Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. 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 σ.
A. 0.11<σ<0.25
B. 0.13<σ<0.22
C. 0.12<σ<0.24
D. 0.15<σ<0.21
The 95% confidence interval for the population standard deviation σ is calculated using the formula:
(√((n-1)s²/χ²_(1-α/2)), √((n-1)s²/χ²_(α/2)))
where n = 19 (sample size), s = 0.16 (sample standard deviation), α = 0.05 (1-0.95), and χ²_(0.025) = 9.59 and χ²_(0.975) = 32.85 (from chi-square table with 18 degrees of freedom).
Plugging in the values, we get:
(√((18)(0.16)²/32.85), √((18)(0.16)²/9.59))
(0.12, 0.24)
Therefore, the 95% confidence interval for the population standard deviation σ is 0.12 < σ < 0.24.
C. 0.12 < σ < 0.24