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 1.3 days per employee with a standard deviation of 1.0 days. Martocchio also estimated that the mean amount of unpaid time lost during a three-month period was 1.4 day per employee with a standard deviation of 1.2 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.)

Answer 1

To find 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, we need to use the Central Limit Theorem. First, we calculate the standard error of the mean using the formula SE = standard deviation / square root of sample size. Next, we calculate the z-score using the formula z = (sample mean - population mean) / SE. Finally, we use the z-score to find the probability using a standard normal distribution table or a calculator.

Step-by-step explanation:

To find 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, we need to use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, as long as the sample size is large enough. In this case, the sample size is 100, which is considered large enough for the Central Limit Theorem to apply.

First, we need to calculate the standard error of the mean. The standard error of the mean (SE) can be calculated using the formula SE = standard deviation / square root of sample size. For paid time lost, the mean is 1.3 days and the standard deviation is 1.0 days. Therefore, the SE of the mean is 1.0 / square root of 100, which is 0.1 days.

Next, we need to calculate the z-score using the formula z = (sample mean - population mean) / SE. The sample mean is 1.3 days and the population mean (estimated by Martocchio) is 1.5 days. Plugging in these values, we get z = (1.3 - 1.5) / 0.1 = -2.

Finally, we can use the z-score to find the probability using a standard normal distribution table or a calculator. Since we are interested in the probability that the average amount of paid time lost will exceed 1.5 days, we want to find the probability in the right tail of the distribution. Looking up the z-score of -2 in the standard normal distribution table, we find that the probability is 0.0228.