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Chapter 10: Problem 96

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.

Short Answer

Expert verified
After the analysis, we make decision based on the statistical results. If there is a statistically significant difference in the number of contacts and fuel consumption, management should consider whether the effect of governors is beneficial. If zero falls within the 90% confidence interval for both metrics, the addition of governors may not significantly impact salespersons' productivity or fuel efficiency.

Step by step solution

01

Calculate Differences

First, calculate the differences in number of contacts and fuel consumption before and after installing governors for each salesperson. Let's denote the difference in number of contacts as \(d_c\) and the difference in fuel consumption as \(d_f\). For example, for salesperson A, \(d_c\) would be 50 - 49 = 1, and \(d_f\) would be 26 - 25 = 1.
02

Perform Hypothesis Testing

Once the differences are calculated, we can perform Hypothesis Testing for each data to check whether the difference is significant or not. Our null hypothesis \(H_0\) is that the mean difference is equal to zero, while the alternative hypothesis \(H_1\) is that the mean difference is not equal to zero. If the p-value associated with t-statistic obtained is less than 0.05, then we reject the null hypothesis.
03

Construct Confidence Interval

After the hypothesis test, construct a 90% confidence interval for the mean difference in contacts and fuel consumption. This is done by finding the mean difference and adding or subtracting the product of the standard deviation and the t-value for a 90% confidence interval. If zero falls within the confidence interval, we fail to reject the null hypothesis.
04

Decision Making

If we fail to reject the null hypothesis for both the number of contacts and fuel consumption, then the governors' influence on these parameters is not significant. Hence, if installing governors does not significantly impact the salespersons' ability to make contacts while still reducing fuel consumption, the governors could be implemented across the board.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
A confidence interval is a range of values, derived from sample data, which is likely to contain the value of an unknown population parameter. In fuel consumption analysis, we are interested in the mean difference between conditions before and after installing governors. By constructing a 90% confidence interval, we can say with 90% confidence that the true mean difference in contacts or fuel consumption lies within this range.

This interval is calculated using the sample mean difference, the standard deviation of the differences, and a critical value from the t-distribution corresponding to the 90% confidence level.
  • If zero is within this interval, it suggests that there is no significant difference in mean fuel consumption or contacts.
  • If zero is not within this interval, there is a significant difference indicating that the installation may have an impact.
Calculating a confidence interval helps to understand the scope and reliability of the observed changes in the data.
Null Hypothesis
When performing hypothesis testing, the first step is to establish a null hypothesis. A null hypothesis posits that there is no effect or no difference. In the context of this exercise:
  • The null hypothesis (\(H_0\)) for number of contacts states that the mean difference between the number of contacts before and after installing governors is zero.
  • Similarly, the null hypothesis for fuel consumption states that the mean difference in fuel consumption is zero.
Testing the null hypothesis helps us determine if observed differences are due to random fluctuations or if they are statistically significant enough to suggest a real effect of the governors.
Fuel Consumption Analysis
Analyzing changes in fuel consumption is an essential part of understanding the implications of installing governors on vehicles. Here, the focus is on evaluating whether the governors lead to a reduction in fuel consumption.

By calculating the mean difference in mpg (miles per gallon) before and after the governors are installed and comparing this with the expected random variation using statistical tools, we can determine if the governors help reduce fuel consumption efficiently.
  • A reduction without affecting the salesperson's performance in making contacts could be considered beneficial.
  • Both quantitative data on fuel consumption and qualitative implications for sales efforts are looked at together to form a comprehensive view.
Statistical Significance
Statistical significance provides a measure of whether results could have happened by chance. When a p-value from hypothesis testing is below a specified level (e.g., 0.05), we may claim statistical significance. In this exercise:
  • If the p-value for the fuel consumption difference is less than 0.05, we have significant evidence that the governors impact fuel consumption.
  • If it is greater than 0.05, the difference isn鈥檛 statistically significant, indicating no important effect.
Essentially, lowercase p-values mean that the observed effects are likely genuine and not simply due to random chance. These findings guide decision-making by indicating whether action should be taken based on the statistical results.
Mean Difference
Mean difference involves calculating the average change in a particular measure from before to after an intervention. In this case, it measures changes in contacts made and fuel consumption across salespersons due to the governors.

