Question

1. The quality control manager at a light-bulb factory needs to estimate the mean life of a new type of light-bulb. The population standard deviation is assumed to be 60 hours. A random sample of 32 light-bulbs shows a sample mean life of 490 hours. Construct and explain a 95% confidence interval estimate of the population mean life of the new light-bulb.

Answer 1

The 95% confidence interval estimate of the population mean life of the new light-bulb is (469.21 hours, 510.79 hours).

This confidence level means that we are 95% sure that the true population mean life of the new light bulb is in this interval.

Explanation:

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

`\alpha = (1-0.95)/(2) = 0.025`

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
`1-0.025 = 0.975`, so
`z = 1.96`

Now we find M as such:

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the length of the sample. So:

`M = 1.96*(60)/(√(32)) = 20.79`

The lower end of the interval is the mean subtracted by M. So it is 490 - 20.79 = 469.21 hours.

The upper end of the interval is the mean added to M. So it is 490 + 20.79 = 510.79 hours.

The 95% confidence interval estimate of the population mean life of the new light-bulb is (469.21 hours, 510.79 hours).

This confidence level means that we are 95% sure that the true population mean life of the new light bulb is in this interval.