Final answer:
To find the probability that the sample mean would differ from the true mean by less than $63, we can use the Z-score formula.
Step-by-step explanation:
To solve this problem, we need to use the concept of the standard deviation and the sample mean. In this case, the mean per capita income is $15,654 and the standard deviation is $570. We want to find the probability that the sample mean would differ from the true mean by less than $63 when a sample of 315 persons is randomly selected.
To find this probability, we need to use the Z-score formula. The Z-score is calculated by subtracting the true mean from the sample mean and then dividing it by the standard deviation divided by the square root of the sample size. In this case, the Z-score is (63 - 0) / (570 / sqrt(315)).
Once we have the Z-score, we can use a Z-table or a statistical calculator to find the probability associated with that Z-score. Round the answer to four decimal places.