Question

A company produces electric bulbs. To estimate the average life of the electric
bulbs, the quality inspector of the company selects 80 bulbs randomly.
She finds that the average life of these bulbs is 50 hours. Find the
population average life of the electric bulbs produced by the company
using 95% confidence limits.

Answer 1

A 95% confidence interval for the population average life of the electric bulbs produced by the company is [2488.85 hours, 2511.15 hours].

Step-by-step explanation:

The correct and complete question is: A company produces electric bulbs. To estimate the average life of the electric  bulbs, the quality inspector of the company selects 80 bulbs randomly.  She finds that the average life of these bulbs is 2500 hours and the standard deviation is 50 hours. Find the  population average life of the electric bulbs produced by the company  using 95% confidence limits.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

                             P.Q.  =  $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$  ~ $t_n_-_1$

where, $\bar X$ = sample average life of bulbs = 2500 hours

            s = sample standard deviation = 50 hours

            n = sample of bulbs = 80

            $\mu$ = population average life of the electric bulbs

Here for constructing a 95% confidence interval we have used One-sample t-test statistics as we don't know about population standard deviation.

So, 95% confidence interval for the population mean, $\mu$ is ;

P(-1.994 < $t_7_9$ < 1.994) = 0.95  {As the critical value of t at 79 degrees of

                                             of freedom are -1.994 & 1.994 with P = 2.5%}  

P(-1.994 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 1.994) = 0.95

P( $-1.994 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $1.994 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

P( $\bar X-1.994 \times {\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+1.994 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

95% confidence interval for $\mu$ = [ $\bar X-1.994 \times {\frac{s}{\sqrt{n} } }$ , $\bar X+1.994 \times {\frac{s}{\sqrt{n} } }$ ]

                                      = [ $2500-1.994 \times {\frac{50}{\sqrt{80} } }$ , $2500+1.994 \times {\frac{50}{\sqrt{80} } }$ ]

                                      = [2488.85 hours, 2511.15 hours]

Therefore, a 95% confidence interval for the population average life of the electric bulbs produced by the company is [2488.85 hours, 2511.15 hours].