Question 1181564: 2. A simple random sample is taken from a left-skewed population. Choose the one statement that
is true.
a. We can never ensure that the distribution of the sample mean is normal or
approximately normal.
b. If the sample mean equals the population mean, then the distribution of the sample
mean must be normal or approximately normal.
c. We can be certain that the sample mean is normal or approximately normal only if the
sample size is greater than 30.
d. For any sample size, the distribution of the sample mean is normal or approximately normal
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! The correct answer is **c. We can be certain that the sample mean is normal or approximately normal only if the sample size is greater than 30.**
Here's why:
The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, *regardless* of the shape of the original population distribution. A commonly used rule of thumb is that a sample size greater than 30 is sufficient for the Central Limit Theorem to apply and for the sample mean to be considered approximately normally distributed.
The other options are incorrect:
* **a.** While a small sample from a skewed population might not have a normally distributed sample mean, *as the sample size increases*, the distribution of the sample mean *will* approach normality.
* **b.** The sample mean being equal to the population mean has no bearing on the *distribution* of the sample mean. The distribution is determined by the Central Limit Theorem and the sample size.
* **d.** This is incorrect; the sample size does matter. For small samples from skewed populations, the distribution of the sample mean will also tend to be skewed.
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