Final answer:
To find the probability that a simple random sample of 100 automobile insurance policies will have a sample mean within $25 of the population mean, we need to use the concept of the sampling distribution of the sample mean. Given the mean annual cost of automobile insurance and the population standard deviation, we can calculate the standard deviation of the sampling distribution and then find the probability using the z-score formula.
Step-by-step explanation:
To find the probability that a simple random sample of 100 automobile insurance policies will have a sample mean within $25 of the population mean, we need to use the concept of the sampling distribution of the sample mean. The sampling distribution of the sample mean follows a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
Given that the mean annual cost of automobile insurance is $939 and the population standard deviation is $245, we can find the standard deviation of the sampling distribution using the formula: standard deviation = population standard deviation / square root of sample size. In this case, the sample size is 100.
After calculating the standard deviation of the sampling distribution, we can use the z-score formula to find the probability. The z-score represents the number of standard deviations a sample mean is away from the population mean. In this case, we want to find the probability that the sample mean is within $25 of the population mean, so we need to find the probability that the z-score is between -25 and 25. Using a standard normal distribution table or a calculator, we can find this probability.