Question

How large a sample n would you need to estimate p with margin of error 0.04 with 95% confidence? Assume that you don’t know anything about the value of p . 1037 256 601 423

Answer 1

`n=(\hat p (1-\hat p))/(((ME)/(z))^2)` (b)

Since we don't know about p we can assume
`\hat p =0.5`. And replacing into equation (b) the values from part a we got:

`n=(0.5(1-0.5))/(((0.04)/(1.96))^2)=600.25`

And rounded up we have that n=601

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\hat p` estimation for the sample proportion

n sample size selected

Confidence =0.95 or 95%

The population proportion have the following distribution

`p \sim N(p,\sqrt{(\hat p(1-\hat p))/(n)})`

Solution to the problem

In order to find the critical values we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by
`\alpha=1-0.95=0.05` and
`\alpha/2 =0.025`. And the critical values would be given by:

`z_(\alpha/2)=-1.96, z_(1-\alpha/2)=1.96`

The margin of error for the proportion interval is given by this formula:

`ME=z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}` (a)

And on this case we have that
`ME =\pm 0.04` and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=(\hat p (1-\hat p))/(((ME)/(z))^2)` (b)

Since we don't know about p we can assume
`\hat p =0.5`. And replacing into equation (b) the values from part a we got:

`n=(0.5(1-0.5))/(((0.04)/(1.96))^2)=600.25`

And rounded up we have that n=601