The police department in Madison, Connecticut, released the...

The police department in Madison, Connecticut, released the following numbers of calls for the different days of the week during a February that had 28 days: Monday (114); Tuesday (152); Wednesday (160); Thursday (164); Friday (179); Saturday (196); Sunday (130). Use a $0.01$ significance level to test the claim that the different days of the week have the same frequencies of police calls. Is there anything notable about the observed frequencies?

Key Concepts

Significance Level

The significance level is the probability threshold used to decide whether to reject the null hypothesis. It represents the risk of a Type I error, and in this test, a very low significance level (0.01) indicates a strict criterion for concluding that the observed differences are statistically significant.

Residual Analysis

Residual analysis involves examining the differences between observed and expected frequencies for each category. This can help identify which categories contribute the most to the overall chi-square statistic and may highlight anomalies or patterns in the data that are noteworthy.

Degrees of Freedom

Degrees of freedom in a chi-square test typically equal the number of categories minus one. This value is important because it affects the shape of the chi-square distribution and the determination of critical values for significance testing.

p-Value

The p-value represents the probability that the observed data would occur, or something more extreme, if the null hypothesis were true. A p-value less than the significance level leads to rejection of the null hypothesis, indicating that the discrepancies between observed and expected frequencies are unlikely to be due to random chance.

Null Hypothesis

In the context of a goodness of fit test, the null hypothesis usually states that there is no difference between the observed frequencies and the frequencies expected under a specific distribution (e.g., a uniform distribution across all categories). It assumes that any observed differences are due solely to random variation.

Chi-Square Goodness of Fit Test

This statistical test is used to determine whether observed frequencies differ significantly from expected frequencies under a specific hypothesis. It compares the calculated chi-square statistic to a critical value, or uses a p?value, to decide whether any discrepancies between the observed and expected frequencies can be attributed to chance.

Expected Frequency

This refers to the frequency that would be expected in each category if the null hypothesis were true. For a uniform distribution, it is calculated by dividing the total count by the number of categories, allowing for a comparison between observed and anticipated counts.

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