Question

Use the confidence level and sample data to find a confidence interval for estimating the population mu. Round your answer to the same number of decimal places as the sample mean. A random sample of 90 light bulbs had a mean life of x overbar equals 592 hours with a standard deviation of sigma equals 25 hours. Construct a​ 90% confidence interval for the mean​ life, mu​, of all light bulbs of this type.

Answer 1

Answer:  $(587.67,596.33)$

Step-by-step explanation:

The confidence interval for population mean is given by :-

$\overline{x}\ \pm\ z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

Given : Sample mean : $\overline{x}= 592$ hours

Standard deviation$\sigma= 25$ hours

Sample size : n=90, which is a large sample(n<30), so we use z-test.

Significance level: $1-0.90=0.1$

Critical value : $z_{\alpha/2}=t_{22,0.025}=1.645$

Then , the confidence interval for population mean will be  :-

$592\ \pm\ (1.645)\dfrac{25}{\sqrt{90}}\\\\\approx592\pm4.33\\\\=(592-4.33,592+4.33)\\\\=(587.67,596.33)$

Hence, the ​ 90% confidence interval for the mean​ life $\mu$ of all light bulbs of this type. is $(587.67,596.33)$