According to Bureau of Labor Statistics, 26.0% of the total part- time workforce in the U.S. was between the ages of 25 and 34 during a recent month. A random sample of 90 part- time employees was selected during this quarter. Using the normal approximation to the binomial distribution, what is the probability that more than 27 people from this sample were between the ages of 25 and 34?

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Quizplus · Accepted Answer

To find the probability that more than 27 people from the sample were between the ages of 25 and 34, we first calculate the mean (μ) and standard deviation (σ) of the distribution. For a binomial distribution, μ = np and σ = sqrt(np(1-p)), where n is the sample size and p is the probability of success.Given n = 90 and p = 0.26, we have:μ = np = 90 * 0.26 = 23.4σ = sqrt(np(1-p)) = sqrt(90 * 0.26 * 0.74) ≈ 4.37To use the normal approximation, we apply the continuity correction by looking for the probability of more than 27.5 people (since we're looking for more than 27, and the next discrete number is 28, we use 27.5 for the continuity correction).The z-score is calculated as:z = (x - μ) / σ = (27.5 - 23.4) / 4.37 ≈ 0.94Using the z-table, the probability of a z-score less than 0.94 is approximately 0.8264. Since we want more than 27.5, we subtract this from 1:P(X > 27) ≈ 1 - 0.8264 = 0.1736However, the provided options do not exactly match this calculation, suggesting a possible miscalculation or misunderstanding in the application of the normal approximation. Among the given options, the closest to the calculated probability (considering possible rounding or calculation differences) is option B) 0.1611, which may indicate a need to reevaluate the exact calculation steps or the use of a different z-score value or table.