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

Answer:

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.

Step-by-step explanation:

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

$\alpha = \frac{1-0.96}{2} = 0.02$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

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

In this problem, we have that:

$\sigma = 40, M = 10$

$M = z*\frac{\sigma}{\sqrt{n}}$

$10 = 2.055*\frac{40}{\sqrt{n}}$

$10\sqrt{n} = 82.2$

$\sqrt{n} = 8.22$

$n = 67.6$

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.