A sample of 23 European countries found that the variance on life expectancy was 7.3 years. What is the 95% confidence interval for the variance of life expectancy in Europe?
A) $27.2

Question

A sample of 23 European countries found that the variance on life expectancy was 7.3 years. What is the 95% confidence interval for the variance of life expectancy in Europe? 
A) $27.2 < \sigma ^ { 2 } < 118.3$
B) $56 < \sigma ^ { 2 } < 10.3$
C) $4.4 < \sigma ^ { 2 } < 14.6$
D) $289 < \sigma ^ { 2 } < 1150$

Quizplus · Accepted Answer

We can use the chi-square distribution to construct a confidence interval for the variance. 
Using the chi-square distribution table with 22 degrees of freedom (df = n-1), we can find the chi-square values that correspond to the upper and lower 2.5% tail areas:

Upper tail area = 0.025, chi-square value = 38.0 
Lower tail area = 0.025, chi-square value = 11.0

Then, we can solve for the confidence interval:

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

$22*7.3/38.0 < \sigma ^ { 2 } < 22*7.3/11.0$

$4.4 < \sigma ^ { 2 } < 14.6$

Therefore, the 95% confidence interval for the variance of life expectancy in Europe is 4.4 to 14.6 years squared. The answer is C.

A Sample of 23 European Countries Found That the Variance 27.2<σ2<118.327.2 < \sigma ^ { 2 } < 118.327.2<σ2<118.3

A Sample of 23 European Countries Found That the Variance $27.2 < \sigma ^ { 2 } < 118.3$