Final answer:
The probability that the mean life of a sample of 71 computers would differ from the population mean by more than 3.37 months can be found using the standard normal distribution after calculating the z-scores based on the given mean, variance, and sample size.
Step-by-step explanation:
The student is asking about the probability that the mean life of a sample of computers would differ from the population mean by more than 3.37 months. Given that the mean life of a computer is 109 months and the variance is 100, with a sample size of 71 computers, we can use the sampling distribution of the sample mean to answer this question.
Firstly, we need to calculate the standard error of the mean, which is the standard deviation of the sampling distribution. The standard error is the square root of the variance divided by the sample size (sqrt(100)/sqrt(71)).
Next, we use the standard normal distribution (as the sample size is large enough for the Central Limit Theorem to apply) to find the z-scores for the mean plus or minus 3.37 months. The z-score is calculated by taking the difference between the observed sample mean and the population mean, then dividing by the standard error.
Finally, we use these z-scores to find the corresponding probabilities from the standard normal distribution table or using a calculator with a normal distribution function. The probability we're looking for is the total area under the standard normal curve that lies beyond both z-scores. This probability is equal to twice the area to the right of the higher z-score since the distribution is symmetric.
The exact calculations and use of a normal distribution table or technology will provide the final answer to the student's question, rounded to four decimal places as requested.