Answer to Question #275154 in Statistics and Probability for Jack

Question

Through careful record-keeping, you have the times to burn out of 250 Bright brand light bulbs. The average time to burn out is 400 hours, with a standard deviation of 14 hours. Give a 95% confidence interval for the meantime to burn out of this brand of the light bulb. Round your answers to 4 decimal places

Answer 1

The critical value for $\alpha = 0.05$ and $df = n-1 = 249$ degrees of freedom is $t_c = z_{1-\alpha/2; n-1} = 1.969537.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-t_c\times\dfrac{s}{\sqrt{n}},\bar{x}+t_c\times\dfrac{s}{\sqrt{n}})$$

$$=(400-1.969537\times\dfrac{14}{\sqrt{250}},400+1.969537\times\dfrac{14}{\sqrt{250}})$$

$$=(398.256,401.744)$$

Therefore, based on the data provided, the 95% confidence interval for the population mean is $398.256 < \mu < 401.744,$ which indicates that we are 95% confident that the true population mean $\mu$ is contained by the interval $(398.256, 401.744).$