Question

Assume that the heights of men are normally distributed. A random sample of 16 men have a mean height of 67.5 inches and a standard deviation of 3.2 inches. Construct a 99% confidence interval for the population standard deviation, σ. (2.2, 5.4) (2.2, 6.0) (1.2, 3.2) (2.2, 5.8)

Answer 1

Answer: (2.2, 5.8)

Explanation:

The confidence interval for standard deviation is given by :-

`\left ( \sqrt{((n-1)s^2)/(\chi^2_((n-1),\alpha/2))} , \sqrt{((n-1)s^2)/(\chi^2_((n-1),1-\alpha/2))}\right )`

Given : Sample size : 16

Mean height :
`\mu=67.5` inches

Standard deviation :
`s=3.2` inches

Significance level :
`1-0.99=0.01`

Using Chi-square distribution table ,

`\chi^2_((15,0.005))=32.80`

`\chi^2_((15,0.995))=4.60`

Then , the 99% confidence interval for the population standard deviation is given by :-

`\left ( \sqrt{((15)(3.2)^2)/(32.80)} , \sqrt{((15)(3.2)^2)/(4.6)}\right )\\\\=\left ( 2.1640071232,5.77852094812\right )\approx\left ( 2.2,5.8 \right )`