Find the expected frequencies, (b) find the critical value...

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 $\alpha=0.05,$ test the claim that the 400 test scores shown in the frequency distribution are normally distributed. (TABLE CAN'T COPY)

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 $\alpha=0.05,$ test the claim that the 400 test scores shown in the frequency distribution are normally distributed.
(TABLE CAN'T COPY)

Key Concepts

Chi-square Goodness-of-Fit Test

This test is used to determine whether a set of observed frequencies differs significantly from the frequencies expected under a given theoretical distribution. It is commonly applied to assess the fit of data to a specific distribution, such as the normal distribution, by comparing observed counts with theoretical counts calculated from the expected model.

Expected Frequency Calculation

Expected frequencies are computed based on the null hypothesis that the data follows a specified theoretical distribution. For each category or interval, an expected count is determined by multiplying the total number of observations by the probability that an observation falls into that interval according to the proposed distribution. This step is critical as it forms the basis for the subsequent chi-square calculation.

Degrees of Freedom

Degrees of freedom in a chi-square goodness-of-fit test are typically calculated as the number of categories minus the number of parameters estimated from the data, minus one. The correct calculation of degrees of freedom is essential because it directly influences the critical value from the chi-square distribution used to assess statistical significance.

Critical Value and Rejection Region

The critical value is a threshold obtained from the chi-square distribution at a specified significance level (alpha) and degrees of freedom. The rejection region is defined as the set of values for the test statistic that are considered too extreme under the null hypothesis. If the computed chi-square test statistic falls into the rejection region, the null hypothesis is rejected.

Chi-square Test Statistic

The chi-square test statistic is calculated by summing the square of the differences between the observed and expected frequencies, divided by the expected frequencies for each category. This statistic measures the discrepancy between the observed data and the distribution expected under the null hypothesis.

Hypothesis Testing and Decision Making

Hypothesis testing involves formulating a null hypothesis (which, in this context, assumes that the data follows a normal distribution) and an alternative hypothesis. Once the test statistic is computed and compared with the critical value, a decision is made to either reject or fail to reject the null hypothesis based on whether the test statistic falls within the rejection region.

Interpretation of Results

After the statistical decision is made regarding the null hypothesis, the results are interpreted in the context of the study. This interpretation explains what the rejection or non-rejection of the null hypothesis implies about the underlying distribution of the data, such as whether there is sufficient evidence to conclude that the data does not follow the proposed theoretical distribution.

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