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Perform the appropriate hypothesis test. A random sample of size \(n=13\) obtained from a population that is normally distributed results in a sample mean of 45.3 and sample standard deviation of \(12.4 .\) An independent sample of size \(n=18\) obtained from a population that is normally distributed results in a sample mean of 52.1 and sample standard deviation of 14.7 . Does this constitute sufficient evidence to conclude that the population means differ at the \(\alpha=0.05\) level of significance?

A random sample of size \(n=41\) results in a sample mean of 125.3 and a sample standard deviation of \(8.5 .\) An independent sample of size \(n=50\) results in a sample mean of 130.8 and sample standard deviation of \(7.3 .\) Does this constitute sufficient evidence to conclude that the population means differ at the \(\alpha=0.01\) level of significance?

The following data represent the muzzle velocity (in feet per second) of rounds fired from a 155 -mm gun. For each round, two measurements of the velocity were recorded using two different measuring devices, with the following data obtained: $$ \begin{array}{ccccccc} \text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline \mathbf{A} & 793.8 & 793.1 & 792.4 & 794.0 & 791.4 & 792.4 \\ \hline \mathbf{B} & 793.2 & 793.3 & 792.6 & 793.8 & 791.6 & 791.6 \\ \hline \end{array} $$ $$ \begin{array}{ccccccc} \text { Observation } & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { A } & 791.7 & 792.3 & 789.6 & 794.4 & 790.9 & 793.5 \\ \hline \text { B } & 791.6 & 792.4 & 788.5 & 794.7 & 791.3 & 793.5 \\ \hline \end{array} $$ (a) Why are these matched-pairs data? (b) Is there a difference in the measurement of the muzzle velocity between device \(A\) and device \(B\) at the \(\alpha=0.01\) level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (c) Construct a \(99 \%\) confidence interval about the population mean difference. Interpret your results. (d) Draw a boxplot of the differenced data. Does this visual evidence support the results obtained in part (b)?

In Problems 9–12, conduct each test at the a = 0.05 level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, (c) the critical value, and (d) the P-value. Assume that the samples were obtained independently using simple random sampling. Test whether \(p_{1}>p_{2}\). Sample data: \(x_{1}=368, n_{1}=541\), \(x_{2}=351, n_{2}=593\)

Walking in the Airport, Part I Do people walk faster in the airport when they are departing (getting on a plane) or when they are arriving (getting off a plane)? Researcher Seth B. Young measured the walking speed of travelers in San Francisco International Airport and Cleveland Hopkins International Airport. His findings are summarized in the table. $$ \begin{array}{lcc} \text { Direction of Travel } & \text { Departure } & \text { Arrival } \\ \hline \text { Mean speed (feet per minute) } & 260 & 269 \\ \hline \begin{array}{l} \text { Standard deviation } \\ \text { (feet per minute) } \end{array} & 53 & 34 \\ \hline \text { Sample size } & 35 & 35 \\ \hline \end{array} $$ (a) Is this an observational study or a designed experiment? Why? (b) Explain why it is reasonable to use Welch's \(t\) -test. (c) Do individuals walk at different speeds depending on whether they are departing or arriving at the \(\alpha=0.05\) level of significance?

A Secchi disk is an 8 -inch-diameter weighted disk that is painted black and white and attached to a rope. The disk is lowered into water and the depth (in inches) at which it is no longer visible is recorded. The measurement is an indication of water clarity. An environmental biologist interested in determining whether the water clarity of the lake at Joliet Junior College is improving takes measurements at the same location on eight dates during the course of a year and repeats the measurements on the same dates five years later. She obtains the following results: $$ \begin{array}{lcccccccc} \text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \text { Date } & \mathbf{5 / 1 1} & \mathbf{6 / 7} & \mathbf{6 / 2 4} & \mathbf{7 / 8} & \mathbf{7 / 2 7} & \mathbf{8 / 3 1} & 9 / 30 & \mathbf{1 0 / 1 2} \\ \hline \begin{array}{l} \text { Initial } \\ \text { depth, } X_{i} \end{array} & 38 & 58 & 65 & 74 & 56 & 36 & 56 & 52 \\ \hline \begin{array}{l} \text { Depth five } \\ \text { years later, } Y_{i} \end{array} & 52 & 60 & 72 & 72 & 54 & 48 & 58 & 60 \\ \hline \end{array} $$ (a) Why is it important to take the measurements on the same date? (b) Does the evidence suggest that the clarity of the lake is improving at the \(\alpha=0.05\) level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (c) Draw a boxplot of the differenced data. Does this visual evidence support the results obtained in part (b)?

Walking in the Airport, Part II Do business travelers walk at a different pace than leisure travelers? Researcher Seth B. Young measured the walking speed of business and leisure travelers in San Francisco International Airport and Cleveland Hopkins International Airport. His findings are summarized in the table. $$ \begin{array}{lcc} \text { Type of Traveler } & \text { Business } & \text { Leisure } \\ \hline \text { Mean speed (feet per minute) } & 272 & 261 \\ \hline \begin{array}{l} \text { Standard deviation (feet per } \\ \text { minute) } \end{array} & 43 & 47 \\ \hline \text { Sample size } & 20 & 20 \\ \hline \end{array} $$ (a) Is this an observational study or a designed experiment? Why? (b) What must be true regarding the populations to use Welch's \(t\) -test to compare the means? (c) Assuming that the requirements listed in part (b) are satisfied, determine whether business travelers walk at a different speed from leisure travelers at the \(\alpha=0.05\) level of significance.

What is the typical age difference between husband and wife? The following data represent the ages of husbands and wives, based on results from the Current Population Survey. (a) What is the response variable in this study? (b) Is the sampling method dependent or independent? Explain. (c) Use the data to estimate the mean difference in age of husband and wives with \(95 \%\) confidence. Explain the technique that you used.

To illustrate the effects of driving under the influence (DUI) of alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine teenagers, the time (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. $$ \begin{array}{lccccccccc} \text { Subject } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline \text { Normal, } X_{i} & 4.47 & 4.24 & 4.58 & 4.65 & 4.31 & 4.80 & 4.55 & 5.00 & 4.79 \\ \hline \text { Impaired, } Y_{i} & 5.77 & 5.67 & 5.51 & 5.32 & 5.83 & 5.49 & 5.23 & 5.61 & 5.63 \\ \hline \end{array} $$ (a) Whether the student had unimpaired vision or wore goggles first was randomly selected. Why is this a good idea in designing the experiment? (b) Use a \(95 \%\) confidence interval to test if there is a difference in braking time with impaired vision and normal vision where the differences are computed as "impaired minus normal." Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Clifford Adelman, a researcher with the Department of Education, followed a cohort of students who graduated from high school in \(1992,\) monitoring the progress the students made toward completing a bachelor's degree. One aspect of his research was to compare students who first attended a community college to those who immediately attended and remained at a four-year institution. The sample standard deviation time to complete a bachelor's degree of the 268 students who transferred to a four-year school after attending community college was \(1.162 .\) The sample standard deviation time to complete a bachelor's degree of the 1145 students who immediately attended and remained at a four-year institution was \(1.015 .\) Assuming the time to earn a bachelor's degree is normally distributed, does the evidence suggest the standard deviation time to earn a bachelor's degree is different between the two groups? Use the \(\alpha=0.05\) level of significance.