Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. He wants to construct the 95% confidence interval with a maximum error of 0.19 reproductions per hour. Assuming that the mean is 6.4 reproductions and the standard deviation is known to be 1.8, what is the minimum sample size required for the estimate? Round your answer up to the next integer.

Answer 1

The minimum sample size required for the estimate is 345.

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`.

That is z with a pvalue of
`1 - 0.025 = 0.975`, so Z = 1.96.

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.

The standard deviation is known to be 1.8.

This means that
`\sigma = 1.8`

What is the minimum sample size required for the estimate?

This is n for which M = 0.19. So

`M = z(\sigma)/(√(n))`

`0.19 = 1.96(1.8)/(√(n))`

`0.19√(n) = 1.96*1.8`

`√(n) = (1.96*1.8)/(0.19)`

`(√(n))^2 = ((1.96*1.8)/(0.19))^2`

`n = 344.8`

Rounding up to the next integer:

The minimum sample size required for the estimate is 345.