Question

In an article in the Journal of Management, Joseph Martocchio studied and estimated the costs of employee absences. Based on a sample of 176 blue-collar workers, Martocchio estimated that the mean amount of paid time lost during a three-month period was 1.4 days per employee with a standard deviation of 1.2 days. Martocchio also estimated that the mean amount of unpaid time lost during a three-month period was 1.2 day per employee with a standard deviation of 1.6 days. Suppose we randomly select a sample of 100 blue-collar workers. Based on Martocchio’s estimates:

(a) What is the probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days? (Use the rounded mean and standard error to compute the rounded Z-score used to find the probability. Round means to 1 decimal place, standard deviations to 2 decimal places, and probabilities to 4 decimal places. Round z-value to 2 decimal places.)

(b) What is the probability that the average amount of unpaid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days? (Use the rounded mean and standard error to compute the rounded Z-score used to find the probability. Round standard deviations to 2 decimal places and probabilities to 4 decimal places. Round z-value to 2 decimal places.)

Answer 1

a)

Here, μ = 1.4, σ = 0.09 and x = 1.5. We need to compute P(X >= 1.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (1.5 - 1.4)/0.09 = 1.11

Therefore,
P(X >= 1.5) = P(z <= (1.5 - 1.4)/0.09)
= P(z >= 1.11)
= 1 - 0.8665 = 0.1335

b)

Here, μ = 1.2, σ = 0.16 and x = 1.5. We need to compute P(X >= 1.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (1.5 - 1.2)/0.16 = 1.88

Therefore,
P(X >= 1.5) = P(z <= (1.5 - 1.2)/0.16)
= P(z >= 1.88)
= 1 - 0.9699 = 0.0301