In a survey of 1063 ​people, 791 people said they voted in a recent presidential election. Voting records show that 72​% of eligible voters actually did vote. Given that 72​% of eligible voters actually did​ vote, (a) find the probability that among 1063 randomly selected​ voters, at least 791 actually did vote.​ (b) What do the results from part​ (a) suggest?

To tackle this problem, we need to use the normal approximation to the binomial distribution. Here’s the step-by-step solution:

### Part (a): Probability Calculation

1. **Define the parameters**:
   - Sample size \( n = 1063 \)
   - Probability of success \( p = 0.72 \)

2. **Calculate the mean and standard deviation**:
   - Mean \( \mu = np = 1063 \times 0.72 = 765.36 \)
   - Standard deviation \( \sigma = \sqrt{np(1-p)} = \sqrt{1063 \times 0.72 \times 0.28} \approx 14.71 \)

3. **Set up the normal approximation**:
   - We are looking for the probability that at least 791 people voted. For the normal approximation, we use a continuity correction by subtracting 0.5.
   - So, we need \( P(X \geq 790.5) \)

4. **Convert to the standard normal variable**:
   - \( Z = \frac{X - \mu}{\sigma} \)
   - For \( X = 790.5 \):
     \[
     Z = \frac{790.5 - 765.36}{14.71} \approx 1.71
     \]

5. **Find the probability**:
   - We use the standard normal distribution table to find the area to the right of \( Z = 1.71 \).
   - \( P(Z > 1.71) \approx 0.0436 \)

So, the probability that at least 791 people actually did vote is approximately \( 0.0436 \) or \( 4.36\% \).

### Part (b): Interpretation of the Result

The results from part (a) suggest that there is a relatively low probability (4.36%) that, given the true proportion of 72% of eligible voters who voted, we would find 791 or more voters out of a sample of 1063. This discrepancy suggests that the survey result (791 out of 1063) is unusually high compared to what would be expected based on the actual voting records. This might indicate a bias in the survey responses, such as overreporting of voting behavior by the respondents.

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Do you want details or have any questions? Here are some related questions for further exploration:

1. What are the assumptions behind using the normal approximation to the binomial distribution?
2. How does the continuity correction improve the accuracy of the normal approximation?
3. What is the central limit theorem, and how is it applied in this context?
4. How would the results change if the sample size were larger or smaller?
5. What other factors could contribute to the discrepancy between the survey results and actual voting records?
6. How can survey bias be minimized in studies like this?
7. What is the impact of nonresponse bias on survey results?
8. How do you interpret the p-value obtained in the context of hypothesis testing?

**Tip:** When working with probabilities and distributions, always verify if the conditions for the approximations (like normal approximation to the binomial distribution) are met to ensure accurate results.

Explore the probability calculation for voting behavior in presidential elections based on survey data. Understand how the normal approximation to the binomial distribution is applied to determine the likelihood of obtaining a certain number of votes. Suitable for undergraduate level students and those interested in statistics and probability.

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