In a study to determine whether counseling could help people lose weight, a sample of people experienced a group-based behavioral intervention, which involved weekly meetings with a trained interventionist for a period of six months. The following data are the numbers of pounds lost for $14$ people, based on means and standard deviations given in the article.

$$\begin{array}{rrrrrrr} \hline 18.2 & 24.8 & 3.9 & 20.0 & 17.1 & 8.8 & 13.4\\ 17.3 & 33.8 & 29.7 & 8.5 & 31.2 & 19.3 & 15.1\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ A boxplot shows these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain.

$\textbf{b.}\hspace{10pt}$ If appropriate, perform a hypothesis test to determine whether the mean weight loss is greater than $10$ pounds. Use the $\alpha = 0.05$ level of significance. What do you conclude?

The following MINITAB output presents the results of a hypothesis test for a population mean $\mu.$ $$ \begin{matrix} \text{One-Sample Z: X}\\ \text{Test of mu = 73.5 vs not = 73.5}\\ \text{The assumed standard deviation = 2.3634}\\ \text{Variable} & \text{N} & \text{Mean} & \text{StDev} & \text{SE Mean} & \text{95\\% CI} & \text{Z} & \text{P}\\ \text{X} & \text{145} & \text{73.2461} & \text{2.3634} & \text{0.1963} & \text{(72.8614, 73.6308)} & \text{−1.29} & \text{0.196}\\ \end{matrix} $$ a. Is this a one-tailed or two-tailed test? b. What is the null hypothesis? c. What is the P-value? d. Use the output and an appropriate table to compute the P-value for the test of $H_{0}: \mu \geq 73.6$ versus $H_{1}: \mu<73.6.$ e. Use the output and an appropriate table to compute a 99% confidence interval for $\mu.$

A psychologist is designing an experiment in which rats will navigate a maze. Ten rats run the maze, and the time it takes for each to complete the maze is recorded. The results are as follows.

$$\begin{array}{rrrrrrrr} \hline 66.3 & 68.1 & 52.5 & 68.3 & 62.6\\ 55.6 & 42.1 & 60.9 & 69.2 & 69.3\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ A boxplot shows these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain.

$\textbf{b.}\hspace{10pt}$ The psychologist hopes that the mean time for a rat to run the maze will be greater than $60$ seconds. If appropriate, perform a hypothesis test to determine whether the mean time is greater than $60$ seconds.

The market research firm Salary.com reported that the mean annual earnings of all family practitioners in the United States was $\$178,258$. A random sample of $55$ family practitioners in Los Angeles that month had mean earnings of $\overline{x} = \$192,340$ with a standard deviation of $\$42,387$. Do the data provide sufficient evidence to conclude that the mean salary for family practitioners in Los Angeles is greater than the national average?

$\textbf{a.}\hspace{10pt}$ State the null and alternate hypotheses.

$\textbf{b.}\hspace{10pt}$ Compute the value of the $t$ statistic. How many degrees of freedom are there?

$\textbf{c.}\hspace{10pt}$ State your conclusion. Use the $\alpha = 0.05$ level of significance.

Are older men shorter than younger men? According to the National Health Statistics Reports, the mean height for U.S. men is $69.4$ inches. In a sample of $300$ men between the ages of $60$ and $69$, the mean height was $\overline{x} = 69.0$ inches. Public health officials want to determine whether the mean height $\mu$ for older men is less than the mean height of all adult men.

$\textbf{a.}\hspace{10pt}$ State the appropriate null and alternate hypotheses.

$\textbf{b.}\hspace{10pt}$ Assume the population standard deviation to be $\sigma = 2.84$ inches. Compute the value of the test statistic.

$\textbf{c.}\hspace{10pt}$ State a conclusion. Use the $\alpha = 0.01$ level of significance

A test will be made of $H_0: \mu = 4$ versus $H_1: \mu \gt 4$, using a sample of size $25$. The population standard deviation is unknown. Find the critical value of the test statistic if the significance level is $\alpha = 0.05$.

