To determine the minimum sample size [tex]\( n \)[/tex] that the political analyst needs to meet the specified requirements, let's break down the necessary components step-by-step:
1. The value of [tex]\( 3 \% \)[/tex] represents the margin of error. The margin of error quantifies the maximum expected difference between the true population parameter and the sample estimate due to sampling variability.
2. In the formula [tex]\( n = \frac{{z^2 \cdot \hat{p} \cdot (1 - \hat{p})}}{{(\text{margin of error})^2 }} \)[/tex], the term [tex]\( (1 - \hat{p}) \)[/tex] represents the proportion of the population that did not prefer candidate A's policy. Specifically, if [tex]\( \hat{p} \)[/tex] is the proportion of people who prefer candidate A's policy, then [tex]\( 1 - \hat{p} \)[/tex] is the proportion of people who prefer either candidate B's policy or have no preference.
3. When [tex]\( n \)[/tex] is calculated and rounded, the result is 1033. This means that, to meet the analyst's precision and confidence level requirements, the minimum sample size required is 1033 respondents.
So, the complete statements are:
1. The value of [tex]\( 3 \% \)[/tex] represents the margin of error.
2. In the formula [tex]\( n = \frac{{z^2 \cdot \hat{p} \cdot (1 - \hat{p})}}{{(\text{margin of error})^2 }} \)[/tex], the term [tex]\( (1 - \hat{p}) \)[/tex] is the proportion of the population that did not prefer candidate A's policy.
3. When [tex]\( n \)[/tex] is calculated and rounded, the result is 1033.
