Question

A company has 1,000 employees. The average number of workdays between absence for illness is 80 with a standard deviation of 11 days. Samples of 80 employees are examined. What is the probability a sample has a mean of workdays with no absence for illness of at least 78 days and at most 84 days?

a) 0.6826

b) 0.4772

c) 0.3413

d) 0.9544

Answer 1

To find the probability that a sample has a mean of workdays with no absence for illness between 78 days and 84 days, we use the Central Limit Theorem and the Z-distribution. By calculating the standard error and converting the sample mean values to Z-scores, we can find the probability using a Z-table or calculator. The probability is approximately 0.9544.

Step-by-step explanation:

To find the probability that a sample has a mean of workdays with no absence for illness between 78 days and 84 days, we need to use the Central Limit Theorem and the Z-distribution.

Step 1: Calculate the standard error of the mean using the formula:

standard error = standard deviation / square root of sample size

standard error = 11 / square root of 80 = 11 / 8.944 = 1.2317

Step 2: Convert the sample mean values to Z-scores using the formula:

Z = (sample mean - population mean) / standard error

Z1 = (78 - 80) / 1.2317 = -1.6231

Z2 = (84 - 80) / 1.2317 = 3.2487

Step 3: Use a Z-table or calculator to find the probability between these Z-scores.

Using the Z-table, the probability between -1.6231 and 3.2487 is approximately 0.9544.

Therefore, the probability a sample has a mean of workdays with no absence for illness between 78 days and 84 days is approximately 0.9544.