Question

Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation sigma. Assume that the population has a normal distribution. Round the confidence interval limits to the same number of decimal places as the sample standard deviation. To find the standard deviation of the diameter of wooden​ dowels, the manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be s=0.16. Find the​ 95% confidence interval for the population standard deviation sigma.

A.0.13<σ<0.22

B.0.11<σ<0.25

C.0.15<σ<0.21

D.0.12<σ<0.24

Answer 1

Solution:

Here, we have to find the confidence interval for the population standard deviation.

Confidence interval for population standard deviation is given as below:

Sqrt[(n – 1)*S² / χ²_{α/2, n– 1} ] < σ < sqrt[(n – 1)*S² / χ²_{1 -α/2, n– 1} ]

We are given

Confidence level = 95%

Sample size = n = 19

Degrees of freedom = n – 1 = 18

Sample standard deviation = S = 0.16

χ²_{α/2, n – 1} = 31.5264

χ²_{1 -α/2, n– 1} = 8.2307

(By using chi square table)

Sqrt[(n – 1)*S² / χ²_{α/2, n – 1}] < σ < Sqrt[(n – 1)*S² / χ²_{1 -α/2, n– 1}]

Sqrt[(19 – 1)*0.16^2 / 31.5264] < σ < Sqrt[(19 – 1)*0.16^2 / 8.2307]

0.1209 < σ < 0.2366

Lower limit = 0.12

Upper limit = 0.24

D.0.12<σ<0.24