Question

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, How large a sample is needed if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean?

Answer 1

A sample of at least 68 bulbs is needed to be 96% confident that our sample mean will be within 10 hours of the true mean.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.96)/(2) = 0.02`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.02 = 0.98`, so
`z = 2.055`

Now, find the margin M as such

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

In this problem, we have that:

`\sigma = 40, M = 10`

`M = z*(\sigma)/(√(n))`

`10 = 2.055*(40)/(√(n))`

`10√(n) = 82.2`

`√(n) = 8.22`

`n = 67.6`

A sample of at least 68 bulbs is needed to be 96% confident that our sample mean will be within 10 hours of the true mean.