91Ó°ÊÓ

The below table gives the probabilities of World Series in baseball lasting \((4,5,6 or 7)\) games if two teams are matched evenly and the actual numbers of times the series lasted \((4,5,6 or 7)\) games.

Define the hypotheses:

\(H_{0}\): The two World Series teams are matched evenly.

\(H_{a}\) : The two World Series teams are not matched evenly.

\(\alpha=5%=0.05\)

Determine the observed frequencies and the chi-square subtotals:

\(\chi ^{2}=7.5943\)

Determine the critical value using Table V with \(df=4-1=3\)

\(\chi^{2}_{0.05}=7.815\)

\(7.5943<7.815 \Rightarrow \) do not reject \(H_{0}\):

Therefore, the two World Series teams are matched evenly at a \(5%\) significance level.

Define the hypotheses:

\(H_{0}\): The two World Series teams are matched evenly.

\(H_{a}\) : The two World Series teams are not matched evenly.

\(\alpha=10%=0.10\)

It was calculated in the above part that the test statistics

\(\chi ^{2}=7.5943\)

Determine the critical value using Table V with \(df=4-1=3\)

\(\chi^{2}_{0.10}=6.251\)

\(7.5943>6.251 \Rightarrow\) Reject \(H_{0}\)

Therefore the two World Series teams are not matched evenly at a \(10%\) significance level.

The assumptions to the Chi-Square goodness-of-fit test are:

• The expected frequencies should be all greater than or equal to \(1\).
• Not more than \(20%\) of the expected frequencies are less than \(5\).
• The sample is drawn from the simple random sampling method.

Explanation:

The sample is not drawn from the simple random sampling method, thus it is not appropriate to use the chi-square goodness-of-fits test to this case.