Step 1 :The problem is asking us to determine whether there is sufficient evidence to support the claim that the variance in the amounts of yeast per loaf is less than 0.27. The null hypothesis is that the variance is equal to 0.27, and the alternative hypothesis is that the variance is less than 0.27.
Step 2 :We are given a sample of 13 loaves, with a sample variance of 0.12. The hypothesized population variance is 0.27. We are also given a significance level of 0.10.
Step 3 :We perform a hypothesis test for the variance. The test statistic is calculated as \( \chi^2 = \frac{(n-1)s^2}{\sigma^2} = \frac{(13-1)0.12}{0.27} = 5.33 \), where \( n = 13 \) is the sample size, \( s^2 = 0.12 \) is the sample variance, and \( \sigma^2 = 0.27 \) is the hypothesized population variance.
Step 4 :The degrees of freedom for the test is \( df = n - 1 = 13 - 1 = 12 \).
Step 5 :We find the p-value associated with the test statistic. The p-value is 0.9459, which is greater than the significance level of 0.10.
Step 6 :Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means that there is insufficient evidence at a 0.10 level of significance to support the claim that the variance in the amounts of yeast per loaf is less than 0.27.
Step 7 :\(\boxed{\text{Final Answer: We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.10 level of significance that the variance in the amounts of yeast per loaf is less than 0.27.}}\)