AP Statistics Help: The Central Limit Theorem

Question 1

1. Because the sample size is large, so the Central Limit Theorem implies $\bar{x}$ is approximately normal
2. Because the population distribution becomes normal when many observations are collected
3. Because the sampling distribution of $\bar{x}$ is normal only when $n$ equals the population size
4. Because any sample size produces a normal sampling distribution for the mean
5. Because the mean and standard deviation of $\bar{x}$ are always equal

Question 2

1. Because the population distribution is normal whenever the sample is random
2. Because with a sufficiently large $n$, the Central Limit Theorem makes the distribution of $\bar{x}$ approximately normal
3. Because the sampling distribution of $\bar{x}$ is always normal even for small $n$
4. Because $\bar{x}$ has the same distribution as the original waiting times
5. Because the standard error of $\bar{x}$ increases as $n$ increases

Question 3

1. Because the sampling distribution of $\bar{x}$ is normal only when the population is normal
2. Because any random sample, regardless of size, produces a normal sampling distribution for $\bar{x}$
3. Because the sample size is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
4. Because the distribution of individual book prices becomes normal once you compute a mean
5. Because the mean of $\bar{x}$ is $\sigma/\sqrt{n}$, which forces normality

Question 4

1. Because the population must be normal whenever we compute a mean
2. Because the sample size is sufficiently large for the Central Limit Theorem to apply
3. Because the sampling distribution of $\bar{x}$ is always normal, even for very small samples
4. Because the sample mean equals the population mean $\mu$ in every sample
5. Because the population standard deviation becomes smaller when $n$ increases

Question 5

1. Because the population distribution must be normal for the sampling distribution of $\bar{x}$ to be normal
2. Because a sample size of $n=35$ is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
3. Because the sample mean is always normally distributed, regardless of sample size
4. Because taking a random sample forces the data in that sample to be symmetric
5. Because increasing $n$ makes the population mean $\mu$ change less from sample to sample

Question 6

1. Because the population distribution is approximately normal, so $\bar{x}$ is approximately normal
2. Because the sample size is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
3. Because any random sample, no matter how small, produces a normal sampling distribution for $\bar{x}$
4. Because the sampling distribution of $\bar{x}$ is normal only if the data have no outliers
5. Because the sampling distribution of $\bar{x}$ has the same shape as the population distribution

Question 7

1. Because the Central Limit Theorem says that with a large random sample size, the sampling distribution of $\bar{x}$ is approximately normal even if the population is not normal
2. Because the sampling distribution of $\bar{x}$ is approximately normal only when the population is exactly normal
3. Because the sample mean is always normally distributed for any sample size
4. Because using a larger sample size makes the population distribution less skewed
5. Because the sampling distribution must be right-skewed whenever the population is right-skewed

Question 8

1. Because the population distribution is left-skewed, the sampling distribution of $\bar{x}$ must be left-skewed too.
2. Because $n=100$ is large, the Central Limit Theorem implies $\bar{x}$ is approximately normal.
3. Because the sample mean is always normally distributed, regardless of sample size or population shape.
4. Because the population must be normal whenever $n$ is large.
5. Because the standard deviation of $\bar{x}$ increases with $n$, making it normal.

Question 9

1. Because the Central Limit Theorem says that for sufficiently large $n$, the distribution of $\bar{x}$ is approximately normal even if the population is not.
2. Because the population distribution must be normal for $\bar{x}$ to be approximately normal.
3. Because the sampling distribution of $\bar{x}$ is approximately normal only when the population is uniform.
4. Because any sample size produces an approximately normal sampling distribution as long as sampling is random.
5. Because the sample mean is computed from counts, it is always exactly normal.

Question 10

1. Because the sampling distribution of $\bar{x}$ is approximately normal for large $n$ by the Central Limit Theorem, even if the population is not normal
2. Because the sampling distribution of $\bar{x}$ is normal only if the population has no skewness at all
3. Because increasing $n$ makes the population distribution itself become normal
4. Because the sample mean always has the same shape as the population distribution
5. Because any random sample mean is approximately normal regardless of how small $n$ is