Using Nonparametric Tests. In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

World Series The last 110 baseball World Series ended with 63 wins by American League teams and 47 wins by National League teams. Use the sign test to test the claim that in each World Series, the American League team has a 0.5 probability of winning.

Out of 110 World Series, 63 were won by American League teams and 47 were won by National League teams.

The sign test is the non-parametric test used to test the difference between the proportions of games won by American League teams and those won by National League teams.

Let p be the proportion of games won by the American League team.

The null hypothesis is as follows:

\({H_0}:p = 0.5\)

The American League team has a 0.5 probability of winning.

The alternative hypothesis is as follows:

\({H_1}:p \ne 0.5\)

The American League team does not have a 0.5 probability of winning.

The test is two-tailed.

The series won by the American League teams are denoted by a negative sign.

The series won by the National League teams are denoted by a positive sign.

The number of positive signs = 47.

The number of negative signs = 63.

Since the number of positive signs and negative signs arenot the same, the observation does not contradict the alternative hypothesis.

The sample size (n) is equal to 110.

Let x be the number of times the less frequent sign occurs.

The less frequent sign is the positive sign corresponding to the series won by the National League teams.

The value of x is equal to 47.

As the sample size n is greater than 25, the value of z is calculated.

The test statistic z is calculated as shown:

\(\begin{array}{c}z = \frac{{\left( {x + 0.5} \right) - \frac{n}{2}}}{{\frac{{\sqrt n }}{2}}}\\ = \frac{{\left( {47 + 0.5} \right) - \frac{{110}}{2}}}{{\frac{{\sqrt {110} }}{2}}}\\ = - 1.430\end{array}\)

The critical value of z from the standard normal table for a two-tailed test with a value of\(\alpha \)equal to 0.05 is equal to 1.96.

The p-value is equal to 0.1527.

Since the absolute value of z equal to 1.430 is less than the critical value, the null hypothesis failes to be rejected.

As the p-value is greater than 0.05, the null hypothesisfailsto be rejected.

It can be concluded that there is not enough evidence to warrant rejection of the claim that the American League team has a 0.5 probability of winning.