Government workers: The Bureau of Labor Statistics reported that \( 15 \% \) of U.S. nonfarm workers are government employees. A randorn sample of 35 workers is drawn. Use the excel spreadsheet as needed. Part: \( 0 / 5 \) Part 1 of 5 (a) Is it appropriate to use the normal approximation to find the probability that less than \( 25 \% \) of the individuals in the sample are government employees? If so, find the probability. If not, explain why not. It (Choose one) \( \mathbf{v} \) appropriate to use the normal curve, since \( n p= \) is not is not

We are told that 15% of nonfarm workers are government employees and that we have a random sample of 35 workers. Let X be the number of government employees in the sample. Then X has a binomial distribution with parameters n = 35 and p = 0.15. Step 1. Check if the normal approximation is appropriate. A common rule‐of‐thumb is that the normal approximation can be used if both   np ≥ 5   and  n(1 – p) ≥ 5. Here,   np = 35 × 0.15 = 5.25   and  n(1 – p) = 35 × 0.85 = 29.75. Thus, both conditions are met (with np just above 5), so it is acceptable to use the normal approximation. Step 2. Find the probability that less than 25% of the sample are government employees. “Less than 25%” means that the sample proportion p̂ < 0.25. Since the sample size is 35, this corresponds to   X < 0.25 × 35 = 8.75. Because X can only take integer values, X < 8.75 means X ≤ 8. For the normal approximation, we find the mean and variance of X:   Mean, μ = np = 5.25.   Variance, σ² = np(1 – p) = 35 × 0.15 × 0.85 = 4.4625.   Standard deviation, σ = √4.4625 ≈ 2.114. We apply a continuity correction to approximate P(X ≤ 8). We use 8 + 0.5 = 8.5. Then, the z-score is   z = (8.5 – 5.25) / 2.114 ≈ 3.25 / 2.114 ≈ 1.54. Using the standard normal distribution, we find   P(Z < 1.54) ≈ 0.938. Thus, the probability that less than 25% of the individuals in the sample are government employees is approximately 93.8%. Final Answer: (a) It is appropriate to use the normal approximation (since np = 5.25 and n(1 – p) = 29.75 are both greater than 5). Using the normal approximation with continuity correction, we find P(p̂ < 0.25) ≈ 0.938.