Question

A company produces a women's bowling ball that is supposed to weigh exactly 14 pounds. Unfortunately, the company has a problem with the variability of the weight. In a sample of 11 of the bowling balls the sample standard deviation was found to be 0.71 pounds. Construct a 95% confidence interval for the variance of the bowling ball weight. Assume normality.

Answer 1

Answer: $0.2461<\sigma^2<1.5511$

Step-by-step explanation:

Given : A company produces a women's bowling ball that is supposed to weigh exactly 14 pounds.

Sample size : n=11

Degree of freedom =n-1=10

Sample standard deviation : s= 0.71 pounds

Significance level for 95% confidence interval :$\alpha=1-0.95=0.05$

We assume that the bowling ball weight is normally distributed.

Using chi-square distribution table, the required critical values are :-

$\chi^2_{df, \alpha/2}=20.48$

$\chi^2_{df, 1-\alpha/2}=3.25$

Then, the 95% confidence interval for the variance of the bowling ball weight will be :

$\dfrac{s^2(n-1)}{\chi_{\alpha/2}}<\sigma^2<\dfrac{s^2(n-1)}{\chi_{1-\alpha/2}}$

$=\dfrac{(0.71)^2(10)}{20.48}<\sigma^2<\dfrac{(0.71)^2(10)}{3.25}\\\\=0.2461<\sigma^2<1.5511$

∴ The 95% confidence interval for the variance of the bowling ball weight will be : $0.2461<\sigma^2<1.5511$