Births Natural births randomly selected from four hospitals in New York State occurred on the days of the week (in the order of Monday through Sunday) with these frequencies: 52, 66, 72, 57, 57, 43, 53. Does it appear that such births occur on the days of the week with equal frequency?

검증된 단계별 안내

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that births occur with equal frequency across all days of the week, while the alternative hypothesis states that births do not occur with equal frequency.

Step 2: Calculate the expected frequency for each day of the week under the assumption of equal frequency. Since there are 7 days in a week and the total number of births is the sum of the given frequencies, divide the total number of births by 7 to find the expected frequency for each day.

Step 3: Use the chi-square test formula to calculate the test statistic. The formula is χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ represents the observed frequency for each day and Eᵢ represents the expected frequency for each day. Perform this calculation for all 7 days and sum the results.

Step 4: Determine the degrees of freedom for the chi-square test. The degrees of freedom are calculated as (number of categories - 1). In this case, there are 7 days, so the degrees of freedom are 7 - 1 = 6.

Step 5: Compare the calculated chi-square test statistic to the critical value from the chi-square distribution table at the chosen significance level (e.g., α = 0.05) and degrees of freedom. If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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주요 개념

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Chi-Square Test of Goodness of Fit

The Chi-Square Test of Goodness of Fit is a statistical method used to determine if observed frequencies differ from expected frequencies. In this context, it helps assess whether the number of births across the days of the week is uniformly distributed or if certain days have significantly more or fewer births than expected.

Null Hypothesis

The null hypothesis is a statement that assumes no effect or no difference, serving as a starting point for statistical testing. In this scenario, the null hypothesis would state that births occur with equal frequency across all days of the week, which can be tested against the observed data.

Expected Frequencies

Expected frequencies are the theoretical frequencies that would occur if the null hypothesis were true. For this question, if births are equally likely on each day, the expected frequency for each day can be calculated by dividing the total number of births by the number of days (7), providing a benchmark for comparison with the observed frequencies.