Question

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who are employed full time. For which of the following situations will the estimate tend to be closest to the actual value of \(p\) ? $$ \begin{array}{ll} \text { i. } & n=500, p=0.6 \\ \text { ii. } & n=450, p=0.7 \\ \text { iii. } & n=400, p=0.8 \end{array} $$

Short answer

The sample proportion estimate will tend to be closest to the actual value of the population proportion in situation i. (n=500, p=0.6).

Step-by-step solution

Analyze the given scenarios

We are given 3 scenarios with different sample sizes (n) and proportions (p). The objective is to determine in which scenario the sample proportion estimate would be closest to the actual population proportion.

Compare sample sizes

The larger the sample size, the closer the sample proportion will be to the population proportion. Comparing the sample sizes among the three scenarios, scenario i. has the largest sample size (n = 500). Therefore, all other things being equal, scenario i. will tend to produce an estimate closer to the actual population proportion.

Compare proportions

The proportion is closer to 0.5 in scenario i. (0.6) compared to the other two scenarios ii. (0.7) and iii. (0.8). This is important as the sampling distribution of the sample proportion is more likely to be normally distributed when the population proportion is close to 0.5, giving a more accurate estimate.

Combine the results from step 2 and 3

After comparing both the sample sizes and proportions, it appears that scenario i. with n = 500 and p = 0.6 will tend to provide the best estimate of the actual population proportion, due to the larger sample size and proportion closer to 0.5.