Final answer:
To find the probability of a worker being sick between 14 and 16 days with a mean of 15 days and a standard deviation of 5 days from a sample of 25, calculate the standard error, convert the sick days to Z-scores, and use the standard normal distribution to determine the probability.
Step-by-step explanation:
The question asks for the probability that a worker will be sick between 14 and 16 days, given a population mean (μ) of 15 days and standard deviation (σ) of 5 days. To solve this, we first calculate the standard error (SE) of the sample mean by dividing the population standard deviation by the square root of the sample size, that is SE = σ / √(n), where n is the sample size. In this case, SE = 5 / √(25) = 5 / 5 = 1. Then, to find the probability that a worker will be sick between 14 and 16 days, we need to use the standard normal distribution (Z-distribution) as the sample size is large enough for the central limit theorem to come into play, although the population size is not taken into account in the calculations.
To find the Z-scores for 14 and 16 days, we use the formula Z = (X - μ) / SE. So, Z = (14 - 15) / 1 = -1 for 14 days, and Z = (16 - 15) / 1 = 1 for 16 days. We then look up these Z-values in the standard normal distribution table or use a Z-score calculator to find the probabilities corresponding to these Z-scores and subsequently calculate the probability of a worker being sick between these values.