Question

Lifetimes of AAA batteries are approximately normally distributed. A manufacturer wants to estimate the standard deviation of the lifetimes of the AAA batteries it produces. A random sample of 23 batteries produced by the manufacturer lasted a mean of 10.3 hours with a standard deviation of 2.4 hours. Find a 95% confidence interval for the population standard deviation of the lifetimes of the batteries produced by the manufacturer.

Answer 1

Answer:

95% confidence interval for the population standard deviation of the lifetimes of the batteries produced by the manufacturer.

(8.889, 11.7106)

Step-by-step explanation:

Step(i):-

Given sample size 'n' = 23

Mean of the sample x⁻ = 10.3

Standard deviation of the sample (s) = 2.4

Level of significance = 0.05

Degrees of freedom = n-1 = 23-1 =22

Step(ii):-

95% confidence interval for the population standard deviation of the lifetimes of the batteries produced by the manufacturer.

$(x^{-} - t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } , x^{-} +t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } )$

$(10.3 - t_{\frac{0.05}{2} } \frac{2.4}{\sqrt{23} } , 10.3 +t_{\frac{0.05}{2} } \frac{2.4}{\sqrt{23} } )$

(10.3 -  2.8188 (0.50043) , 10.3 + 2.8188(0.50043)

(10.3-1.4106 , 10.3+1.4106)

(8.889, 11.7106)

final answer:-

95% confidence interval for the population standard deviation of the lifetimes of the batteries produced by the manufacturer.

(8.889, 11.7106)