Question

A researcher wishes to​ estimate, with 99​% ​confidence, the population proportion of adults who think the president of their country can control the price of gasoline. Her estimate must be accurate within 3​% of the true proportion. ​a) No preliminary estimate is available. Find the minimum sample size needed. ​b) Find the minimum sample size​ needed, using a prior study that found that 26​% of the respondents said they think their president can control the price of gasoline. ​c) Compare the results from parts​ (a) and​ (b).

Answer 1

Answer:

a) $n=\frac{0.5(1-0.5)}{(\frac{0.03}{2.58})^2}=1849$  

And rounded up we have that n=1849

b) $n=\frac{0.26(1-0.26)}{(\frac{0.03}{2.58})^2}=1422.99$  

And rounded up we have that n=1423

c) For this case we can see that if we have a prior estimate the minimum sample size required for the margin of error desired would be less as we can see in part b we reduce the sample size compared to the part a

Step-by-step explanation:

Part a

The critical value for a confidence level of 99% is for this case $z_{\alpha/2} =2.58$

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 $ME =\pm 0.03$ 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)  

Since we don't know any estimate for the true proportion we can use $\hat p =0.5$ as a godd estimator. And replacing into equation (b) the values from part a we got:

$n=\frac{0.5(1-0.5)}{(\frac{0.03}{2.58})^2}=1849$  

And rounded up we have that n=1849

Part b

For this case we have a prior estimate $\hat p =0.26$ and replacing we got:

$n=\frac{0.26(1-0.26)}{(\frac{0.03}{2.58})^2}=1422.99$  

And rounded up we have that n=1423

Part c

For this case we can see that if we have a prior estimate the minimum sample size required for the margin of error desired would be less as we can see in part b we reduce the sample size compared to the part a