Question

17. An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. How large a sample is need it if we wish to be 98% confident that our sample mean will be within 4 hours of the true mean

Answer 1

Answer:

A sample of at least 541 is needed if we wish to be 98% confident that our sample mean will be within 4 hours of the true mean.

Step-by-step explanation:

We are given that an electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours.

We have to find a sample such that we are 98% confident that our sample mean will be within 4 hours of the true mean.

As we know that the Margin of error formula is given by;

The margin of error =  $Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }$

where, $\sigma$ = standard deviation = 40 hours

            n = sample size

            $\alpha$ = level of significance = 1 - 0.98 = 0.02 or 2%

Now, the critical value of z at ($\frac{0.02}{2}$ = 1%) level of significance n the z table is given as 2.3263.

So, the margin of error =  $Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }$

                 $4=2.3263 \times \frac{40}{\sqrt{n} }$

                 $\sqrt{n}= \frac{40 \times 2.3263}{ 4}$

                  $\sqrt{n}=23.26$

                   n = $23.26^{2}$ = 541.03 ≈ 541

Hence, a sample of at least 541 is needed if we wish to be 98% confident that our sample mean will be within 4 hours of the true mean.