Question

for a sample of 42 rabbits, the mean weight is 5 pounds and the standard deviation of weights is 3 pounds. which of the following is most likely true about the weights for the rabbits in this sample? a the distribution of weights is approximately normal because the sample size is 42, and therefore the central limit theorem applies. b the distribution of weights is approximately normal because the standard deviation is less than the mean. c the distribution of weights is skewed to the right because the least possible weight is within 2 standard deviations of the mean. d the distribution of weights is skewed to the left because the least possible weight is within 2 standard deviations of the mean. e the distribution of weights has a median that is greater than the mean.

Explanation:

Step1: Recall central limit theorem

The central - limit theorem states that if the sample size \(n\) is large (usually \(n\geq30\)), the sampling distribution of the sample mean is approximately normal. Here, \(n = 42\geq30\).

Step2: Analyze each option

• Option A: Since \(n = 42\geq30\), the central limit theorem applies and the distribution of sample means (and under certain conditions, the distribution of the data itself) can be approximated as normal. This is a correct application of the central - limit theorem.
• Option B: The relationship between the standard deviation and the mean (\(3<5\) here) does not determine the normality of the distribution.
• Option C: Knowing that the least possible weight is within 2 standard deviations of the mean does not imply right - skewness.
• Option D: Knowing that the least possible weight is within 2 standard deviations of the mean does not imply left - skewness.
• Option E: There is no information to suggest that the median is greater than the mean. In a normal distribution, the mean and median are equal.

Answer:

A. The distribution of weights is approximately normal because the sample size is 42, and therefore the central limit theorem applies.