The statement that is not true is:

D) The sampling distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is small enough.

Explanation:

1. A) True: The sampling distribution of the sample mean, $\bar{x}$, is normal if the population distribution is normal, regardless of sample size.
2. B) True: The mean of the sampling distribution of $\bar{x}$ equals the population mean ($\mu_{\bar{x}} = \mu$).
3. C) True: The standard deviation of the sampling distribution (known as the standard error) is smaller than the population standard deviation because it is scaled by $\sqrt{n}$: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
4. D) False: The shape of the sampling distribution becomes approximately normal only if the sample size, $n$, is large enough according to the Central Limit Theorem (CLT). If $n$ is too small and the population distribution is not normal, the sampling distribution may not be normal.
5. E) False: Since D is false, E cannot be true.

Conclusion:

The correct answer is D.

Would you like a deeper explanation or an example illustrating the Central Limit Theorem?

Related Questions:

1. What conditions need to be satisfied for the Central Limit Theorem to apply?
2. How does sample size influence the standard error of the mean?
3. How can we determine if a population distribution is normal based on sample data?
4. Why is the sampling distribution important in statistical inference?
5. What happens to the sampling distribution of the mean if the sample size is very large?

Tip:

Always consider whether the population distribution or sample size affects the assumptions of normality for the sampling distribution.