"In Problems 7–10, determine (c) the critical value using.
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 type of test being conducted. Since the hypotheses involve proportions across multiple categories, this is a Chi-square goodness-of-fit test.
2. Determine the degrees of freedom (df) for the Chi-square distribution. The formula for degrees of freedom in this context is \(df = k - 1\), where \(k\) is the number of categories. Here, \(k = 5\) (A, B, C, D, E), so \(df = 5 - 1\).
3. Choose the significance level (\(\alpha\)) for the test. This is typically given in the problem or assumed (commonly 0.05).
4. Using the degrees of freedom and the significance level, find the critical value from the Chi-square distribution table. The critical value is the value that the test statistic must exceed to reject the null hypothesis.
5. Interpret the critical value in the context of the problem: if the calculated Chi-square statistic is greater than this critical value, reject \(H_0\); otherwise, do not reject \(H_0\).

Key Concepts

Chi-Square Test for Goodness of Fit

This test evaluates whether observed categorical data 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. The null hypothesis assumes all category proportions are equal or follow a specified distribution.

Critical Value in Hypothesis Testing

The critical value is a threshold that defines the boundary for rejecting the null hypothesis. It depends on the significance level (alpha) and the degrees of freedom in the test. If the test statistic exceeds this value, the null hypothesis is rejected, indicating significant evidence against it.

Degrees of Freedom in Chi-Square Tests

Degrees of freedom (df) represent the number of independent values that can vary in the calculation after constraints are applied. For a goodness-of-fit test, df equals the number of categories minus one. This value is essential for determining the critical value from the chi-square distribution table.