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Chapter 10: Problem 61
Construct a \(95 \%\) confidence interval for \(p_{1}-p_{2}\) for the following. $$ n_{1}=100, \quad \hat{p}_{1}=.81, \quad n_{2}=150, \quad \hat{p}_{2}=.77 $$
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A factory that emits airbome pollutants is testing two different brands of filters for its smokestacks. The factory has two smokestacks. One brand of filter (Filter I) is placed on one smokestack, and the other brand (Filter II) is placed on the second smokestack. Random samples of air released from the smokestacks are taken at different times throughout the day. Pollutant concentrations are measured from both stacks at the same time. The following data represent the pollutant concentrations (in parts per million) for samples taken at 20 different times after passing through the filters. Assume that the differences in concentration levels at all times are approximately normally distributed. \begin{tabular}{cccccc} \hline Time & Filter I & Filter II & Time & Filter I & Filter II \\ \hline 1 & 24 & 26 & 11 & 11 & 9 \\ 2 & 31 & 30 & 12 & 8 & 10 \\ 3 & 35 & 33 & 13 & 14 & 17 \\ 4 & 32 & 28 & 14 & 17 & 16 \\ 5 & 25 & 23 & 15 & 19 & 16 \\ 6 & 25 & 28 & 16 & 19 & 18 \\ 7 & 29 & 24 & 17 & 25 & 27 \\ 8 & 30 & 33 & 18 & 20 & 22 \\ 9 & 26 & 22 & 19 & 23 & 27 \\ 10 & 18 & 18 & 20 & 32 & 31 \\ \hline \end{tabular} a. Make a \(95 \%\) confidence interval for the mean of the population paired differences, where a paired difference is equal to the pollutant concentration passing through Filter I minus the pollutant concentration passing through Filter II. b. Using a \(5 \%\) significance level, can you conclude that the average paired difference for concentration levels is different from zero??
The owner of a mosquito-infested fishing camp in Alaska wants to test the effectiveness of two rival brands of mosquito repellents, \(\mathrm{X}\) and \(\mathrm{Y}\). During the first month of the season, eight people are chosen at random from those guests who agree to take part in the experiment. For each of these guests, Brand \(\bar{X}\) is randomly applied to one arm and Brand \(\mathrm{Y}\) is applied to the other arm. These guests fish for 4 hours, then the owner counts the number of bites on each arm. The table below shows the number of bites on the arm with Brand \(X\) and those on the arm with Brand \(Y\) for each guest. \begin{tabular}{l|rrrrrrrr} \hline Guest & A & B & C & D & E & F & G & H \\ \hline Brand X & 12 & 23 & 18 & 36 & 8 & 27 & 22 & 32 \\ \hline Brand Y & 9 & 20 & 21 & 27 & 6 & 18 & 15 & 25 \\ \hline \end{tabular} a. Construct a \(95 \%\) confidence interval for the mean \(\mu_{d}\) of population paired differences, where a paired difference is defined as the number of bites on the arm with Brand \(X\) minus the number of bites on the arm with Brand \(Y\). b. Test at a \(5 \%\) significance level whether the mean number of bites on the arm with Brand \(\mathrm{X}\) and the mean number of bites on the arm with Brand \(Y\) are different for all such guests.
Manufacturers of two competing automobile models, Gofer and Diplomat, each claim to have the lowest mean fuel consumption. Let \(\mu_{1}\) be the mean fuel consumption in miles per gallon (mpg) for the Gofer and \(\mu_{2}\) the mean fuel consumption in mpg for the Diplomat. The two manufacturers have agreed to a test in which several cars of each model will be driven on a 100 -mile test run. Then the fuel consumption, in mpg, will be calculated for each test run. The average of the \(m p g\) for all 100 -mile test runs for each model gives the corresponding mean. Assume that for each model the gas mileages for the test runs are normally distributed with \(\sigma=2 \mathrm{mpg} .\) Note that each car is driven for one and only one 100 -mile test run. a. How many cars (i.e., sample size) for each model are required to estimate \(\mu_{1}-\mu_{2}\) with a \(90 \%\) confidence level and with a margin of error of estimate of \(1.5 \mathrm{mpg}\) ? Use the same number of cars (i.c., sample size) for each model. b. If \(\mu_{1}\) is actually \(33 \mathrm{mpg}\) and \(\mu_{2}\) is actually \(30 \mathrm{mpg}\), what is the probability that five cars for each model would yield \(\bar{x}_{1} \geq \bar{x}_{2}\) ?
Refer to Exercise \(10.95 .\) Suppose Gamma Corporation decides to test govemors on seven cars. However, the management is afraid that the speed limit imposed by the governors will reduce the number of contacts the salespersons can make each day. Thus, both the fuel consumption and the number of contacts made are recorded for each car/salesperson for each week of the testing period, both before and after the installation of governors. \begin{tabular}{c|cc|cc} \hline \multirow{2}{*} { Salesperson } & \multicolumn{2}{|c|} { Number of Contacts } & \multicolumn{2}{|c} { Fuel Consumption (mpg) } \\ \cline { 2 - 5 } & Before & After & Before & After \\ \hline A & 50 & 49 & 25 & 26 \\ B & 63 & 60 & 21 & 24 \\ C & 42 & 47 & 27 & 26 \\ D & 55 & 51 & 23 & 25 \\ E & 44 & 50 & 19 & 24 \\ F & 65 & 60 & 18 & 22 \\ G & 66 & 58 & 20 & 23 \\ \hline \end{tabular} Suppose that as a statistical analyst with the company, you are directed to prepare a brief report that includes statistical analysis and interpretation of the data. Management will use your report to help decide whether or not to install governors on all salespersons' cars. Use \(90 \%\) confidence intervals and 05 significance levels for any hypothesis tests to make suggestions. Assume that the differences in fuel consumption and the differences in the number of contacts are both normally distributed.
Explain what conditions must hold true to use the \(t\) distribution to make a confidence interval and to test a hypothesis about \(\mu_{1}-\mu_{2}\) for two independent samples selected from two populations with unknown but equal standard deviations.
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