Check the conditions required to complete this test. Specifically, is the sampling distribution of average differences approximately normally distributed?

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BlackTom AI · Accepted Answer

To start, we need to evaluate what the question is asking: whether the sampling distribution of average differences is approximately normal under the given test conditions.
Option 1: 'Yes, the distribution of the differences appears fairly normally distributed and the sample size is greater than 30.' This aligns with the central limit theorem intuition: with a sufficiently large sample size (commonly n > 30), the distribution of the sample mean differences tends to be close to normal, even if the underlying data are not perfectly normal.
Option 2: 'Yes, both nphat and n(1-phat) are greater than 10.' This statement mixes concepts from hypothesis testing about proportions (nphat and n(1-phat) are sample counts for success and failure) and is not a direct criterion for the normality of the sampling distribution of average differences. It is more about conditions for a z- or chi-square approximation in proportion testing than about the normality of sample means.
Option 3: 'No, the distribution of the differences appears heavily right skewed, which means a larger sample size doesn't necessarily make the distribution of average differences normally distributed.' If the observed distribution is heavily skewed, the normal approximation may be questionable for small samples, but with large samples the CLT suggests convergence toward normality. Saying a larger sample size doesn’t help is inaccurate unless there is extreme skewness or dependence; this option is overly pessimistic and contradicts the CLT principle in typical sampling scenarios.
Option 4: 'No, the distribution of the differences appears heavily left skewed, which means a larger sample size doesn't necessarily make the distribution of average differences normally distributed.' This mirrors the same issue as Option 3 but with left skew. The direction of skewness matters for the exact shape, but again, increasing sample size generally encourages the sampling distribution of the mean toward normality per the CLT, making this claim incorrect in standard contexts.
In summary, Option 1 correctly ties the idea of a fairly normal sampling distribution to having a sample size greater than 30, which is a common rule of thumb for the CLT-based normal approximation. The other options introduce mismatched criteria (proportions-based conditions) or assert that skewness permanently prevents normality with larger samples, both of which are inconsistent with standard sampling theory and the typical interpretation of the CLT.

FA25_QTM_100_3 7.20 High School and Beyond, Part I. - Requires Respondus LockDown Browser

Check the conditions required to complete this test. Specifically, is the sampling distribution of average differences approximately normally distributed?

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