Questions: If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of (barX1-barX2) is approximately normal only if both sample sizes are sufficiently large-that is, when Multiple Choice - (n1+n2=30) - (n1+n2 geq 30) - (n1=30) and (n2=30) - (n1 geq 30) and (n2 geq 30)

If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of (barX1-barX2) is approximately normal only if both sample sizes are sufficiently large-that is, when

Multiple Choice

- (n1+n2=30)
- (n1+n2 geq 30)
- (n1=30) and (n2=30)
- (n1 geq 30) and (n2 geq 30)

Transcript text: 2 If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of $\bar{X}_{1}-\bar{X}_{2}$ is approximately normal only if both sample sizes are sufficiently large-that is, when $\qquad$ 0.25 points Multiple Choice ebook $\square$ References \[ n_{1}+n_{2}=30 \] \[ n_{1}+n_{2} \geq 30 \] \[ n_{1}=30 \text { and } n_{2}=30 \] \[ n_{1} \geq 30 \text { and } n_{2} \geq 30 \] ( 6

Solution

Solution Steps

Step 1: Understanding the Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean $ \bar{X} $ will be approximately normal if the sample size is sufficiently large, regardless of the shape of the population distribution. This is particularly important when comparing the means of two independent populations.

Step 2: Conditions for Normality

For the difference in sample means $ \bar{X}_1 - \bar{X}_2 $ to be approximately normal, both sample sizes must be large enough. Specifically, the condition is that both $ n_1 $ and $ n_2 $ should be at least 30. This ensures that the sampling distribution of the difference in means approaches normality.

Step 3: Conclusion

Based on the conditions derived from the Central Limit Theorem, the correct condition for the sampling distribution of $ \bar{X}_1 - \bar{X}_2 $ to be approximately normal is:

\[ n_1 \geq 30 \quad \text{and} \quad n_2 \geq 30 \]