Question

the table below lists the number of games played in a yearly best - of - seven baseball championship series, along with the expected proportions for the number of games played with teams of equal abilities. use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions.
determine the null and alternative hypotheses.
( h_0 ): the observed frequencies agree with the expected proportions.
( h_1 ): at least one of the observed frequencies do not agree with the expected proportions.
calculate the test statistic, ( chi^2 ).
( chi^2 = 8.387 ) (round to three decimal places as needed.)
calculate the p - value.
p - value = ( square ) (round to four decimal places as needed.)

Explanation:

Step1: Identify the test and degrees of freedom

This is a chi - square goodness - of - fit test. The number of categories (number of different games played) \(k = 4\) (for 4, 5, 6, 7 games). The degrees of freedom \(df=k - 1=4 - 1 = 3\). The test statistic \(\chi^{2}=6.367\).

Step2: Use chi - square distribution to find P - value

We need to find \(P(\chi^{2}>6.367)\) with \(df = 3\). Using a chi - square distribution table or a statistical software (such as a calculator with chi - square distribution functions or software like R, Python's scipy.stats.chi2.sf), we can calculate the P - value.

Using the formula for the survival function (1 - cumulative distribution function) of the chi - square distribution: \(P - value=\chi^{2}.sf(6.367,3)\) (in Python's scipy.stats notation).

Calculating this, we know that for a chi - square distribution with \(df = 3\), the cumulative probability \(P(\chi^{2}\leq6.367)\) can be found from tables or software. The survival function (which gives the P - value for a right - tailed test) is \(1 - P(\chi^{2}\leq6.367)\).

Using a chi - square table: Looking at the chi - square table for \(df = 3\), we see that \(\chi^{2}_{0.10,3}=6.251\) and \(\chi^{2}_{0.09,3}\approx6.367\) (more accurately, using a calculator or software like scipy in Python: from scipy.stats import chi2
p_value = chi2.sf(6.367, 3)
print(p_value)
The result is approximately 0.0957.

Answer:

\(0.0957\)