Answer to Question #344804 in Statistics and Probability for Christian

Question

Through careful record keeping, you have the times to burn out of 100 Bright brand light bulbs. The average time to burn out is 200 hours, with a standard deviation of 12 hours. Give a 95% confidence interval for the mean time to burn out of this brand of light bulb.

Answer 1

The critical value for $\alpha = 0.05$ and $df = n-1 = 99$ degrees of freedom is (using critical values table)

$t_c = z_{1-\alpha/2; n-1} = 1.984$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-t_c\times\dfrac{s}{\sqrt{n}},\bar{x}+t_c\times\dfrac{s}{\sqrt{n}})$$ $$=(200-1.984\times\dfrac{12}{\sqrt{100}},200+1.984\times\dfrac{12}{\sqrt{100}})$$ $$=(197.62,202.38)$$

Therefore, based on the data provided, the 95% confidence interval for the population mean is $197.62 < \mu < 202.38,$ which indicates that we are 95% confident that the true population mean $\mu$ is contained by the interval $197.62, 202.38).$