Question

97. A baseball fan wanted to know if there is a difference between the number of games played in a World Series when the American League won the series versus when the National League won the series. From 1922 to 2012 , the population standard deviation of games won by the American League was 1.14, and the population standard deviation of games won by the National League was 1.11. Of 19 randomly selected World Series games won by the American League, the mean number of games won was 5.76. The mean number of 17 randomly selected games won by the National League was 5.42. Conduct a hypothesis test.

Short answer

(a) The null hypothesis: $H_0:{\mu }_1={\mu }_2$
(b) The alternate hypothesis: $H_a:{\mu }_1\ne {\mu }_2$
(c) The number of mean games in the World Series won by the American League is different from that of the National League is denotes by $\overline{X_1}-\overline{X_2}$.
(d)The distribution is normal. .
(e) The test statistics:$0.9056$

(f) The $p$-value is $0.3650$.
(g) (i) $\alpha =0.05$
(ii) Decision: do not reject the null hypothesis.
(iii) Reason for Decision: $\alpha <p$-value.
(iv) As a result: There is insufficient evidence to determine that the American League's mean number of World Series victories differs from the National League's at the $5\%$level of significance.

Step-by-step solution

Given information

To conduct a hypothesis test. Let, the mean number of $17$randomly selected games won by the National League was $5.42$.

Explanation

(a) The null hypothesis is indicated as:
$H_0:{\mu }_1={\mu }_2$
(b) Alternate hypothesis is indicated as:
$H_a:{\mu }_1\ne {\mu }_2$
(c) The number of mean games won by the American League in the World Series differs from the amount won by the National League is denotes by $\overline{X_1}-\overline{X_2}$.
(d) The distribution is normal.
(e) To find determine the static as follows:
To access the stat list editor, click STAT and then 1.

Then ENTER all values the OUTPUT will be:

Hence the test statistics is $0.9056$.

Explanation

(f) The $p$-value from the output is determined as $0.3650$.
(g) Obtain a clear picture of the situation using the information from the last task. The horizontal axis should be clearly labelled and scaled, and the region(s) corresponding to the p-value should be shaded on the graph.

(i) $\alpha =0.05$
(ii) Decision: do not reject the null hypothesis.
(iii) Reason for Decision: $\alpha <p$-value.
(iv) As a result: There is insufficient evidence to determine that the American League's mean number of World Series victories differs from the National League's at the $5\%$level of significance.