"In Problems 7–10, determine (a) the chi-square test statistic.
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: Understand 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: Identify the observed frequencies from the table: \(O_A = 38\), \(O_B = 45\), \(O_C = 41\), \(O_D = 33\), and \(O_E = 43\). The expected frequencies under \(H_0\) are all equal to 40 for each category.
3. Step 3: Use the chi-square test statistic 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 \(i\).
4. Step 4: Calculate the chi-square components for each category by subtracting the expected frequency from the observed frequency, squaring the result, and dividing by the expected frequency. That is, compute \(\frac{(38-40)^2}{40}\), \(\frac{(45-40)^2}{40}\), \(\frac{(41-40)^2}{40}\), \(\frac{(33-40)^2}{40}\), and \(\frac{(43-40)^2}{40}\).
5. Step 5: Sum all the components calculated in Step 4 to get the chi-square test statistic value. This value will be used to determine whether to reject the null hypothesis based on the chi-square distribution with degrees of freedom \(df = k - 1 = 5 - 1 = 4\).

Key Concepts

Chi-Square Test Statistic

The chi-square test statistic measures how much the observed frequencies deviate from the expected frequencies under the null hypothesis. It is calculated by summing the squared differences between observed and expected counts, divided by the expected counts for each category. This statistic helps determine if observed data fits a specified distribution.

Null and Alternative Hypotheses in Goodness-of-Fit Test

The null hypothesis (H0) assumes that all category proportions are equal, indicating no difference among groups. The alternative hypothesis (H1) suggests that at least one category proportion differs. These hypotheses guide the chi-square test to assess if observed data significantly deviates from expected equal proportions.

Expected Frequencies

Expected frequencies represent the counts we would anticipate in each category if the null hypothesis is true. They are calculated based on the total sample size and the hypothesized proportions. Comparing observed frequencies to expected frequencies is essential for computing the chi-square statistic and evaluating the fit.