Final answer:
To find the probability that the mean clock life would be less than 13.8 years, we use the central limit theorem and the standard normal distribution. The probability is approximately 0.0228, or 2.28%.
Step-by-step explanation:
To answer this question, we need to use the central limit theorem. The central limit theorem states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. In this case, we are given that the population mean life of the wall clocks is 15 years with a standard deviation of 5 years.
The distribution of the sample mean follows a normal distribution with the same mean as the population mean but with a standard deviation equal to the population standard deviation divided by the square root of the sample size. So, in this case, the standard deviation of the sample mean is 5 / sqrt(37).
To find the probability that the mean clock life would be less than 13.8 years, we need to standardize the value using the z-score. The z-score is calculated as (x - mean) / standard deviation, where x is the value we want to find the probability for. So, in this case, the z-score is (13.8 - 15) / (5 / sqrt(37)). Once we have the z-score, we can look up the probability associated with it in the standard normal distribution table. The probability represents the area under the curve to the left of the z-score. Using the table or a calculator, we find that the probability is approximately 0.0228, or 2.28%.