Final answer:
To calculate the probability that the sample mean would differ from the true mean by greater than $38, if a sample of 208 persons is randomly selected, we need to use the Central Limit Theorem. First, we determine the standard error of the mean (SEM) using the formula SEM = standard deviation / square root of sample size. Then, we calculate the Z-score using the formula Z = (sample mean - true mean) / SEM. Finally, we find the probability associated with the Z-score using a Z-table or calculator.
Step-by-step explanation:
To calculate the probability that the sample mean would differ from the true mean by greater than $38, if a sample of 208 persons is randomly selected, we need to use the Central Limit Theorem.
According to the Central Limit Theorem, the distribution of sample means will be approximately normal regardless of the shape of the population distribution, as long as the sample size is large enough.
Since the sample size is greater than 30, we can assume that the distribution of sample means will be approximately normal.
To calculate the probability, we first need to determine the standard error of the mean (SEM), which is the standard deviation divided by the square root of the sample size. In this case, the SEM = $397 / √208.
Next, we calculate the Z-score using the formula Z = (sample mean - true mean) / SEM = ($38 - 0) / ($397 / √208). Finally, we can use a Z-table or calculator to find the probability associated with the Z-score.
In this case, it is the probability that Z is greater than the calculated Z-score. Hence, the probability that the sample mean would differ from the true mean by greater than $38 is the probability that Z is greater than the calculated Z-score.