A normal distribution is a common assumption underlying many statistical tests.

Question

A normal distribution is a common assumption underlying many statistical tests. Although the assumption is usually not important, it is possible to test whether it is plausible. Specifically, the unit normal table lists proportions for individual sections of a normal distribution, and a chi-square test for goodness of fit can be used to evaluate whether a specific distribution fits the proportions. For example, if a normal distribution is divided into sections using z-score values, then proportions in each section should be as follows: Use these proportions to test whether the following cord sample of n = 90 scores is significantly different from a normal distribution. Test with alpha = .05.

Explanation / Answer

Null Hypothesis : The given scores are similar to that of normal distribution

Alternative Hypothesis : The given scores are significantly different from normal distribution

We will test this hypothesis at 0.05 significance level.

The observed values are given in the problem. To find the expected values for each group we will multiply the given percentages with 90 and then divide by 100.

The degree of freedom is the defined as the number of groups - 1.

df = 5 - 1 =4

The critical value for 4 degrees of freedom and 0.05 signifcance level is 11.1. Since the test statistic(2.91) is smaller than the critical value , we fail to reject the null hypothesis. Thus the evidence suggests that the given scores follow a normal distribution.

z - range percentage expected frequency observed (O) frequency expected (E ) (O-E)2/E z<-1.5 6.68% 8 6.0 0.66 -1.5<z<-0.5 24.17% 19 21.8 0.35 -0.5<z<0.5 38.30% 31 34.5 0.35 0.5<z<1.5 24.17% 23 21.8 0.07 z>1.5 6.68% 9 6.0 1.49 Total 100.00% 90 90 2.91

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