Final answer:
To find the probability that the sample mean is within 3.37 months of the population mean, we calculate the standard error, find the corresponding z-scores, and use these to determine the probability using a standard normal distribution. As the sample size is greater than 30, the Central Limit Theorem justifies the use of normal distribution.
Step-by-step explanation:
To solve the mathematical problem completely, we need to use the Central Limit Theorem (CLT), which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (n > 30 is a common rule of thumb). Since we have a sample of 71 computers, the CLT applies. The formula for the standard error of the mean (SEM) is √(variance/n), which in this case will be √(100/71).
We want to know the probability that the sample mean is within 3.37 months of the population mean of 109 months. We can convert this into a z-score, which tells us how many standard deviations an element is from the mean. The z-score is calculated as (X - μ) / SEM, where X is the value we are checking, μ is the population mean, and SEM is the standard error of the mean.
Let's calculate the z-scores for 109 + 3.37 and 109 - 3.37. After finding the z-scores, we will use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores and subtract the smaller probability from the larger one to find the probability that the sample mean is within 3.37 months of the population mean.
For instance, if the calculated z-scores are Z1 and Z2, then the probability we seek is P(Z1 < Z < Z2), which equals P(Z < Z2) - P(Z < Z1).