In a survey of 1322 people, 841 people said they voted in a recent presidential election. Voting records show that \( 61 \% \) of eligible voters actually did vote. Given that \( 61 \% \) of eligible voters actually did vote, (a) find the probability that among 1322 fandomly selected voters, at least 841 actually did vote. (b) What do the results from part (a) suggest? (a) \( \mathrm{P}(\mathrm{X} \geq 841)=\square \) (Round to four decimal places as needed)

To solve this problem, we want to determine the probability that at least 841 out of 1322 randomly selected voters actually voted when it is given that \( 61\% \) is the proportion of voters who did vote. We can model this situation using the normal approximation to the binomial distribution since the sample size is large. The relevant parameters are: - \( n = 1322 \) (the number of trials), - \( p = 0.61 \) (the probability of success, meaning a person actually voted). First, we calculate the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the binomial distribution: \[ \mu = n \cdot p = 1322 \cdot 0.61 = 806.42 \] \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{1322 \cdot 0.61 \cdot 0.39} \approx 13.66 \] Next, we convert our X value (841) to a Z-score using: \[ Z = \frac{X - \mu}{\sigma} = \frac{841 - 806.42}{13.66} \approx 2.54 \] Now we will use the Z-score to find the probability \( P(X \geq 841) \). We look up the Z-score in the standard normal distribution table (or use a calculator): \[ P(Z \geq 2.54) = 1 - P(Z < 2.54) \approx 1 - 0.9945 = 0.0055 \] Thus, rounding to four decimal places, we find: (a) \( P(X \geq 841) = 0.0055 \) (b) The result of part (a) suggests that observing 841 or more people voting among the 1322 surveyed is quite unlikely under the assumption that \( 61\% \) of the population voted. This low probability (0.0055) indicates that the survey results could be an anomaly or suggest a higher voting participation than the stated \( 61\% \) in the population. It could prompt a reevaluation of voting outreach, demographics of the surveyed population, or the accuracy of the voting records.