(a) identify the claim and state H0 and Ha,(b) find the...

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology. The table shows the costs per mile (in cents) for a sample of automobiles. At $\alpha=0.01,$ can you conclude that at least one mean cost per mile is different from the others? (Adapted from American Automobile Association) $$\begin{array}{|c|c|c|c|c|}\hline \text { Small sedan } & \text { Medium sedan } & \text { Large sedan } & \text { SUV 4WD } & \text { Minivan } \\\hline 43 & 67 & 64 & 87 & 66 \\41 & 49 & 73 & 66 & 76 \\49 & 63 & 83 & 75 & 59 \\ 54 & 59 & 75 & 78 & 51 \\46 & 64 & 80 & & 70 \\& 52 & 72 & & \\\hline\end{array}$$

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology.
The table shows the costs per mile (in cents) for a sample of automobiles. At $\alpha=0.01,$ can you conclude that at least one mean cost per mile is different from the others? (Adapted from American Automobile Association)
$$\begin{array}{|c|c|c|c|c|}\hline \text { Small sedan } & \text { Medium sedan } & \text { Large sedan } & \text { SUV 4WD } & \text { Minivan } \\\hline 43 & 67 & 64 & 87 & 66 \\41 & 49 & 73 & 66 & 76 \\49 & 63 & 83 & 75 & 59 \\
54 & 59 & 75 & 78 & 51 \\46 & 64 & 80 & & 70 \\& 52 & 72 & & \\\hline\end{array}$$

Key Concepts

Decision Making in Hypothesis Testing

Decision making in hypothesis testing involves comparing the computed test statistic to the critical value. If the test statistic exceeds the critical value, or equivalently if the p-value is less than the significance level, the null hypothesis is rejected, leading to the conclusion that there is statistically significant evidence against H0.

Test Statistic

The test statistic, specifically the F-statistic in ANOVA, is calculated as the ratio of the mean square between groups to the mean square within groups. This statistic follows an F-distribution under the null hypothesis and is compared to the critical value to decide whether to reject H0.

Assumptions in ANOVA

ANOVA relies on several key assumptions: the observations must be independent, the populations from which samples are drawn should be normally distributed, and the variances across the groups should be equal. Violations of these assumptions can affect the reliability and validity of the test outcomes.

Hypothesis Testing

Hypothesis testing involves formulating a null hypothesis (H0) that typically states there is no effect or no difference among groups, and an alternative hypothesis (Ha) that indicates the presence of a significant effect or difference. In the context of ANOVA, H0 posits that all group means are equal, while Ha asserts that at least one group mean differs.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to test the hypothesis that three or more population means are equal. It partitions the total variability in the data into components related to the variability between groups and within groups, enabling evaluation of whether observed differences among group means are too large to be attributed to random chance.

Critical Value and Rejection Region

The critical value is a threshold value derived from a statistical distribution (in this case, the F-distribution for ANOVA tests) at a chosen significance level (?). The rejection region consists of values beyond this critical value where, if the test statistic falls into this region, the null hypothesis is rejected.

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