Question

How large a sample must be drawn so that a 99.5% confidence interval for µ will have a margin of error equal to 1.1? Round up the answer to the nearest integer. (Round the critical value to no less than three decimal places.) A sample size of________ is needed to be drawn in order to obtain a 99.5% confidence interval with a margin of error equal to 1.1.

Answer 1

A sample size of 2018 is needed to be drawn in order to obtain a 99.5% confidence interval with a margin of error equal to 1.1.

Explanation:

Missing data:

The value of population standard deviation is 17.6

We have been provided with the following data:

Confidence Level = 99.5%

Margin of error = M.E = 1.1

Population standard Deviation =
`\sigma` = 17.6

What we have to do is find the minimum sample size required to keep the margin of error upto 1.1 with 99.5% confidence level.

Since, the value of population standard deviation is known, we will use the formulas of z-distribution to solve the question.

Formula of Margin of Error is:

`M.E=z_{(\alpha )/(2)} * (\sigma)/(√(n) )`

Here,

`z_{(\alpha)/(2)}` is the z critical value for the given confidence level. From the z-table the z-value for 99.5% confidence level comes out to be 2.807

Using the values in above formula, we get:

`1.1=2.807 * (17.6)/(√(n)) \\\\√(n)=2.807 * (17.6)/(1.1)\\\\ n=((2.807 * 17.6)/(1.1) )^(2)\\\\ n=2017.1`

Rounding up to the nearest integer, the minimum sample size required is 2018.

A sample size of 2018 is needed to be drawn in order to obtain a 99.5% confidence interval with a margin of error equal to 1.1.