alpha$ equal to $.10$, $.05$, $.01$, and $.001$. How much evidence is there that $\mu_d$ differs from $0$? What does this say about how $\mu_1$ and $\mu_2$ compare?

Suppose a sample of $49$ paired differences that have been randomly selected from a normally distributed population of paired differences yields a sample mean of $\bar{d}=5$ and a sample standard deviation of $s_d=7$

The $p$-value for testing $H_0: \mu_d \leq 3$ versus $H_a: \mu_d>3$ equals .0256. Use the $p$-value to test these hypotheses with $\alpha$ equal to $.10$, $.05$, $.01$, and $.001$. How much evidence is there that $\mu_d$ exceeds $3$? What does this say about the size of the difference between $\mu_1$ and $\mu_2$?

A random sample of $50$ perceived age estimates for a model in a cigarette advertisement showed that the sample mean and sample standard deviation were $26.22$ years and $3.7432$ years, respectively.

Remembering that the cigarette industry requires that models must appear at least $25$ years old, does the confidence interval make us $95$ percent confident that the population mean perceived age estimate is at least $25$? Is the mean perceived age estimate much more than $25$? Explain.

Which of the following is not a component of a time series?

a. trend

b. seasonality

c. cycles

d. all of the above are components of a time series

Consolidated Power, a large electric power utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is $60^{\circ} \mathrm{F}$ or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of $100$ temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly (the mean temperature of waste water discharged is $60^{\circ} \mathrm{F}$ or cooler), then the plant must be shut down and appropriate actions must be taken to correct the problem.

Suppose that Consolidated Power decides to use a level of significance of $\alpha = .05$, and suppose a random sample of $100$ temperature readings is obtained. If the sample mean of the $100$ temperature readings is $60.482$, test $H_0$ versus $H_a$ and determine whether the power plant should be shut down and the cooling system repaired. Perform the hypothesis test by using a critical value and a $p$-value. Assume that the population standard deviation equals $2$.