[NOW WORK] The NHL In his book Outliers, Malcolm Gladwell claims that more hockey players are born in January through March than in October through December. The following data show the number of players in the National Hockey League in the 2018–2019 season according to their birth month. Is there evidence to suggest that professional hockey players’ birth dates are not uniformly distributed throughout the year at the alpha = 0.05 level of significance?

Verified step by step guidance

1. Step 1: Define the hypotheses for the chi-square goodness-of-fit test. The null hypothesis \(H_0\) states that the birth months of NHL players are uniformly distributed across the four quarters of the year, meaning each quarter has an equal proportion of players. The alternative hypothesis \(H_a\) states that the birth months are not uniformly distributed.
2. Step 2: Calculate the expected frequency for each quarter under the assumption of uniform distribution. Since there are 4 quarters, each should have \(\frac{1}{4}\) of the total players. First, find the total number of players by summing all observed frequencies: \$273 + 263 + 211 + 206\(. Then, calculate the expected frequency for each quarter as \)\text{Expected frequency} = \frac{\text{Total players}}{4}$.
3. Step 3: Use the chi-square test statistic formula to compare observed and expected frequencies: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each quarter.
4. Step 4: Determine the degrees of freedom for the test. Since there are 4 categories (quarters), the degrees of freedom is \(df = 4 - 1 = 3\).
5. Step 5: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table at \(\alpha = 0.05\) and \(df = 3\). If the test statistic exceeds the critical value, reject the null hypothesis, indicating that birth months are not uniformly distributed. Otherwise, do not reject the null hypothesis.

Key Concepts

Chi-Square Goodness-of-Fit Test

This test determines if observed categorical data differ significantly from expected frequencies under a specific distribution. In this case, it tests whether birth months of NHL players are uniformly distributed across the year. The test compares observed counts to expected counts assuming equal distribution, using the chi-square statistic and significance level.

Null and Alternative Hypotheses

The null hypothesis (H0) assumes that birth months are uniformly distributed, meaning each quarter has an equal number of players. The alternative hypothesis (H1) suggests that birth months are not uniformly distributed. Hypothesis testing evaluates evidence against H0 to determine if observed differences are statistically significant.

Significance Level and P-Value

The significance level (alpha = 0.05) is the threshold for rejecting the null hypothesis. If the p-value from the chi-square test is less than alpha, it indicates strong evidence against H0, suggesting birth months are not uniform. Otherwise, there is insufficient evidence to reject H0, implying uniform distribution.