Testing for Normality Using a chi-square goodness-of-fit test, you can decide, with some degree of certainty, whether a variable is normally distributed. In all chi-square tests for normality, the null and alternative hypotheses are as listed below.


H₀: The variable has a normal distribution.


Hₐ: The variable does not have a normal distribution.


To determine the expected frequencies when performing a chi-square test for normality, first estimate the mean and standard deviation of the frequency distribution. Then, use the mean and standard deviation to compute the z-score for each class boundary. Then, use the z-scores to calculate the area under the standard normal curve for each class. Multiplying the resulting class areas by the sample size yields the expected frequency for each class.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.


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.


Test Scores At α=0.05, test the claim that the 400 test scores shown in the frequency distribution are normally distributed.

Verified step by step guidance

1. Step 1: Calculate the mean and standard deviation of the test scores. Use the formula for the mean: μ = Σ(f * midpoint) / Σf, where midpoint is the average of the class boundaries for each interval. For standard deviation, use the formula: σ = sqrt(Σ(f * (midpoint - μ)^2) / Σf).
2. Step 2: Compute the z-scores for each class boundary using the formula: z = (X - μ) / σ, where X is the class boundary, μ is the mean, and σ is the standard deviation.
3. Step 3: Use the z-scores to find the area under the standard normal curve for each class interval. This can be done using a standard normal distribution table or software tools.
4. Step 4: Multiply the area under the curve for each class interval by the total sample size (400) to calculate the expected frequencies for each class.
5. Step 5: Perform the chi-square test for goodness-of-fit using the formula: χ² = Σ((O - E)^2 / E), where O is the observed frequency and E is the expected frequency for each class. Compare the test statistic to the critical value at α = 0.05 to determine whether to reject or fail to reject the null hypothesis.

Key Concepts

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical method used to determine if a sample distribution matches an expected distribution. In the context of testing for normality, it compares the observed frequencies of data within specified intervals (or classes) to the expected frequencies derived from a normal distribution. A significant result indicates that the sample does not follow the expected distribution.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H₀) represents a statement of no effect or no difference, while the alternative hypothesis (Hₐ) suggests that there is an effect or a difference. For the chi-square test for normality, H₀ states that the variable is normally distributed, and Hₐ posits that it is not. The outcome of the test will help determine which hypothesis is supported by the data.

Expected Frequencies

Expected frequencies are the theoretical frequencies of occurrences in each class if the null hypothesis is true. They are calculated using the mean and standard deviation of the sample data to determine the area under the normal curve for each class interval. These expected values are crucial for the chi-square test, as they are compared against the observed frequencies to assess the goodness of fit.