Question

A study is going to be conducted in which a mean of a lifetime of batteries produced by a certain method will be estimated using a 90% confidence interval. The estimate needs to be within +/- 2 hours of the actual population mean. The population standard deviation s is estimated to be around 25. The necessary sample size should be at least _______.

Answer 1

The necessary sample size should be at least 423.

Explanation:

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

`\alpha = (1-0.9)/(2) = 0.05`

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
`1-0.05 = 0.95`, so
`z = 1.645`

Now, find the margin of error 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.

In this problem, we have that:

`M = 2, \sigma = 25`. So

`2 = 1.645*(25)/(√(n))`

`2√(n) = 41.125`

`√(n) = 20.5625`

`√(n)^(2) = (20.5625)^(2)`

`n = 422.81`

The necessary sample size should be at least 423.