"In Problems 7–10, determine (d) test the hypothesis at the level of significance.
H0: pA=pB=pC=pD=pE=1/5
H1: At least one of the proportions is different from the others.

Verified step by step guidance

1. Step 1: Identify the hypotheses. The null hypothesis \(H_0\) states that all proportions are equal, i.e., \(p_A = p_B = p_C = p_D = p_E = \frac{1}{5}\). The alternative hypothesis \(H_1\) states that at least one proportion is different.
2. Step 2: Calculate the test statistic for the chi-square goodness-of-fit test using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each category.
3. Step 3: Substitute the observed and expected values from the table into the formula. For each category (A, B, C, D, E), compute the squared difference between observed and expected, divide by the expected, and sum all these values.
4. Step 4: Determine the degrees of freedom for the test. Since there are 5 categories, the degrees of freedom is \(df = 5 - 1 = 4\).
5. Step 5: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table at the given significance level and \(df=4\). If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

Key Concepts

Chi-Square Goodness-of-Fit Test

This test evaluates whether the observed frequencies in categories differ significantly from expected frequencies under a specific hypothesis. It compares observed counts to expected counts to determine if deviations are due to chance or indicate a real difference.

Null and Alternative Hypotheses

The null hypothesis (H0) states that all category proportions are equal, implying no difference among groups. The alternative hypothesis (H1) claims that at least one category proportion differs, which the test aims to detect.

Level of Significance and Decision Rule

The level of significance (commonly denoted as alpha) is the threshold for rejecting H0, often set at 0.05. If the test statistic exceeds the critical value from the chi-square distribution, H0 is rejected, indicating significant differences among proportions.