use a normal approximation to find the probability of the Indicated number of voters In this case, assume that 144 aged 18-24 are randomly selected showed that among eligible voters aged 18-24, 22% of them voted Probability that exactly 35 voted asnwer rounded to 4 decimal places

To find the probability that exactly 35 out of 144 voters aged 18-24 voted, using a normal approximation to the binomial distribution, follow these steps:

1. **Identify the binomial distribution parameters**:
   - \( n = 144 \) (the number of trials)
   - \( p = 0.22 \) (the probability of success on each trial)

2. **Calculate the mean (\(\mu\)) and standard deviation (\(\sigma\))**:
   - \(\mu = n \times p = 144 \times 0.22 = 31.68\)
   - \(\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{144 \times 0.22 \times 0.78} = \sqrt{24.7392} \approx 4.974\)

3. **Convert the binomial probability to a normal probability**:
   - The continuity correction factor is used to improve the approximation. For the probability of exactly 35 voters, we use the range from 34.5 to 35.5.
   - Standardize these values using the z-score formula: \( z = \frac{x - \mu}{\sigma} \)

4. **Calculate the z-scores**:
   - For \( x = 34.5 \): \( z = \frac{34.5 - 31.68}{4.974} \approx 0.5674 \)
   - For \( x = 35.5 \): \( z = \frac{35.5 - 31.68}{4.974} \approx 0.7693 \)

5. **Find the probability using the standard normal distribution**:
   - Use the z-scores to find the probabilities from the standard normal distribution table or a calculator.
   - \( P(Z \leq 0.7693) \approx 0.7794 \)
   - \( P(Z \leq 0.5674) \approx 0.7146 \)

6. **Calculate the probability**:
   - The probability that exactly 35 voters voted is the difference between these probabilities:
   \[
   P(34.5 \leq X \leq 35.5) = P(Z \leq 0.7693) - P(Z \leq 0.5674) = 0.7794 - 0.7146 = 0.0648
   \]

Thus, the probability that exactly 35 out of 144 voters aged 18-24 voted is approximately \( \boxed{0.0648} \).

Would you like more details or have any questions about this problem?

Here are some related questions you might consider:

1. How would the probability change if the sample size was 200 instead of 144?
2. What is the probability that at most 35 voters out of 144 voted?
3. What is the probability that more than 40 voters out of 144 voted?
4. How would the result change if the voting percentage was 25% instead of 22%?
5. Can we use the normal approximation for a sample size of 50?

**Tip:** When using the normal approximation for the binomial distribution, ensure that \(np\) and \(n(1-p)\) are both greater than 5 to justify the approximation.

Learn how to use normal approximation to find the probability that exactly 35 out of 144 voters aged 18-24 voted. This example covers binomial distribution, normal approximation, and uses key formulas for mean, variance, and standard deviation. Suitable for undergraduate students studying probability theory.

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