1. Happy Acres Farm is considering spraying their apple trees with a plant hormone intended to stabilize the size of apples the tree produces. Very small and very large apples cannot be sold as fresh fruit because the distributor will cull them out as seconds for apple cider. Happy Acres knows that thei species of tree will produce apples of mean diameter $8.3 \mathrm{~cm}$. with a variance of $2.89 \mathrm{~cm}^{2}$ After spraying a sample typical test tree with the hormone, they found that a random sample of 71 apples gar a sample variance of $2.25 \mathrm{~cm}^{2}$. for apple diameter. Use a $5 \%$ significance level to test the claim that th hormone has made a difference either way.
Answer
\(\boxed{\text{Final Answer:}}\) Since the test statistic (54.50) falls between the critical values (48.76 and 95.02), we fail to reject the null hypothesis. Therefore, we cannot conclude that the hormone has made a significant difference in the variance of apple diameters at a 5% significance level.
Step 1 ：State the null hypothesis (H0): \(\sigma^2 = 2.89\) cm², and the alternative hypothesis (H1): \(\sigma^2 \neq 2.89\) cm².
Step 2 ：Calculate the test statistic: \(\chi^2 = (n - 1) * s^2 / \sigma^2 = (71 - 1) * 2.25 / 2.89 = 54.50\)
Step 3 ：Determine the critical values for df = 70 and \(\alpha/2 = 0.025\): \(\chi^2_{lower} = 48.76\) and \(\chi^2_{upper} = 95.02\)
Step 4 ：\(\boxed{\text{Final Answer:}}\) Since the test statistic (54.50) falls between the critical values (48.76 and 95.02), we fail to reject the null hypothesis. Therefore, we cannot conclude that the hormone has made a significant difference in the variance of apple diameters at a 5% significance level.
