Question

An environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of 55​% and 9090​% confidence. To gauge the size of the​ problem, the agency first picks 7070 cars and finds 1414 with faulty emissions systems. How many should be sampled for a full​ investigation?

Answer 1

`n=(0.2(1-0.2))/(((0.05)/(1.64))^2)=172.13`

And rounded up we have that n=173

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".

`p` represent the real population proportion of interest

`\hat p` represent the estimated proportion for the sample

n is the sample size required (variable of interest)

`z` represent the critical value for the margin of error

Solution to the problem

The population proportion have the following distribution

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

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 90% of confidence, our significance level would be given by
`\alpha=1-0.90=0.10` and
`\alpha/2 =0.05`. And the critical value would be given by:

`z_(\alpha/2)=-1.64, z_(1-\alpha/2)=1.64`

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 0.05` 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)

We can assume that the estimates proportion is
`\hat p=(14)/(70)=0.2`. And replacing into equation (b) the values from part a we got:

`n=(0.2(1-0.2))/(((0.05)/(1.64))^2)=172.13`

And rounded up we have that n=173