Voter Intent. A poll for the presidential campaign sampled...

Voter Intent. A poll for the presidential campaign sampled 491 potential voters in June. A primary purpose of the poll was to obtain an estimate of the proportion of potential voters who favored each candidate. Assume a planning value of $p^*=.50$ and a $95 \%$ confidence level. a. For $p^*=.50$, what was the planned margin of error for the June poll? b. Closer to the Nowember election, better precision and smaller margins of error are desired. Assume the following margins of error are requested for surveys to be conducted during the presidential campaign. Compute the recommended sample size for each survey. $$ \begin{array}{lc} \text { Survey } & \text { Margin of Error } \\ \text { September } & .04 \\ \text { October } & .03 \\ \text { Early November } & .02 \\ \text { Pre-Election Day } & .01 \end{array} $$

Voter Intent. A poll for the presidential campaign sampled 491 potential voters in June. A primary purpose of the poll was to obtain an estimate of the proportion of potential voters who favored each candidate. Assume a planning value of $p^*=.50$ and a $95 \%$ confidence level.
a. For $p^*=.50$, what was the planned margin of error for the June poll?
b. Closer to the Nowember election, better precision and smaller margins of error are desired. Assume the following margins of error are requested for surveys to be conducted during the presidential campaign. Compute the recommended sample size for each survey.
$$
\begin{array}{lc}
\text { Survey } & \text { Margin of Error } \\
\text { September } & .04 \\
\text { October } & .03 \\
\text { Early November } & .02 \\
\text { Pre-Election Day } & .01
\end{array}
$$

Key Concepts

Margin of Error

The margin of error quantifies the uncertainty inherent in survey estimates and reflects the maximum expected difference between the sample proportion and the true population proportion. It is computed using the formula involving the z-score (which corresponds to the chosen confidence level) and the standard error of the proportion, effectively encapsulating the variability inherent in sampling.

Confidence Level

The confidence level indicates how sure we are that the interval we calculate from our sample data contains the true population parameter. For example, a 95% confidence level means that if the same survey were repeated multiple times, we would expect the resulting confidence intervals to contain the true population proportion 95% of the time.

Planning Value (p*)

The planning value, often denoted as p*, is an assumed value for the population proportion used during the design phase of a study. A common choice is p* = 0.50 because it maximizes the product p*(1-p) and thereby represents the worst-case scenario in terms of variance, ensuring that the sample size determined will be sufficiently large to meet the desired margin of error.

Sample Size Determination for Proportions

Determining the necessary sample size to achieve a specified margin of error involves rearranging the margin of error formula for proportions. The formula, n = (z-score^2 * p*(1-p))/(margin of error^2), enables researchers to calculate the number of respondents required to estimate a population proportion with the desired precision given a particular confidence level.

Z-Score for Confidence Level

The z-score or critical value associated with a specified confidence level, such as 1.96 for a 95% confidence interval, is a statistic from the normal distribution. It represents the number of standard deviations a point is from the mean and is crucial for converting the standard error into an actual margin of error in the context of constructing confidence intervals.

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