Find the critical value or values for the following values of the significance level $\alpha$, sample size $n$, and alternate hypothesis $H_1$.

$$\textbf{a.}\hspace{10pt} \alpha = 0.05, n = 27, H_1: \mu \ne \mu_0$$

$$\textbf{b.}\hspace{10pt} \alpha = 0.01, n = 61, H_1: \mu \gt \mu_0$$

$$\textbf{c.}\hspace{10pt} \alpha = 0.10, n = 16, H_1: \mu \ne \mu_0$$

$$\textbf{d.}\hspace{10pt} \alpha = 0.05, n = 25, H_1: \mu \lt \mu_0$$

A study conducted in $2015$ by the technology company Rovi (now TiVo) showed that the mean time spent per day browsing the video streaming service Netflix for something to watch was $19.3$ minutes. Assume the standard deviation is $\sigma = 8$. Suppose a simple random sample of $100$ visits taken this year has a sample mean of $\overline{x} = 21.5$ minutes. A social scientist is interested to know whether the mean time browsing Netflix has increased.

a. State the appropriate null and alternate hypotheses.

b. Compute the value of the test statistic.

c. State a conclusion. Use the $\alpha = 0.05$ level of significance.

Find the $P$-value for the following values of the test statistic $t$, sample size $n$, and alternate hypothesis $H_1$. If you use the table of Critical values for the Students's $t$ distribution, you may specify that $P$ is between two values.

a. $t = 2.584, n = 12, H_1: \mu \gt \mu_0$

b. $t = −1.741, n = 21, H_1: \mu \lt \mu_0$

c. $t = 3.031, n = 14, H_1: \mu \ne \mu_0$

d. $t = −2.584, n = 31, H_1: \mu \ne \mu_0$

A fast-food restaurant chain has 600 outlets in the United States. The following table categorizes cities by size and location, and presents the number of restaurants in cities of each category. A restaurant is to be chosen at random from the 600 to test market a new style of chicken.

$$\begin{array}{lrrrr} \hline & & & {\text { Region }} \\ \hline \text { Population of City } & \text { NE } & \text { SE } & \text { SW } & \text { NW } \\ \hline \text { Under } 50,000 & 30 & 35 & 15 & 5 \\ 50,000-500,000 & 60 & 90 & 70 & 30 \\ \text { Over } 500,000 & 150 & 25 & 30 & 60 \\ \hline \end{array}$$

a. If the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast? b. If the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 50,000? c. If the restaurant is located in the Southwest, what is the probability that it is in a city with a population of 500,000 or less? d. If the restaurant is located in a city with a population of 500,000 or less, what is the probability that is in the Southwest? e. If the restaurant is located in the South (either SE or SW), what is the probability that it is in a city with a population of 50,000 or more?

For a sample of nine automobiles, the mileage (in 1000s of miles) at which the original front brake pads were worn to 10% of their original thickness was measured, as was the mileage at which the original rear brake pads were worn to 10% of their original thickness. The results are given in the following table.

$$\begin{matrix} \text{Automobile} & \text{Front} & \text{Rear}\\ \text{1} & \text{32.8} & \text{41.2}\\ \text{2} & \text{26.6} & \text{35.2}\\ \text{3} & \text{35.6} & \text{46.1}\\ \text{4} & \text{36.4} & \text{46.0}\\ \text{5} & \text{29.2} & \text{39.9}\\ \text{6} & \text{40.9} & \text{51.7}\\ \text{7} & \text{40.9} & \text{51.6}\\ \text{8} & \text{34.8} & \text{46.1}\\ \text{9} & \text{36.6} & \text{47.3}\\ \end{matrix}$$

Find a 95% confidence interval for the difference in mean lifetime between the front and rear brake pads.

A lot of 10 components contains 3 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective. a. Find P(A). b. Find P(B|A). c. Find $P(A \cap B).$ d. Find $P\left(A^{c} \cap B\right).$ e. Find P(B). f. Are A and B independent? Explain.

Approximate the sample variance and standard deviation given the following frequency distribution:

$$\begin{array}{cr} \hline \text{Class} & \text{Frequency}\\ \hline 0 - 15 & 2\hspace{15pt}\\ 16 - 31 & 14\hspace{15pt}\\ 32 - 47 & 6\hspace{15pt}\\ 48 - 63 & 13\hspace{15pt}\\ 64 - 79 & 15\hspace{15pt}\\ \hline \end{array}$$

A normal distribution has mean $\mu = 60$ and standard deviation $\sigma = 20$. Find and interpret the z-score for $x = 75$.