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	$9$	$9$	$11$	$10$	$19$	$10$	$14$	$10$

Set up the null and alternative hypotheses to test whether there is a difference in the population mean study time before and after the midterm exam.

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 $\mu_2$?

Suppose a sample of $11$ paired differences that has been randomly selected from a normally distributed population of paired differences yields a sample mean of $103.5$ and a sample standard deviation of $5.$

Test the null hypothesis $H_0: \mu_d \geq 110$ versus $H_a: \mu_d<110$ by setting $\alpha$ equal to $.05$ and $.01$. How much evidence is there that $\mu_d=\mu_1-\mu_2$ is less than $110$?

Suppose a sample of $11$ paired differences that has been randomly selected from a normally distributed population of paired differences yields a sample mean of $103.5$ and a sample standard deviation of $5.$

Test the null hypothesis $H_0: \mu_d \leq 100$ versus $H_a: \mu_d>100$ by setting $\alpha$ equal to $.05$ and $.01$. How much evidence is there that $\mu_d=\mu_1-\mu_2$ exceeds $100$?

Suppose a sample of $11$ paired differences that has been randomly selected from a normally distributed population of paired differences yields a sample mean of $103.5$ and a sample standard deviation of $5.$

Calculate $95$ percent and $99$ percent confidence intervals for $\mu_d=\mu_1-\mu_2$.

Recall that the cigarette industry requires that models in cigarette ads must appear to be at least $25$ years old. Also recall that a sample of $50$ people is randomly selected at a shopping mall. Each person in the sample is shown a "typical cigarette ad" and is asked to estimate the age of the model in the ad.

Suppose that a random sample of $50$ perceived age estimates gives a mean of $23.663$ years and a standard deviation of $3.596$ years. Use these sample data and critical values to test the hypotheses of part $a$ at the $.10$, $.05$, $.01$, and $.001$ levels of significance.

Recall that the cigarette industry requires that models in cigarette ads must appear to be at least $25$ years old. Also recall that a sample of $50$ people is randomly selected at a shopping mall. Each person in the sample is shown a "typical cigarette ad" and is asked to estimate the age of the model in the ad.

Let $\mu$ be the mean perceived age estimate for all viewers of the ad, and suppose we consider the industry requirement to be met if $\mu$ is at least $25$. Set up the null and alternative hypotheses needed to attempt to show that the industry requirement is not being met.

The Crown Bottling Company has just installed a new bottling process that will fill $16$-ounce bottles of the popular Crown Classic Cola soft drink. Both overfilling and underfilling bottles are undesirable: Underfilling leads to customer complaints and overfilling costs the company considerable money. In order to verify that the filler is set up correctly, the company wishes to see whether the mean bottle fill, $\mu$, is close to the target fill of $16$ ounces. To this end, a random sample of $36$ filled bottles is selected from the output of a test filler run. If the sample results cast a substantial amount of doubt on the hypothesis that the mean bottle fill is the desired $16$ ounces, then the filler's initial setup will be readjusted.

In the context of this situation, interpret making a Type I error; interpret making a Type II error.

The Crown Bottling Company has just installed a new bottling process that will fill $16$-ounce bottles of the popular Crown Classic Cola soft drink. Both overfilling and underfilling bottles are undesirable: Underfilling leads to customer complaints and overfilling costs the company considerable money. In order to verify that the filler is set up correctly, the company wishes to see whether the mean bottle fill, $\mu$, is close to the target fill of $16$ ounces. To this end, a random sample of $36$ filled bottles is selected from the output of a test filler run. If the sample results cast a substantial amount of doubt on the hypothesis that the mean bottle fill is the desired $16$ ounces, then the filler's initial setup will be readjusted.

The bottling company wants to set up a hypothesis test so that the filler will be readjusted if the null hypothesis is rejected. Set up the null and alternative hypotheses for this hypothesis test.

Consider the cigarette ad situation discussed before. Using the sample information given in that exercise, the $p$-value for testing $H_0$ versus $H_a$ can be calculated to be $.0057$.

Describe how much evidence we have that the industry requirement is not being met.

The Crown Bottling Company has just installed a new bottling process that will fill $16$-ounce bottles of the popular Crown Classic Cola soft drink. Both overfilling and underfilling bottles are undesirable: Underfilling leads to customer complaints and overfilling costs the company considerable money. In order to verify that the filler is set up correctly, the company wishes to see whether the mean bottle fill, $\mu$, is close to the target fill of $16$ ounces. To this end, a random sample of $36$ filled bottles is selected from the output of a test filler run. If the sample results cast a substantial amount of doubt on|the hypothesis that the mean bottle fill is the desired $16$ ounces, then the filler's initial setup will be readjusted.

The bottling company wants to set up a hypothesis test so that the filler will be readjusted if the null hypothesis is rejected. Set up the null and alternative hypotheses for this hypothesis test.