Find the 95% confidence interval for the variance of the heights of maple trees if a sample of 11 trees has a standard deviation of 12.2 feet.
A) 11.2

Question

Find the 95% confidence interval for the variance of the heights of maple trees if a sample of 11 trees has a standard deviation of 12.2 feet.
A) 11.2 < < 13.2 
B) 8.5 < < 21.4 
C) 137.8 < < 159.8 
D) 72.7 < < 458.4

Quizplus · Accepted Answer

We know that for a sample of size n, the statistic $\frac{(n-1)S^2}{\sigma^2}$ follows a chi-squared distribution with n-1 degrees of freedom. We can use this fact to make a confidence interval for the true variance $\sigma^2$.

Using a chi-squared distribution table, we find the critical values that correspond to the 2.5% and 97.5% percentiles for a chi-squared distribution with 10 degrees of freedom (n-1 = 10, since our sample size is 11). Using these critical values, we can find the confidence interval as follows:

$\frac{(n-1)S^2}{\chi^2_{0.025,10}} < \sigma^2 < \frac{(n-1)S^2}{\chi^2_{0.975,10}}$

Plugging in our values, we get:

$\frac{(11-1)(12.2)^2}{18.307} < \sigma^2 < \frac{(11-1)(12.2)^2}{3.940} $

Simplifying this gives:

72.7 < $\sigma^2$ < 458.4

Therefore, the 95% confidence interval for the variance of the heights of maple trees is (72.7, 458.4). The answer is choice D.

Find the 95% Confidence Interval for the Variance of the Heights