For the manufacturing plant discussed in Exercise 8.10, the union president and the human resources director jointly select a simple random sample of 36 employees to engage in a discussion with regard to the company’s work rules and overtime policies. What is the probability that the average number of overtime hours last year for members of this sample would have been less than 65.0 hours? Between 55.0 and 65.0 hours?

(8.10) Employees in a large manufacturing plant worked and average of 62.0 hours of overtime last year, with a standard deviation of 15.0 hours. For a simple random sample of n=36 employees and x=the number of overtime hours worked last year, determine the z-score corresponding to each of the following sample means:
a. ȭ= 55.0 hours b. ȭ=60.0 hours
c. ȭ=65.0 hours d. ȭ= 47.0 hours

To solve this problem, we can use the concept of the standard normal distribution and z-scores. The z-score measures how many standard deviations an observation is from the mean.

First, we need to calculate the population mean (μ) and the population standard deviation (σ) based on the information given in Exercise 8.10.

μ = 62.0 hours (average number of overtime hours last year)
σ = 15.0 hours (standard deviation of overtime hours last year)

Next, we find the z-scores corresponding to the given sample means, using the formula:
z = (x - μ) / (σ / √n)

where x is the sample mean (in this case, the given values of ȭ), μ is the population mean, σ is the population standard deviation, and n is the sample size.

a. ȭ = 55.0 hours:
z = (55.0 - 62.0) / (15.0 / √36) = -1.40

b. ȭ = 60.0 hours:
z = (60.0 - 62.0) / (15.0 / √36) = -0.40

c. ȭ = 65.0 hours:
z = (65.0 - 62.0) / (15.0 / √36) = 0.80

d. ȭ = 47.0 hours:
z = (47.0 - 62.0) / (15.0 / √36) = -3.40

Now, to find the probabilities for the average number of overtime hours, we need to use a standard normal distribution table or a calculator that provides the area under the curve.

a. To find the probability that the average number of overtime hours would be less than 65.0 hours, we need to find the area to the left of the z-score of 0.80, which corresponds to ȭ = 65.0 hours.

b. To find the probability that the average number of overtime hours would be between 55.0 and 65.0 hours, we need to find the area between the z-scores of -1.40 and 0.80, which correspond to ȭ = 55.0 hours and ȭ = 65.0 hours.

By using a standard normal distribution table or a calculator, you can look up the values for the z-scores and find the corresponding probabilities.