Question 12.1: Miles per Gallon A researcher wishes to see if there is a di......

Step 1 State the hypotheses and identify the claim.

H_{0}: \mu_{1}=\mu_{2}=\mu_{3} (claim)

H_{1} : At least one mean is different from the others

Step 2 Find the critical value.

\begin{aligned}\quad N & =12 \quad k=3 \\\text { d.f.N. } & =k-1=3-1=2 \\\text { d.f.D. } & =N-k=12-3=9\end{aligned}

The critical value from Table H in Appendix A with \alpha=0.05 is 4.26.

Step 3 Compute the test value.

a. Find the mean and variance for each sample. (Use the formulas in Chapter 3.)

\begin{array}{lll}\text { For the small cars: } & \bar{X}=37.25 & s^{2}=20.917 \\\text { For the sedans: } & \bar{X}=35.4 & s^{2}=37.3 \\\text { For the luxury cars: } & \bar{X}=26 & s^{2}=7\end{array}

b. Find the grand mean.

\bar{X}_{\mathrm{GM}}=\frac{\Sigma X}{N}=\frac{36+44+34+\cdots+24}{12}=\frac{404}{12}=33.667

c. Find the between-group variance.

\begin{aligned}s_{B}^{2} & =\frac{\sum n\left(\bar{X}_{i}-\bar{X}_{\mathrm{GM}}\right)^{2}}{k-1} \\& =\frac{4(37.25-33.667)^{2}+5(35.4-33.667)^{2}+3(26-33.667)^{2}}{3-1} \\& =\frac{242.717}{2}=121.359\end{aligned}

d. Find the within-group variance.

\begin{aligned}s_{W}^{2} & =\frac{\Sigma\left(n_{i}-1\right) s_{i}^{2}}{\Sigma\left(n_{i}-1\right)}=\frac{(4-1)(20.917)+(5-1)(37.3)+(3-1) 7}{(4-1)+(5-1)+(3-1)} \\& =\frac{225.951}{9}=25.106\end{aligned}

e. Find the F test value.

F=\frac{s_{B}^{2}}{s_{W}^{2}}=\frac{121.359}{25.106}=4.83

Step 4 Make the decision. The test value 4.83>4.26, so the decision is to reject the null hypothesis. See Figure 12-1.

Step 5 Summarize the results. There is enough evidence to conclude that at least one mean is different from the others.

The ANOVA summary table is shown in Table 12-2.