Q$4.$H$.$W A researcher wants to estimate the mean lifespan of a certain light bulbs. Suppose that the distribution is normal with standard deviation of $5$ hours.

Determine the sample size needed on order that the researcher will be $90\%$ confident that the error will not exceed $2$ hours when he uses the sample mean as a point estimate for the true mean.

Suppose that the researcher selected a random sample of $49$ bulbs and found that the sample mean is $390$ hours.

$($i$)$ Find a good point estimate for the true mean $m.$

$($ii$)$ Find a $95\%$ confidence interval for the true mean $m.$