Questions: Suppose that 20% of the people in a large city have used a hospital emergency room in the past year. If a random sample of 150 people from the city is taken, approximate the probability that at least 27 used an emergency room in the past year. Use the normal approximation to the binomial with a correction for continuity. Round your answer to at least three decimal places. Do not round any intermediate steps.

Solution Steps

For a binomial distribution with parameters \( n = 150 \) and \( p = 0.20 \), we calculate the mean \( \mu \) and standard deviation \( \sigma \) as follows:

\[ \mu = n \cdot p = 150 \cdot 0.20 = 30.0 \]

\[ q = 1 - p = 0.80 \]

\[ \sigma^2 = n \cdot p \cdot q = 150 \cdot 0.20 \cdot 0.80 = 24.0 \]

\[ \sigma = \sqrt{npq} = \sqrt{24.0} \approx 4.899 \]

To find the probability that at least 27 people used the emergency room, we apply a continuity correction. We consider \( x = 26.5 \) for our calculations.

The z-score is calculated using the formula:

\[ z = \frac{X - \mu}{\sigma} = \frac{26.5 - 30.0}{4.899} \approx -0.7144 \]

Using the z-score, we find the probability that at least 27 people used the emergency room:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-0.7144) \approx 0.7625 \]

Final Answer