Question

An electrical firm which manufactures a certain type of bulb wants to estimate its mean life. The life of the light bulb is normally distributed with a standard deviation of 40 hours. A random sample of 36 bulbs resulted in a mean of 200 hours.

a) (3 points) Construct a 92% confidence interval for the mean life of all light bulbs the firm manufactures.

b) (4 points) How many bulbs should be tested so that we can be 92% confident that the estimate of the mean will not differ from the true mean life by more than 10 hours?

Answer 1

a) The sample mean is $\overline{x}=200$ , the sample standard deviation is $\sigma =40$ and sample size is $n=36$ .

The confidence interval for true mean based on the sample mean is

(1 - a)100%

$\overline{x} \pm z_{1-\alpha /2}\frac{\sigma }{\sqrt{n}}\\$

The 92% CI for mean $\left (\alpha =0.08 \right )$ is

$200 \pm z_{1-0.08 /2}\frac{40}{\sqrt{36}}\\ {\color{Blue} (188.33, 211.67)}$

a) We need

$z_{1-0.08 /2}\frac{40}{\sqrt{n}}<10\\ n> z^2_{1-0.08 /2}40^2/10^2=49.03843$

The required sample size is ${\color{Blue} 50}$ .