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 exactly 23 people from this sample were between the ages of 25 and 34?

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Quizplus · Accepted Answer

To find the probability that exactly 23 people from the sample were between the ages of 25 and 34, we use the normal approximation to the binomial distribution. The mean ($\mu$) of the distribution is $n \times p = 90 \times 0.26 = 23.4$ and the standard deviation ($\sigma$) is $\sqrt{n \times p \times (1-p)} = \sqrt{90 \times 0.26 \times 0.74} \approx 4.37$. To find the probability of exactly 23, we need to find the probability of the sample being between 22.5 and 23.5 (to approximate the "exactly" part using continuity correction). We convert these to Z-scores: $Z = \frac{X - \mu}{\sigma}$. The Z-score for 22.5 is $\frac{22.5 - 23.4}{4.37} \approx -0.206$, and for 23.5 is $\frac{23.5 - 23.4}{4.37} \approx 0.023$. Looking these Z-scores up in a standard normal distribution table or using a calculator, we find the probabilities and subtract the smaller from the larger to find the probability of exactly 23, which is closest to option B, 0.0951.