Final answer:
To calculate the probability that the sample mean differs from the true mean by more than $28, the standard error of the mean is computed followed by the z-score for a $28 difference. Using a standard normal distribution table or calculator, we find that option (a) 0.0448 is the correct probability.
Step-by-step explanation:
The question at hand involves finding the probability that the sample mean of per capita income would differ from the true mean by more than $28 when a sample of 55 persons is randomly selected, given that the true mean is $20,908 and the standard deviation is $407 per annum.
To solve this problem, we will use the concept of the standard error of the mean (SEM), which is the standard deviation of the sampling distribution of the sample mean. The formula for the SEM is σ/√n, where σ is the standard deviation and n is the sample size. The z-score can then be calculated to find the standard deviation units the required difference of $28 is from the mean. The probability can be found using the standard normal distribution table or a calculator.
Calculating the SEM, we have SEM = 407/√55 ≈ 54.86. The z-score for a difference of $28 is z = 28/54.86 ≈ 0.51. Referring to standard normal distribution tables or using a calculator for areas under the normal curve, we find that the probability of observing a sample mean within ±0.51 standard deviations from the true mean is high, and hence the probability of observing a sample mean more than 0.51 standard deviations from the true mean (i.e., differing by more than $28) is low. Checking the options given in the question, the correct answer is (a) 0.0448.