Question

g A researcher wants to construct a 95% confidence interval for the proportion of children aged 8-10 living in Atlanta who own a cell phone. Estimate the sample size needed if no estimate of p is available so that the confidence interval will have a margin of error of 0.02.

Answer 1

`n=(0.5(1-0.5))/(((0.02)/(1.96))^2)=2401`

And rounded up we have that n=2401

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".

Solution to the problem

In order to find the critical value 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 value 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 0.02` 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)

For this case we can use an estimator for the population proportion as
`\hat p = 0.5`. And replacing into equation (b) the values from part a we got:

`n=(0.5(1-0.5))/(((0.02)/(1.96))^2)=2401`

And rounded up we have that n=2401

Answer 2

The sample size needed is 2401.

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

The margin of error of the interval is:

`M = z\sqrt{(\pi(1-\pi))/(n)}`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

Estimate the sample size needed if no estimate of p is available so that the confidence interval will have a margin of error of 0.02.

We have to find n, for which M = 0.02.

There is no estimate of p available, so we use
`\pi = 0.5`

`M = z\sqrt{(\pi(1-\pi))/(n)}`

`0.02 = 1.96\sqrt{(0.5*0.5)/(n)}`

`0.02√(n) = 1.96*0.5`

`√(n) = (1.96*0.5)/(0.02)`

`(√(n))^(2) = ((1.96*0.5)/(0.02))^(2)`

`n = 2401`

The sample size needed is 2401.