In Exercises 17 and 18,(a) find the expected frequencies,...

In Exercises 17 and $18,(a)$ find the expected frequencies, $(b)$ find the critical value and identify the rejection region, $(c)$ find the chi-square test statistic, $(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. At $\alpha=0.01,$ test the claim that the 200 test scores shown in the frequency distribution are normally distributed. (TABLE CAN'T COPY)

In Exercises 17 and $18,(a)$ find the expected frequencies, $(b)$ find the critical value and identify the rejection region, $(c)$ find the chi-square test statistic, $(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.
At $\alpha=0.01,$ test the claim that the 200 test scores shown in the frequency distribution are normally distributed.
(TABLE CAN'T COPY)

Key Concepts

Interpretation in Context

The final step involves interpreting the outcome of the hypothesis test within the framework of the original claim. For a test of normality, if the null hypothesis is rejected, it indicates that the test scores do not appear to follow a normal distribution; otherwise, there is insufficient evidence to conclude a departure from normality.

Critical Value and Rejection Region

The critical value is the threshold from the chi-square distribution corresponding to the chosen significance level (alpha) and the calculated degrees of freedom. The rejection region is defined by chi-square values greater than this critical value, leading to the decision to reject the null hypothesis if the test statistic falls in this region.

Degrees of Freedom

Degrees of freedom in a goodness-of-fit test are typically calculated as the number of categories minus the number of estimated parameters minus one. They are essential for determining the appropriate critical value from the chi-square distribution.

Hypothesis Testing Decision

After calculating the chi-square test statistic and comparing it with the critical value, the decision is made to either reject or fail to reject the null hypothesis. This decision informs whether there is sufficient evidence to conclude that the data do not follow the specified distribution.

Expected Frequencies

In the context of a goodness-of-fit test, expected frequencies are calculated based on the theoretical probability of occurrence for each category, multiplied by the total sample size. These values represent what would be anticipated if the null hypothesis were true.

Chi-Square Goodness-of-Fit Test

This test is used to determine whether a set of observed frequencies matches those expected under a specified theoretical distribution. It compares observed counts in different categories to the counts predicted by the distribution assumed under the null hypothesis.

Chi-Square Test Statistic

The chi-square test statistic is computed by summing the squared differences between the observed (O) and expected (E) frequencies, each divided by the expected frequency. It quantifies the overall discrepancy between the observed data and the expected model.

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