Answer to Question #305078 in Statistics and Probability for mohamed MD

Question

Suppose that the standard deviation of the tube life of a particular brand of TV picture

tube is known to be 500, the population of tube life cannot be assumed to be normally

distributed. However, the sample mean of x = 8900 is based on a sample of n = 35.

Construct the 95% confidence interval for estimating the population mean.

Answer 1

By the Central Limit Theorem if $n$ is sufficiently large, $\bar{X}$ has approximately a normal distribution with $\mu_{\bar{X}}=\mu$ and $\sigma_{\bar{X}}^2=\sigma^2/n.$

The larger the value of $n,$ the better the approximation.

The Central Limit Theorem can generally be used if $n>30.$

The critical value for $\alpha = 0.05$ is $z_c = z_{1-\alpha/2} = 1.96.$

The corresponding confidence interval is computed as shown below:

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

$$=(8900-1.96\times\dfrac{500}{\sqrt{35}}, 8900+1.96\times\dfrac{500}{\sqrt{35}})$$

$$=(8734.35, 9065.65)$$

Therefore, based on the data provided, the 95% confidence interval for the population mean is $8734.35<\mu<9065.65,$ which indicates that we are 95% confident that the true population mean $\mu$ is contained by the interval  $(8734.35, 9065.65).$