Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. He wants to construct the 99% confidence interval with a maximum error of 0.1 reproductions per hour. Assuming that the mean is 6.1 reproductions and the variance is known to be 5.76, 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 of 3820.

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 - 0.99)/(2) = 0.005`

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 - 0.005 = 0.995`, so Z = 2.575.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

Variance is known to be 5.76

This means that
`\sigma = √(5.76) = 2.4`

What is the minimum sample size required for the estimate?

Maximum error of 0.1 means that we find n for which M = 0.1. So

`M = z(\sigma)/(√(n))`

`0.1 = 2.575(2.4)/(√(n))`

`0.1√(n) = 2.575*2.4`

Multiplying both sides by 10

`√(n) = 2.575*24`

`(√(n))^2 = (2.575*24)^2`

`n = 3819.2`

Rounding up:

The minimum sample size required for the estimate is of 3820.