Question

A researcher wishes to estimate the number of households with two cars. how large a sample is needed in order to be 95% confident that the sample proportion will not differ from the true proportion by more than 5%? a previous study indicates that the proportion of households with two cars is 24%

Answer 1

n = 2553

According to one source, the answer to this question is by determining the sample size of the given values. When encountered with questions that relate on how large a sample should yield a significant result is answered by determining the sample size. The sample size is the determination of how much sample is needed for the study to reveal confidence levels of 95% or higher. 

Answer 2

Answer:

$n=\frac{0.24(1-0.24)}{(\frac{0.05}{1.96})^2}=280.283$  

And rounded up we have that n=281

Step-by-step 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".  

The population proportion have the following distribution

$p \sim N(p,\sqrt{\frac{p(1-p)}{n}})$

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:  

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

And on this case we have that $ME \leq \pm 0.05$ and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

$n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}$   (b)  

And replacing into equation (b) the values from part a we got:

$n=\frac{0.24(1-0.24)}{(\frac{0.05}{1.96})^2}=280.283$  

And rounded up we have that n=281