Question

According to a study, 53% of all males between the ages of 18 and 24 live at home. (Unmarried college students living in a dorm are counted as living at home.) Suppose that a survey is administered and 132 of 231 respondents indicated that they live at home. (a) Use the normal approximation to the binomial to approximate the probability that at least 132 respondents live at home? (b) Do the results from part (a) contradict the study?

P(X ≥ 132) = (Round to four decimal places as needed.)

Answer 1

The probability that at least 132 respondents live at home is calculated by using the normal approximation to the binomial distribution. Calculate mean and standard deviation, convert to a z-score, and find the probability using a z-table or calculator.

Step-by-step explanation:

To find the probability that at least 132 respondents live at home, we use the normal approximation to the binomial distribution. First, calculate the mean (μ) and the standard deviation (σ) of the binomial distribution.

Mean, μ = np = 231 * 0.53 = 122.43

Standard deviation, σ = sqrt(np(1-p)) = sqrt(231 * 0.53 * (1 - 0.53)) = sqrt(122.43 * 0.47) ≈ 7.75

To find the probability P(X ≥ 132), we first convert X to a z-score using the formula:

z = (X - μ) / σ = (132 - 122.43) / 7.75 ≈ 1.23

Now, look up the z-score in a standard normal distribution table or use a calculator to get the probability. This probability represents the area to the right of 132, denoted as P(Z ≥ 1.23).

If the calculated probability is substantially lower than the expected probability of 0.53, it might indicate a contradiction to the study. However, without the actual calculated probability, we cannot conclusively say if the result contradicts the study.

The final step is to use a standard normal distribution table or calculator to find: P(Z ≥ 1.23). Remember to round the result to four decimal places, as needed.