Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
No. Three is not a reasonable-sized sample.
To determine whether it would be unusual for the mean of a sample of 3 to be 115 or more, we can calculate the sampling distribution of sample means using the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size, the sampling distribution of sample means will be approximately normally distributed, regardless of the shape of the original population. In this case, a sample size of 3 is relatively small, but we can still apply the Central Limit Theorem as an approximation.
First, we need to calculate the standard error of the mean (SEM), which is equal to the population standard deviation divided by the square root of the sample size.
SEM = 15 / √3 ≈ 8.66
Next, we need to calculate the z-score, which represents the number of standard deviations a sample mean is away from the population mean. The z-score can be calculated using the formula:
z = (sample mean - population mean) / SEM
z = (115 - 100) / 8.66 ≈ 1.73
To determine whether it would be unusual, we can consult a z-table to find the probability associated with a z-score of 1.73. Looking up the z-score in the table, we find that the corresponding probability is approximately 0.958.
Since the z-score corresponds to a probability of 0.958, it means that only approximately 5.8% (100% - 95.8%) of sample means would be 115 or more. Therefore, it would be considered unusual for the mean of a sample of 3 to be 115 or more, given the population parameters provided.