Question

The population standard deviation for the typing speeds for secretaries is 4 words per minute. If we want to be 90% confident that the sample mean is within 1 word per minute of the true population mean, what is the minimum sample size that should be taken

Answer 1

44

Explanation:

The formula for sample size isn=z2σ2EBM2

In this formula,z=zα2=z0.05=1.645 because the confidence level is 90%. From the problem, we know that

σ=4 and EBM=1.

Therefore, n=z2σ2EBM2=(1.645)2(4)212

≈43.30

Use n=44 to ensure that the sample size is large enough.

Answer 2

The minimum sample size that should be taken is 62.

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 size of the sample.

If we want to be 90% confident that the sample mean is within 1 word per minute of the true population mean, what is the minimum sample size that should be taken

This is n when
`M = 1, \sigma = 4`. So

`M = z*(\sigma)/(√(n))`

`1 = 1.96*(4)/(√(n))`

`√(n) = 4*1.96`

`(√(n))^(2) = (4*1.96)^(2)`

`n = 61.5`

The minimum sample size that should be taken is 62.