AP Statistics 5.8 Sampling Distributions for Differences in Sample Means - MCQs - Exam Style Questions

1. Analyze the given statistics:
– Mean (\(\mu\)) = \(5\) pounds.
– Standard deviation (\(\sigma\)) = \(3\) pounds.
– A key constraint is that rabbit weight cannot be negative. The lowest possible weight is \(0\) pounds.

2. Check for normality:
– Let’s see how many standard deviations the minimum possible value (\(0\)) is from the mean.
– Distance from mean to \(0\): \(5 – 0 = 5\) pounds.
– Number of standard deviations: \(\frac{5}{3} \approx 1.67\).
– A value of \(0\) is only \(1.67\) standard deviations below the mean. For a distribution to be approximately normal, we would expect the data to spread out about \(3\) standard deviations in both directions from the mean. Since the distribution is cut off on the left so close to the mean, it cannot be symmetric or normal.

3. Determine the skewness:
– The left tail is constrained by the hard limit of \(0\) pounds, which is less than \(2\) standard deviations from the mean. This means the data cannot extend far to the left. To balance this, the right tail must be long. A distribution with a long right tail is skewed to the right.
– In a right-skewed distribution, the mean is typically greater than the median.

4. Evaluate the options:
– (A) The Central Limit Theorem (CLT) applies to the sampling distribution of the *sample mean*, not the distribution of the sample itself.
– (C) This aligns with our finding. The distribution is skewed right because the lower bound is close to the mean.
– (E) In a right-skewed distribution, mean > median, so this is incorrect.
✅ Answer: (C)