AP Statistics 5.3 The Central Limit Theorem- MCQs - Exam Style Questions
▶️ Answer/Explanation
1. Define the Central Limit Theorem (CLT):
The CLT states that for a population with any distribution, the sampling distribution of the sample mean ($\bar{x}$) will be approximately normal, provided the sample size ($n$) is sufficiently large (typically $n \geq 30$).
2. Identify What the CLT Describes:
The CLT specifically describes the shape of the distribution of sample means, not the distribution of a single sample or the original population. This eliminates options (A), (B), and (C).
3. Check the Condition for the CLT:
The condition for the CLT is that the sample size must be large enough. The sample size here is $n=100$, which is greater than 30. The population mean (400) is irrelevant to the CLT condition.
4. Conclusion:
The correct statement must say that the distribution of the sample means is approximately normal because the sample size of 100 is greater than 30.
✅ Answer: (D)
▶️ Answer/Explanation
This question describes the sampling distribution of the sample mean, \(\bar{x}\). According to the Central Limit Theorem (CLT):
1. Shape: Since the sample size \(n=64\) is large (\(\ge 30\)), the sampling distribution will be approximately normal, even though the population is skewed.
2. Mean: The mean of the sampling distribution (\(\mu_{\bar{x}}\)) is equal to the population mean (\(\mu\)), so \(\mu_{\bar{x}} = 10\) gallons.
3. Standard Deviation: The standard deviation of the sampling distribution (\(\sigma_{\bar{x}}\)) is \(\frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{64}} = \frac{4}{8} = 0.5\) gallons.
The distribution of the sample means is approximately normal with a mean of \(10\) and a standard deviation of \(0.5\).
✅ Answer: (D)