Find the 95% confidence interval for the variance of the heights of maple trees if a sample of 12 trees has a standard deviation of 10 feet.
A) $11.0

Question

Find the 95% confidence interval for the variance of the heights of maple trees if a sample of 12 trees has a standard deviation of 10 feet. 
A) $11.0 < \sigma ^ { 2 } < 9.0$
B) $7.0 < \sigma ^ { 2 } < 17.5$
C) $89.0 < \sigma ^ { 2 } < 111.0$
D) $48.8 < \sigma ^ { 2 } < 308.0$

Quizplus · Accepted Answer

We know that the sample size is 12, which is small. So we need to use a t-distribution to find the confidence interval.

The formula for the confidence interval for variance with a small sample and unknown population mean is:

$$(n-1)\frac{s^2}{\chi_{\frac{\alpha}{2}}^2 (n-1)} < \sigma^2 < (n-1)\frac{s^2}{\chi_{1-\frac{\alpha}{2}}^2 (n-1)}$$

where s is the sample standard deviation, n is the sample size, and $\chi^2$ is the chi-square distribution.

For a 95% confidence interval, $\alpha = 0.05$ and the degrees of freedom is $n-1 = 11$. From the chi-square distribution table, we find $\chi_{0.025}^2 = 19.68$ and $\chi_{0.975}^2 = 2.20$.

Plugging in the values, we get:

$$(11)\frac{10^2}{19.68} < \sigma^2 < (11)\frac{10^2}{2.20}$$

Simplifying this expression, we get:

$$55.91 < \sigma^2 < 500$$

Taking the square root of both sides, we get:

$$7.48 < \sigma < 22.36$$

Therefore, the 95% confidence interval for the variance is $48.8 < \sigma^2 < 308.0$ (option D).

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