The Probability of Volunteers Receiving Certificate of Merit

The Distribution of Number of Hours Worked by Volunteers

The distribution of number of hours worked by volunteers last year at a large hospital is approximately normal with mean 80 and standard deviation 7. Volunteers in the top 20 percent of hours worked will receive a certificate of merit. If a volunteer from last year is selected at random, which of the following is closest to the probability that the volunteer selected will receive a certificate of merit given that the number of hours the volunteer worked is less than 90?

The probability that a randomly selected volunteer who worked less than 90 hours will receive a certificate of merit is approximately 0.22, or 22%.

How to Calculate the Probability

How can you find the probability of a randomly selected volunteer receiving a certificate of merit given that they worked less than 90 hours?

Calculate the z-score for 90 hours:

z = (90 - mean) / standard deviation

z = (90 - 80) / 7

z ≈ 1.43

Find the proportion of volunteers who worked less than 90 hours:

This represents the area to the left of the z-score 1.43 in the standard normal distribution. Using a z-table or calculator, we find this area to be approximately 0.9236.

Determine the percentile of volunteers who receive a certificate of merit:

Since the top 20% of volunteers receive the certificate, the percentile cut-off for receiving the certificate is 80%.

Calculate the probability of being in the top 20% given working less than 90 hours:

This is the proportion of the area between the z-score 1.43 and the 80th percentile (which corresponds to a z-score of approximately 0.84) in the standard normal distribution. Subtracting this area from 1 (the total area under the curve) and then dividing by the proportion of volunteers who worked less than 90 hours (0.9236) gives us the desired probability:

p(top 20% | < 90 hours) = (1 - p(80th percentile | < 90 hours)) / p(< 90 hours)

≈ (1 - 0.7881) / 0.9236

≈ 0.2166

Therefore, the probability that a randomly selected volunteer who worked less than 90 hours will receive a certificate of merit is approximately 0.22, or 22%.