Chapter 8: Problem 42

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 November 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.

Short Answer

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a. The planned margin of error for the June poll is approximately 0.069 or 6.9%. b. The recommended sample sizes for each requested margin of error can be calculated using the formula: \[n = \frac{(1.96 \sqrt{.50(1-.50)})^2}{E^2}\] By plugging in the values for each requested margin of error, we can calculate the corresponding sample sizes.

Step by step solution

Determine the z-score

For a 95% confidence level, the z-score is \(1.96\). This value can be found using a standard normal table or calculator.

Calculate the margin of error

Now, we plug in the given values into the margin of error formula: \[E = 1.96 \sqrt{\frac{.50(1-.50)}{491}}\]

Simplify and compute the margin of error

Simplify the equation and calculate the margin of error: \[E = 1.96 \sqrt{\frac{.25}{491}} = 1.96(.0353) \approx 0.069\] The planned margin of error for the June poll is approximately 0.069 or 6.9%. #b. Finding the recommended sample size for each survey#

Rearrange the margin of error formula

Rearrange the margin of error formula to solve for the sample size \(n\): \[n = \frac{(z \sqrt{p^{*}(1-p^{*})})^2}{E^2}\]

Calculate the sample size for each requested margin of error

For each requested margin of error, plug in the values into the formula and solve for \(n\). Round up to the nearest whole number as it represents the number of voters. The recommended sample sizes for each requested margin of error are as follows: 1. Margin of Error 1: \[n_1 = \frac{(1.96 \sqrt{.50(1-.50)})^2}{E_1^2}\] 2. Margin of Error 2: \[n_2 = \frac{(1.96 \sqrt{.50(1-.50)})^2}{E_2^2}\] 3. Margin of Error 3: \[n_3 = \frac{(1.96 \sqrt{.50(1-.50)})^2}{E_3^2}\] 4. Margin of Error 4: \[n_4 = \frac{(1.96 \sqrt{.50(1-.50)})^2}{E_4^2}\] Calculate the sample sizes and substitute the values for each requested margin of error to get the recommended sample sizes for each survey.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error Calculation

Understanding the margin of error in polling is crucial for interpreting survey results accurately. It represents the range within which the true value of the population parameter is expected to fall, with a certain level of confidence. To calculate it, one must identify the confidence level (typically expressed as a z-score), the sample proportion (\( p^* \)), and the sample size (\( n \)).

The formula used for the margin of error calculation is: \[ E = z \sqrt{\frac{p^*(1-p^*)}{n}} \]
In our example, with a sample size of 491, a sample proportion (\( p^* \_ .50 \)), and a z-score corresponding to a 95% confidence level (1.96), the margin of error was determined to be approximately 6.9%. Knowing this allows researchers and the audience to understand that the true proportion of potential voters favoring a candidate could range 6.9% above or below the poll's results.

Sample Size Determination

When conducting a poll or survey, selecting the correct sample size is fundamental to obtaining valid and precise results. The sample size influences the margin of error; a larger sample size provides a smaller margin of error, suggesting more precise results. To determine the sample size needed to achieve a specific margin of error, the formula is restructured as follows:

\[ n = \left(\frac{z \sqrt{p^*(1-p^*)}}{E}\right)^2 \]
Here, \( n \) represents the desired sample size, \( z \) is the z-score based on the confidence level, \( p^* \) is the sample proportion, and \( E \) is the margin of error.

For various margins of error requested in the presidential campaign's surveys, we apply the same formula by inserting different values of \( E \) for each scenario. The calculated sample sizes for each survey will ensure the margin of error is tight enough to reflect the precision required as the election approaches.

Confidence Level Analysis

The confidence level in statistics signifies the degree to which one can be sure that the sample results reflect the actual population, within the margin of error. Commonly chosen levels are 90%, 95%, and 99%, with higher percentages indicating greater certainty. The confidence level corresponds to a z-score in the normal distribution, which can be obtained from statistical tables or calculators.

For a poll with a 95% confidence level, as in the presidential campaign example, the corresponding z-score is 1.96. This reflects that if the same poll were repeated numerous times, we could expect that 95% of the calculated confidence intervals would contain the true population parameter. Consequently, the confidence level is directly related to the z-score used in margin of error and sample size calculations, playing a role in determining the reliability and accuracy of survey results.