The Central Limit Theorem
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AP Statistics › The Central Limit Theorem
The distribution of individual commute times in a large city is not normal; it is right-skewed due to occasional traffic jams. A researcher repeatedly takes random samples of $n=80$ commuters and computes the sample mean commute time $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because any sample size produces a normal sampling distribution for the mean
Because the sampling distribution of $\bar{x}$ is normal only when $n$ equals the population size
Because the mean and standard deviation of $\bar{x}$ are always equal
Because the sample size is large, so the Central Limit Theorem implies $\bar{x}$ is approximately normal
Because the population distribution becomes normal when many observations are collected
Explanation
This question assesses understanding of the Central Limit Theorem's application. Commute times are right-skewed due to traffic jams (not normal), yet when we take samples of n=80 and compute sample means, the sampling distribution becomes approximately normal. This occurs because n=80 is well above the typical threshold of n≥30 for the CLT to apply. The CLT ensures that for large sample sizes, the sampling distribution of x̄ will be approximately normal regardless of the population distribution's shape. Option B incorrectly suggests the population becomes normal. Option C wrongly requires n to equal population size. Options D and E misstate the CLT's implications.
Individual waiting times at a busy clinic are not normally distributed; they are right-skewed with a few extremely long waits. The clinic repeatedly selects random samples of $n=55$ patients and computes the sample mean waiting time $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because the sampling distribution of $\bar{x}$ is always normal even for small $n$
Because with a sufficiently large $n$, the Central Limit Theorem makes the distribution of $\bar{x}$ approximately normal
Because the standard error of $\bar{x}$ increases as $n$ increases
Because the population distribution is normal whenever the sample is random
Because $\bar{x}$ has the same distribution as the original waiting times
Explanation
This problem evaluates knowledge of the Central Limit Theorem. Waiting times are right-skewed with extreme values (not normal), but the sampling distribution of x̄ for samples of n=55 is approximately normal. This happens because n=55 is sufficiently large for the CLT to take effect. The CLT states that when sample size is large enough (typically n≥30), the sampling distribution of the sample mean becomes approximately normal regardless of the population's shape. Option A incorrectly links randomness to population normality. Option C overgeneralizes - small samples don't guarantee normality. Options D and E misunderstand the CLT - it doesn't make x̄ have the same distribution as individuals, and standard error decreases with larger n.
The distribution of individual book prices sold online is not normal; it is right-skewed because most books are inexpensive but some collectibles are very costly. An analyst repeatedly takes random samples of $n=70$ book prices and computes the sample mean price $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because any random sample, regardless of size, produces a normal sampling distribution for $\bar{x}$
Because the distribution of individual book prices becomes normal once you compute a mean
Because the sample size is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
Because the sampling distribution of $\bar{x}$ is normal only when the population is normal
Because the mean of $\bar{x}$ is $\sigma/\sqrt{n}$, which forces normality
Explanation
This question evaluates understanding of the Central Limit Theorem. Book prices are right-skewed due to expensive collectibles (not normal), yet when we take samples of n=70 and compute sample means, the sampling distribution becomes approximately normal. This occurs because n=70 is well above the typical threshold of n≥30 for the CLT to apply. The CLT ensures that for large sample sizes, the sampling distribution of x̄ will be approximately normal regardless of the population's shape. Option A incorrectly requires population normality. Option B wrongly claims any sample size works. Options D and E confuse the CLT with other concepts - computing a mean doesn't change the population distribution, and the mean of x̄ is μ, not σ/√n.
The distribution of individual household electricity use in a city is left-skewed due to a small number of very low-usage homes and many moderate-usage homes; it is not normal. A utility company repeatedly takes random samples of $n=40$ households and records the sample mean usage $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because the sampling distribution of $\bar{x}$ is always normal, even for very small samples
Because the population standard deviation becomes smaller when $n$ increases
Because the population must be normal whenever we compute a mean
Because the sample mean equals the population mean $\mu$ in every sample
Because the sample size is sufficiently large for the Central Limit Theorem to apply
Explanation
This question assesses knowledge of when the Central Limit Theorem applies. The population distribution is left-skewed (not normal), yet the sampling distribution of x̄ for samples of size n=40 is approximately normal. This occurs because n=40 exceeds the typical threshold of n≥30 for the CLT to take effect. The CLT ensures that as sample size increases, the sampling distribution of the mean approaches normality regardless of the population distribution's shape. Option A incorrectly claims the population must be normal. Option C wrongly states the sampling distribution is always normal. Options D and E confuse different statistical concepts - the CLT doesn't change population parameters.
A store’s individual purchase amounts are right-skewed because many customers buy low-cost items and a few spend a lot; the population is not normal. The manager repeatedly selects random samples of $n=35$ purchases and computes the sample mean purchase amount $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because a sample size of $n=35$ is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
Because the population distribution must be normal for the sampling distribution of $\bar{x}$ to be normal
Because increasing $n$ makes the population mean $\mu$ change less from sample to sample
Because the sample mean is always normally distributed, regardless of sample size
Because taking a random sample forces the data in that sample to be symmetric
Explanation
This problem tests knowledge of when the Central Limit Theorem applies. Purchase amounts are right-skewed (not normal), but the sampling distribution of x̄ for samples of n=35 is approximately normal. This happens because n=35 exceeds the typical threshold of n≥30 for the CLT to work. The CLT states that for sufficiently large samples, the sampling distribution of the mean approaches normality regardless of the population's shape. Option A incorrectly requires the population to be normal. Option C overstates the theorem - small samples don't guarantee normality. Options D and E misunderstand what the CLT actually does - it doesn't change the population or its parameters.
