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The sample proportion is a crucial idea in statistics, especially when interpreting survey results. In this particular survey, 70% of 1002 people said they voted in the recent presidential election. We call this 70% the sample proportion, denoted as \(\bar{p}\). The formula for the sample proportion is simple: \[ \bar{p} = \frac{x}{n} \] where \(x\) is the number of people who responded a certain way (in this case, people who said they voted), and \(n\) is the total number of people surveyed. Understanding the sample proportion helps us generalize the survey findings to a larger population, but we must also consider the sample's reliability and error.

The standard error (SE) measures the accuracy with which a sample proportion represents the entire population. In our example, the SE gives us an idea of how much the proportion of people who said they voted might vary from the true proportion of all eligible voters.

We calculate SE using the formula: \[ \text{SE} = \sqrt{ \frac{ \bar{p}(1 - \bar{p}) }{ n } } \]

For our survey:
\[ \text{SE} = \sqrt{ \frac{0.70 \times (1 - 0.70)}{1002} } \approx 0.0144 \]

A smaller SE indicates that the sample proportion is a more accurate reflection of the population proportion. Conversely, a larger SE suggests more variability and potentially less reliability in the sample.

The margin of error (ME) provides a range within which we expect the true proportion of the population to fall. This measure takes into account the SE and the critical value \(z^*\) from the standard normal distribution (z-table).

For a 98% confidence level, \(z^*\) is approximately 2.326. The formula for ME is: \[ \text{ME} = z^* \times \text{SE} \]

Using our survey data: \[ \text{ME} = 2.326 \times 0.0144 \approx 0.0335 \]

The margin of error allows us to construct a confidence interval around the sample proportion, indicating that we're fairly confident (98% in this case) that the true population proportion lies within this range.

Voter turnout represents the percentage of eligible voters who actually cast their votes in an election. In our exercise, the actual voter turnout is noted to be 61%, or 0.61.

To determine if survey results are consistent with actual voter turnout, we compare the actual turnout to our confidence interval. If the true voter turnout falls within this interval, we can say the survey is consistent with the actual turnout.

However, in this case, the 98% confidence interval for the proportion of people who said they voted is (0.6665, 0.7335). Since 0.61 is not within this range, the survey results are not consistent with the actual voter turnout. This discrepancy can result from various factors such as sampling error, response bias, or inaccuracies in self-reporting.