Questions: Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 182 eligible voters aged 18 - 24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that exactly 44 voted The probability that exactly 44 of 182 eligible voters voted is (Round to four decimal places as needed.)

Solution Steps

To approximate the probability of exactly 44 voters out of 182 voting, we first calculate the mean \( \mu \) and standard deviation \( \sigma \) of the binomial distribution.

The mean is given by: \[ \mu = n \cdot p = 182 \cdot 0.22 = 40.04 \]

The variance \( \sigma^2 \) is calculated as: \[ \sigma^2 = n \cdot p \cdot q = 182 \cdot 0.22 \cdot (1 - 0.22) = 31.2312 \]

Thus, the standard deviation is: \[ \sigma = \sqrt{npq} = \sqrt{31.2312} \approx 5.5885 \]

Next, we apply the continuity correction and calculate the z-scores for \( x = 44.5 \) and \( x = 43.5 \).

For \( x = 44.5 \): \[ z = \frac{X - \mu}{\sigma} = \frac{44.5 - 40.04}{5.5885} \approx 0.7981 \]

For \( x = 43.5 \): \[ z = \frac{X - \mu}{\sigma} = \frac{43.5 - 40.04}{5.5885} \approx 0.6191 \]

Using the z-scores, we find the probability that the number of voters is exactly 44 by calculating the difference in the cumulative distribution function values at these z-scores.

Let \( \Phi(z) \) denote the cumulative distribution function of the standard normal distribution. The probability is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.7981) - \Phi(0.6191) \approx 0.0555 \]

Final Answer