"In Problems 7–10, determine (b) the degrees of freedom.
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. Identify the number of categories involved in the problem. Here, the categories are A, B, C, D, and E, so there are 5 categories in total.
2. Recall that the degrees of freedom (df) for a chi-square goodness-of-fit test is calculated as the number of categories minus 1.
3. Write the formula for degrees of freedom: \(\text{df} = k - 1\), where \(k\) is the number of categories.
4. Substitute the number of categories into the formula: \(\text{df} = 5 - 1\).
5. Simplify the expression to find the degrees of freedom for this test.

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 data to a theoretical distribution, such as equal proportions, to determine if deviations are due to chance or indicate a real difference.

Degrees of Freedom in Chi-Square Tests

Degrees of freedom (df) represent the number of independent values that can vary in the calculation after certain constraints are applied. For a goodness-of-fit test with k categories, df is calculated as k - 1, reflecting the number of categories minus one parameter estimated from the data.

Null and Alternative Hypotheses for Proportions

The null hypothesis (H0) states that all category proportions are equal, implying no preference or difference among categories. The alternative hypothesis (H1) claims that at least one category proportion differs, indicating a deviation from equal distribution.