Question

A random sample of n = 6 observations from a normal distribution resulted in the data shown in the table. Compute a 95% confidence interval for ${\sigma }^2$

Short answer

The 95% confidence interval for ${\sigma }^2$is (4, 2689, 65.9044).

Step-by-step solution

Given information

Given data is as follows,

8 2 3 7 11 6

Calculating the Confidence interval

The 95% confidence interval can be calculated using the formula,

$\frac{\left(n-1\right)s^2}{{\chi }_{\frac{\alpha }{2}}^2}⩽{\sigma }^2⩽\frac{\left(n-1\right)s^2}{{\chi }_{\left(1-\frac{\alpha }{2}\right)}^2}$

From the given data, $n=6$and$s=3.31$

From the table values, at the 0.05 level of significance and 5 degrees of freedom, the value for 5 is 12.832 and

the value for$\left(\frac{\left(n-1\right)s^2}{{\chi }_{0.005}^2}⩽{\sigma }^2⩽\frac{\left(n-1\right)s^2}{{\chi }_{0.995}^2}\right)=\left(\frac{\left(100-1\right)6^2}{140.169}⩽{\sigma }^2⩽\frac{\left(100-1\right)6^2}{67.3276}\right)$ is 0.8312.

Substitute the values to get the required confidence interval

$$\begin{gathered} \left(\frac{\left(6-1\right){3.31}^2}{12.8325}⩽{\sigma }^2⩽\frac{\left(6-1\right){3.31}^2}{0.8312}\right) \\ \left(4.2689⩽{\sigma }^2⩽65.9044\right) \end{gathered}$$

Therefore, the 95% confidence interval for${\sigma }^2$is (4, 2689, 65.9044).