To find the mean difference:
  • Calculate the difference for each salesperson.
  • Average these differences to find the mean difference for the group.
The resulting values help assess the overall effect of the intervention, providing an easy-to-understand metric for management teams analyzing impacts, like whether the governors perhaps influence fuel efficiency positively while not drastically reducing contacts.

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Most popular questions from this chapter

Gamma Corporation is considering the installation of govemors on cars driven by its sales staff. These devices would limit the car speeds to a preset level, which is expected to improve fuel economy. The company is planning to test several cars for fuel consumption without governors for 1 week. Then governors would be installed in the same cars, and fuel consumption will be monitored for another week. Gamma Corporation wants to estimate the mean difference in fuel consumption with a margin of error of estimate of 2 mpg with a 90 \% confidence level. Assume that the differences in fuel consumption are normally distributed and that previous studies suggest that an estimate of \(s_{d}=3 \mathrm{mpg}\) is reasonable. How many cars should be tested? (Note that the critical value of \(t\) will depend on \(n\), so it will be necessary to use trial and error.)

The Bath Heritage Days, which take place in Bath, Maine, have been popular for, among other things, an cating contest. In 2009 , the contest switched from blueberry pie to a Whoopie Pie, which consists of two large, chocolate cake-like cookies filled with a large amount of vanilla cream. Suppose the contest involves eating nine Whoopie Pies, each weighing \(1 / 3\) pound. The following data represent the times (in seconds) taken by cach of the 13 contestants (all of whom finished all nine Whoopie Pies) to eat the first Whoopie Pie and the last (ninth) Whoopie Pie. \begin{tabular}{l|rrrrrrrrrrrrr} \hline Contestant & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) & \(\mathbf{5}\) & \(\mathbf{6}\) & \(\mathbf{7}\) & \(\mathbf{8}\) & \(\mathbf{9}\) & \(\mathbf{1 0}\) & \(\mathbf{1 1}\) & \(\mathbf{1 2}\) & \(\mathbf{1 3}\) \\ \hline First pie & 49 & 59 & 66 & 49 & 63 & 70 & 77 & 59 & 64 & 69 & 60 & 58 & 71 \\ \hline Last pie & 49 & 74 & 92 & 93 & 91 & 73 & 103 & 59 & 85 & 94 & 84 & 87 & 111 \\ \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 time taken to eat the ninth pie (which is the last pie) minus the time taken to cat the first pie. b. Using a \(10 \%\) significance level, can you conclude that the average time taken to eat the ninth pie (which is the last pie) is at least 15 seconds more than the average time taken to eat the first pie.

The global recession has led more and more people to move in with relatives, which has resulted in a large number of multigenerational households. An October 2011 Pew Research Center poll showed that \(11.5 \%\) of people living in multigenerational households were living below the poverty level, and 14.6\% of people living in other types of households were living below the poverty level (www. pewsocialtrends.org/201 1/10/03/fighting-poverty-in-a-bad- cconomy-americans-move-in-with-relatives/? sre-pre-headline). Suppose that these results were based on samples of 1000 people living in multigenerational households and 2000 people living in other types of households. a. Let \(p_{1}\) be the proportion of all people in multigenerational households who live below the poverty level and \(p_{2}\) be the proportion of all people in other types of households who live below the poverty level. Construct a 98\% confidence interval for \(p_{1}-p_{2}\) - b. Using a \(2.5 \%\) significance level, can you conclude that \(p_{1}\) is less than \(p_{2}\) ? Use the critical-value approach. c. Repeat part b using the \(p\) -value approach.

Maine Mountain Dairy claims that its 8-ounce low-fat yogurt cups contain, on average, fewer calories than the 8 -ounce low-fat yogurt cups produced by a competitor. A consumer agency wanted to check this claim. A sample of 27 such yogurt cups produced by this company showed that they contained an average of 141 calories per cup. A sample of 25 such yogurt cups produced by its competitor showed that they contained an average of 144 calories per cup. Assume that the two populations are normally distributed with population standard deviations of \(5.5\) and \(6.4\) calories, repectively. a. Make a \(98 \%\) confidence interval for the difference between the mean number of calories in the 8-ounce low-fat yogurt cups produced by the two companies. b. Test at a \(1 \%\) significance level whether Maine Mountain Dairy's claim is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=05\) ? What if \(\alpha=.025\) ?

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the population means. b. Test at a 2.5\% significance level whether the claim of the management of the New Century Bank is truc. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=01 ?\) What if \(\alpha=05\) ?
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