91Ó°ÊÓ

The Insurance Institute for Highway Safety regularly tests cars for various safety factors. In one such test, the institute tests the bumpers in 5 -mile per hour (mph) crashes. The following data represent the cost of repairs (in dollars) after four different 5 -mph crashes on small utility vehicles. The institute blocks by location of crash, and the treatment is car model. $$ \begin{array}{lcccc} & \text { Jeep } & \text { Saturn } & \text { Toyota } & \text { Hyundai } \\ & \text { Cherokee } & \text { VUE } & \text { RAV4 } & \text { Santa Fe } \\ \hline \text { Front into } & 652 & 416 & 489 & 539 \\ \text { flat barrier } & & & & \\ \hline \begin{array}{l} \text { Rear into } \\ \text { flat barrier } \end{array} & 824 & 556 & 1897 & 1504 \\ \hline \begin{array}{l} \text { Front into } \\ \text { angle barrier } \end{array} & 1448 & 1179 & 1151 & 1578 \\ \hline \text { Rear into pole } & 1553 & 1335 & 2377 & 1988 \end{array} $$ (a) Normal probability plots for each treatment indicate that the requirement of normality is satisfied. Verify that the requirement of equal population variances for each treatment is satisfied. (b) Is there sufficient evidence that the mean cost of repairs is different among the four SUVs at the \(\alpha=0.05\) level of significance? (c) If the null hypothesis from part (b) was rejected, use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05 .\)

Step by step solution

01

Calculate the Mean and Variance

Calculate the mean and variance of repair costs for each car model (Jeep Cherokee, Saturn VUE, Toyota RAV4, and Hyundai Santa Fe) to verify the requirement of equal population variances. Formulas to use:Mean: \[ \bar{x} = \frac{\text{sum of all values}}{\text{number of values}} \]Variance: \[ s^2 = \frac{\text{sum of } (x_i - \bar{x})^2}{n-1} \]Use the given data to calculate:

02

Jeep Cherokee Calculations

Repair Costs for Jeep Cherokee: 652, 824, 1448, 1553Mean: \[ \bar{x}_{Jeep} = \frac{652 + 824 + 1448 + 1553}{4} = 1119.25 \] Variance: \[ s^2_{Jeep} = \frac{(652 - 1119.25)^2 + (824 - 1119.25)^2 + (1448 - 1119.25)^2 + (1553 - 1119.25)^2}{3} = 160230.92 \]

03

Saturn VUE Calculations

Repair Costs for Saturn VUE: 416, 556, 1179, 1335Mean: \[ \bar{x}_{Saturn} = \frac{416 + 556 + 1179 + 1335}{4} = 871.5 \] Variance: \[ s^2_{Saturn} = \frac{(416 - 871.5)^2 + (556 - 871.5)^2 + (1179 - 871.5)^2 + (1335 - 871.5)^2}{3} = 196254.33 \]

04

Toyota RAV4 Calculations

Repair Costs for Toyota RAV4: 489, 1897, 1151, 2377Mean: \[ \bar{x}_{Toyota} = \frac{489 + 1897 + 1151 + 2377}{4} = 1478.5 \] Variance: \[ s^2_{Toyota} = \frac{(489 - 1478.5)^2 + (1897 - 1478.5)^2 + (1151 - 1478.5)^2 + (2377 - 1478.5)^2}{3} = 609237.67 \]

05

Hyundai Santa Fe Calculations

Repair Costs for Hyundai Santa Fe: 539, 1504, 1578, 1988Mean: \[ \bar{x}_{Hyundai} = \frac{539 + 1504 + 1578 + 1988}{4} = 1402.25 \] Variance: \[ s^2_{Hyundai} = \frac{(539 - 1402.25)^2 + (1504 - 1402.25)^2 + (1578 - 1402.25)^2 + (1988 - 1402.25)^2}{3} = 399637.58 \]

06

Verify Equal Population Variances

Compare the variances calculated in Step 1. If the variances are approximately equal, the requirement for equal population variances is satisfied.Variances:Jeep Cherokee: 160230.92Saturn VUE: 196254.33Toyota RAV4: 609237.67Hyundai Santa Fe: 399637.58

07

ANOVA Test

Use ANOVA (Analysis of Variance) to determine if there is a significant difference in the mean repair costs among the four SUVs. The null hypothesis is that all means are equal, and the alternative hypothesis is that at least one mean is different. Use a significance level of \( \alpha = 0.05 \). Check the ANOVA table for the F-value and p-value.

08

Tukey's Test (If Necessary)

If the null hypothesis is rejected in the ANOVA test, apply Tukey's Test to determine which pairs of means are significantly different. Use a familywise error rate of \( \alpha = 0.05 \). Calculate the Tukey HSD (Honestly Significant Difference) and compare pairwise mean differences to this value.

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Key Concepts

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

Mean and Variance Calculations

Understanding mean and variance calculations is fundamental in statistics, especially for ANOVA analysis. The mean is simply the average value, calculated by summing all data points and dividing by the number of points. The formula for the mean (\bar{x}) is
\(\bar{x} = \frac{\text{sum of all values}}{\text{number of values}}\).

Variance measures how spread out the data points are around the mean. A higher variance indicates more spread. The variance formula is
\[ s^2 = \frac{\text{sum of } (x_i - \bar{x})^2}{n-1} \].

Let's consider the repair costs for the Jeep Cherokee:
Mean: \(\bar{x}_{Jeep} = \frac{652 + 824 + 1448 + 1553}{4} = 1119.25\)
Variance: \(\frac{(652 - 1119.25)^2 + (824 - 1119.25)^2 + (1448 - 1119.25)^2 + (1553 - 1119.25)^2}{3} = 160230.92\).

Similar steps are followed for Saturn VUE, Toyota RAV4, and Hyundai Santa Fe to find their respective means and variances. Comparing these variances helps verify the assumption of equal variances for ANOVA.

Hypothesis Testing

Hypothesis testing in ANOVA helps determine if there are any statistically significant differences among the means of different groups. The null hypothesis (H0) states that all group means are equal. The alternative hypothesis (H1) suggests that at least one mean is different.

In this problem, we want to know if the mean repair costs among the four SUVs are different. We set our significance level (\alpha) at 0.05. Using the ANOVA table, we calculate the F-value and p-value. ANOVA works by comparing the variation between group means to the variation within the groups. If the p-value is less than \(\alpha = 0.05\), we reject the null hypothesis, indicating a significant difference among the means.

The ANOVA formula involves dividing the Mean Square Between (MSB) by the Mean Square Within (MSW): \[\text{F} = \frac{\text{MSB}}{\text{MSW}}\]. You can check the F-distribution tables or software outputs for the critical value. Rejecting H0 leads us to conduct further analysis with post-hoc tests.

Tukey's Test

Tukey's Honestly Significant Difference (HSD) test is a post-hoc analysis used to discover which specific means are different after an ANOVA indicates significant differences. It compares all possible pairs of means while controlling the familywise error rate to \(\alpha = 0.05\).

The formula for Tukey's HSD is: \[ HSD = q \sqrt{\frac{MSW}{n}} \],
where q is the studentized range statistic, MSW is the Mean Square Within groups, and n is the number of observations per group. This test helps identify where the significant differences lie among paired group means.

For our SUV repair costs, if ANOVA results reject the null hypothesis, Tukey's test will reveal which specific SUV pairs have significantly different repair costs. This helps in pinpointing the exact differences, ensuring detailed and specific insights into the cost variances among the models evaluated in the crash tests.