A regional package delivery company is considering changing...

A regional package delivery company is considering changing from full-size vans to minivans. The company sampled minivans from each of three manufacturers. The number sampled represents the number the manufacturer was able to provide for the test. Each minivan was driven for 5,000 miles, and the operating cost per mile was computed. The operating costs, in cents per mile, for the 12 are provided in the data file called Delivery: a. Perform an analysis of variance on these data. Assume a significance level of 0.05 . Do the experimental data provide evidence that the average operating costs per mile for the three types of minivans are different? Use a $p$-value approach. b. Referring to part a, based on the sample data and the appropriate test for multiple comparisons, what conclusions should be reached concerning which type of car the delivery company should adopt? Discuss and prepare a report to the company CEO. Use $\alpha=0.05$. c. Provide an estimate of the maximum and minimum difference in average savings per year if the CEO chooses the "best" versus the "worst" minivan using operating costs as a criterion. Assume that minivans are driven 30,000 miles a year. Use a $90 \%$ confidence interval.

A regional package delivery company is considering changing from full-size vans to minivans. The company sampled minivans from each of three manufacturers. The number sampled represents the number the manufacturer was able to provide for the test. Each minivan was driven for 5,000 miles, and the operating cost per mile was computed. The operating costs, in cents per mile, for the 12 are provided in the data file called Delivery:
a. Perform an analysis of variance on these data. Assume a significance level of 0.05 . Do the experimental data provide evidence that the average operating costs per mile for the three types of minivans are different? Use a $p$-value approach.
b. Referring to part a, based on the sample data and the appropriate test for multiple comparisons, what conclusions should be reached concerning which type of car the delivery company should adopt? Discuss and prepare a report to the company CEO. Use $\alpha=0.05$.
c. Provide an estimate of the maximum and minimum difference in average savings per year if the CEO chooses the "best" versus the "worst" minivan using operating costs as a criterion. Assume that minivans are driven 30,000 miles a year. Use a $90 \%$ confidence interval.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of three or more groups to determine if at least one group mean differs significantly from the others. It partitions the overall variation in the data into variation between groups and within groups, allowing analysts to assess group differences with respect to a quantitative outcome. This method is essential in evaluating whether observed differences are likely due to random variation or to actual differences in group performance.

Hypothesis Testing Framework

This concept refers to the process of formulating a null hypothesis, which typically states that there is no difference among group means, and an alternative hypothesis that suggests at least one group mean is different. In the context of ANOVA, hypothesis testing involves calculating an F-statistic and comparing it with a critical value or using a p-value to decide whether to reject the null hypothesis. This framework is fundamental for statistically validating differences observed in experimental data.

p-value Approach

The p-value approach involves computing the probability of obtaining an effect at least as extreme as the one observed, assuming the null hypothesis is true. In the context of ANOVA, a p-value less than the chosen level of significance (commonly alpha = 0.05) indicates strong evidence against the null hypothesis, suggesting that differences among the group means are statistically significant. This approach provides a continuous measure of evidence and is widely used in statistical decision-making.

Multiple Comparisons Procedures

When an overall test like ANOVA indicates significant differences among groups, multiple comparisons procedures are used to identify which specific groups differ from each other. These methods adjust for the increased risk of Type I errors that occur when making several pairwise comparisons. Techniques such as Tukey's HSD, Bonferroni correction, or Scheffé's method ensure that the conclusions drawn about differences between individual groups are statistically valid.

Confidence Intervals

Confidence intervals provide a range of plausible values for a population parameter, such as a difference in means, and offer insight into the precision of an estimate. In comparing groups, confidence intervals help quantify the uncertainty associated with estimated differences, and when constructed with a certain level of confidence (e.g., 90%), they allow decision-makers to assess the practical significance of the differences observed. This is particularly useful when evaluating economic implications or potential savings in operational settings.

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