Question

The lifetime of a certain brand of electric light bulb is known to have a standard deviation of 49 hours. Suppose that a random sample of 90 bulbs of this brand has a mean lifetime of 500 hours. Find a 90% confidence interval for the true mean lifetime of all light bulbs of this brand. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. What is the lower limit of the 90% confidence interval? What is the upper limit of the 90% confidence interval? x 5 ?

Answer 1

Solution :

Given that,

Point estimate = sample mean = $\bar x$ = 500

Population standard deviation =   $\sigma$ = 49
Sample size = n =90

At 90% confidence level

$\alpha$ = 1 - 90%  

$\alpha$ = 1 - 0.90 =0.10

$\alpha$/2 = 0.05

Z$\alpha$/2 = Z_{0.05 = 1.645}

Margin of error = E = Z$\alpha$/2 * ( $\sigma$ /$\sqrt$n)

= 1.645 * (49 /  $\sqrt$90 )

= 8.497
At 90% confidence interval estimate of the population mean
is,

$\bar x$ - E < $\mu$ < $\bar x$ + E

500- 8.497 <  $\mu$ <500 + 8.497

491.5 <  $\mu$ < 508.5

(lower limit =491.5 ,upper limit = 508.5)