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 22%.
A)264
B)186
C)339
D)4

Question

Quizplus · Accepted Answer

The sample size needed for estimating a proportion with a specific margin of error (E) at a certain confidence level can be calculated using the formula: $n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}$, where $Z$ is the Z-score corresponding to the desired confidence level (for 95% confidence, $Z = 1.96$), $p$ is the estimated proportion (0.22 in this case), and $E$ is the margin of error (0.05). Plugging in the values: $n = \frac{1.96^2 \cdot 0.22 \cdot (1-0.22)}{0.05^2}$, which calculates to approximately 264. This means a sample size of 264 is needed to meet the criteria.

A Researcher Wishes to Estimate the Number of Households with Two