Question

In Exercises 5–8, use the given information to find the number of degrees of freedom, the critical values ${\chi }_L^2$and${\chi }_R^2$, and the confidence interval estimate of $\sigma$. The samples are from Appendix B and it is reasonable to assume that a simple random sample has been selected from a population with a normal distribution.

Weights of Pennies 95% confidence;n= 37,s= 0.01648 g.

Short answer

The degrees of freedom is 36.

The critical values are${\chi }_L^2=$21.3359 and${\chi }_R^2=$54.4373.

The 95% confidence interval estimate of $\sigma$is 0.013 g <$\sigma$<0.021 g.

Step-by-step solution

Given information

The size of the sample is $n=37$.

The sample standard deviation is $s=0.01648$g.

The level of confidence is 95%.

Compute the degrees of freedom, critical values and confidence interval estimate of σ

The degrees of freedom is computed as,

$$\begin{gathered} df=n-1 \\ =37-1 \\ =36 \end{gathered}$$

Using the Chi-square table, the critical values are ${\chi }_L^2=21.3359$and${\chi }_R^2=54.4373$.

The 95% confidence interval estimate of $\sigma$is computed as,

$$\begin{gathered} \sqrt{\frac{\left(n-1\right)s^2}{{\chi }_R^2}}<\sigma <\sqrt{\frac{\left(n-1\right)s^2}{{\chi }_L^2}} \\ \sqrt{\frac{\left(37-1\right){0.01648}^2}{54.4373}}<\sigma <\sqrt{\frac{\left(37-1\right){0.01648}^2}{21.3359}} \\ 0.013<\sigma <0.021 \end{gathered}$$

Therefore, the 95% confidence interval estimate of role="math" localid="1648104834948" $\sigma$ isrole="math" localid="1648104851448" $0.013g<\sigma <0.021g$