Question

A simple random sample of 20 women’s hair lengths prior to entering a local hair salon is found to have a standard deviation of 1.2 inches. Use a 0.01 significance level to test the claim that the standard deviation of all women’s hair lengths prior to entering the hair salon is greater than 1.0 in. Find the critical value(s) needed to test the claim. Could anyone help me solve this problem?

Answer 1

To find the critical value we need to know 3 things, degrees of freedom, significance or confidence level, whether 1- or 2-tail test.

First, what is the claim: σ (population standard deviation) > 1 inch, and the counterclaim is σ≤1.

The null hypothesis, H0: σ=1 (based on the counterclaim which includes σ=1 as well as σ<1); the alternative hypothesis, H1: σ>1 (claim). This is a right tail test (1-tail) because the right tail of the distribution deals with standard deviations above the mean, that is, greater than.

The critical value is found from a t-table because we only know the sample standard deviation, s. We have dof=19, ɑ=0.01, 1-tail test, and the critical value from the table is 2.54. Now we need our test statistic:

(s-σ)/(s/√n)=(1.2-1)/(1.2/√20)=0.745. Since 0.745<2.54, we fail to reject H0, which means we have insufficient evidence to support the claim or counterclaim.