A sample of 9 days over the past six months showed that...

A sample of 9 days over the past six months showed that Philip Sherman, DDS, treated the following numbers of patients at this dental clinic: $22,25,20,18,15,22,24,19,$ and $26 .$ If the number of patients seen per day is normally distributed, would an analysis of these sample data reject the hypothesis that the variance in the number of patients seen per day is equal to $10 ?$ Use a. 10 level of significance. What is your conclusion?

Key Concepts

Significance Level and Critical Values

The significance level (alpha) defines the probability of a Type I error, which is rejecting the null hypothesis when it is true. Critical values from the chi-square distribution, determined by the chosen significance level and the degrees of freedom, provide the boundaries of the acceptance region for the null hypothesis. If the test statistic lies outside these critical values, the null hypothesis is rejected.

Hypothesis Testing

Hypothesis testing is a statistical procedure that evaluates a claim about a population parameter using sample data. In this context, it involves testing whether the sample evidence (sample variance) is compatible with a specified population variance, by comparing the calculated test statistic to the theoretical distribution.

Chi?Square Distribution

In tests of variance for a normally distributed population, the chi?square distribution is used because the quantity (n-1)*s²/??² follows a chi-square distribution with n-1 degrees of freedom under the null hypothesis. This property is key in determining whether the observed variability in the sample is consistent with the hypothesized population variance.

Test Statistic for Variance

The test statistic for variance is calculated as (n-1)*s²/??², where s² is the sample variance and ??² is the variance under the null hypothesis. This statistic follows a chi-square distribution under the null hypothesis, allowing comparison with critical values to decide if the null hypothesis can be rejected.

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