Questions: Suppose 40% of the students in a university are baseball players. If a sample of 781 students is selected, what is the probability that the sample proportion of baseball players will be greater than 43%? Round your answer to four decimal places.

Suppose 40% of the students in a university are baseball players. If a sample of 781 students is selected, what is the probability that the sample proportion of baseball players will be greater than 43%? Round your answer to four decimal places.

Transcript text: Suppose $40 \%$ of the students in a university are baseball players. If a sample of 781 students is selected, what is the probability that the sample proportion of baseball players will be greater than $43 \%$ ? Round your answer to four decimal places.

Solution

Solution Steps

Step 1: Define the Problem

We are given that $ 40\% $ of the students in a university are baseball players. We want to find the probability that the sample proportion of baseball players in a sample of $ n = 781 $ students is greater than $ 43\% $.

Step 2: Calculate the Mean and Standard Deviation

The mean $ \mu $ of the sampling distribution of the sample proportion is given by: \[ \mu = p = 0.40 \] The standard deviation $ \sigma $ of the sampling distribution is calculated using the formula: \[ \sigma = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.40 \times (1 - 0.40)}{781}} = \sqrt{\frac{0.40 \times 0.60}{781}} \approx 0.022 \]

Step 3: Calculate the Z-scores

To find the probability that the sample proportion is greater than $ 43\% $, we first convert $ 0.43 $ to a Z-score using the formula: \[ Z = \frac{\hat{p} - \mu}{\sigma} = \frac{0.43 - 0.40}{0.022} \approx 1.7114 \] The Z-score for the upper limit (infinity) is: \[ Z_{end} = \infty \]

Step 4: Calculate the Probability

Using the Z-scores, we can find the probability: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(1.7114) \approx 1 - 0.0435 = 0.0435 \]