Question 13.11: Baseball All-Star Winners The data show the winners of the b......

Step 1 State the hypotheses and identify the claim.

H_{0} : The winners occur at random (claim).

H_{1} : The winners do not occur at random.

Step 2 Determine the actual values.

Since n_{1}>20 and n_{2}>20, Table E is used. At \alpha=0.05, the critical values are \pm 1.96.

Step 3 Find the test value.

n_{1}(\text { National })=28 \quad n_{2}(\text { American })=22

The number of runs is

1. A

2. NNNNNNNN

3. A

4. NNNNNNNNNNN

5. A

6. NN

7. A

8. N

9. AAAAAA

10. NNN

11. AAAAAAAAAAAA

12. NNN

There are G=12 runs.

\begin{aligned} \mu_{G} & =\frac{2 n_{1} n_{2}}{n_{1}+n_{2}}+1 \\ & =\frac{2(28)(22)}{28+22}+1=25.64 \\ \sigma_{G} & =\sqrt{\frac{2 n_{1} n_{2}\left(2 n_{1} n_{2}-n_{1}-n_{2}\right)}{\left(n_{1}+n_{2}\right)^{2}\left(n_{1}+n_{2}-1\right)}} \\ & =\sqrt{\frac{2(28)(22)[2(28)(22)-28-22]}{(28+22)^{2}(28+22-1)}}=3.448 \\ z & =\frac{G-\mu_{G}}{\sigma_{G}} \\ & =\frac{12-25.64}{3.448} \\ & =-3.96 \end{aligned}

Step 4 Make the decision. Since -3.96<-1.96, the decision is to reject the null hypothesis.

Step 5 Summarize the results. There is enough evidence to reject the claim that the sequence of winners occurs at random.