Daily rainfall amounts in a region have a distribution that is heavily right-skewed with many zeros and a few very large values; it is not normal. Meteorologists repeatedly take random samples of $n=45$ days and compute the sample mean rainfall $\bar{x}$. The sampling distribution of $\bar{x}$ is approximately normal. Why is the sampling distribution approximately normal?
Because the population distribution is approximately normal, so $\bar{x}$ is approximately normal
Because the sampling distribution of $\bar{x}$ is normal only if the data have no outliers
Because the sampling distribution of $\bar{x}$ has the same shape as the population distribution
Because the sample size is large enough for the Central Limit Theorem to make $\bar{x}$ approximately normal
Because any random sample, no matter how small, produces a normal sampling distribution for $\bar{x}$
Explanation
This question evaluates understanding of the Central Limit Theorem. The rainfall distribution is heavily right-skewed with many zeros (not normal), yet the sampling distribution of x̄ for samples of size n=45 is approximately normal. This occurs because n=45 is large enough for the CLT to take effect. The CLT ensures that when sample size is sufficiently large (typically n≥30), the sampling distribution of the sample mean becomes approximately normal regardless of the population distribution. Option A incorrectly assumes the population is normal. Option C wrongly claims any sample produces normality. Options D and E confuse the CLT with other concepts.
A university dining hall tracks the number of cookies students take at lunch. The population distribution is heavily right-skewed because most students take 0–2 cookies, but a few take many. The manager repeatedly selects random samples of $n=80$ students and computes the sample mean number of cookies $\bar{x}$. He considers the sampling distribution of $\bar{x}$. Why is the sampling distribution approximately normal?
Because the sample mean is always normally distributed for any sample size
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
Because using a larger sample size makes the population distribution less skewed
Because the sampling distribution must be right-skewed whenever the population is right-skewed
Because the sampling distribution of $\bar{x}$ is approximately normal only when the population is exactly normal
Explanation
In AP Statistics, this question addresses the Central Limit Theorem for the sampling distribution of the sample mean from a heavily right-skewed population of cookie counts. With n=80, a large sample, the CLT makes the distribution of
A wildlife biologist studies the weights of a certain fish species. The population distribution of fish weights is not normal: it is left-skewed because of a minimum size limit and a few unusually light fish. A random sample of $n=100$ fish is taken, and the sample mean weight is calculated. Why is the sampling distribution of the sample mean approximately normal?
Because the population must be normal whenever $n$ is large.
Because $n=100$ is large, the Central Limit Theorem implies $\bar{x}$ is approximately normal.
Because the sample mean is always normally distributed, regardless of sample size or population shape.
Because the population distribution is left-skewed, the sampling distribution of $\bar{x}$ must be left-skewed too.
Because the standard deviation of $\bar{x}$ increases with $n$, making it normal.
Explanation
This question assesses understanding of the CLT with left-skewed data and a large sample size. With n=100 fish, this sample size is well above the threshold needed for the Central Limit Theorem to ensure the sampling distribution of the sample mean is approximately normal. The CLT works regardless of the direction of skewness in the population - whether right-skewed or left-skewed as in this case. Choice A incorrectly assumes the sampling distribution inherits the population's left skew, but the CLT tells us that large samples produce approximately normal sampling distributions. Choice C overstates by claiming the sample mean is always normal, when we actually need large samples for non-normal populations. Choice E contains a mathematical error - the standard deviation of the sample mean actually decreases with larger n (it equals σ/√n).
A delivery service records the number of packages delivered per driver per day. The population distribution is discrete and not normal, with a long right tail during peak seasons. A random sample of $n=40$ driver-days is selected, and the sample mean number of packages is computed. Why is the sampling distribution of $\bar{x}$ approximately normal?
Because the sample mean is computed from counts, it is always exactly normal.
Because the sampling distribution of $\bar{x}$ is approximately normal only when the population is uniform.
Because the population distribution must be normal for $\bar{x}$ to be approximately normal.
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.
Because any sample size produces an approximately normal sampling distribution as long as sampling is random.
Explanation
This question tests understanding of the Central Limit Theorem's core principle with discrete, right-tailed data. The sample size of n=40 exceeds the typical threshold of 30, allowing the CLT to apply. The CLT states that for sufficiently large sample sizes, the sampling distribution of the sample mean becomes approximately normal regardless of the population distribution's shape - even if it's discrete with a long right tail as described. Choice B incorrectly requires the population to be normal, which contradicts the fundamental purpose of the CLT. Choice C wrongly claims the CLT only works for uniform populations, when it actually works for any population shape. Choice D overstates by suggesting any sample size works, and Choice E incorrectly claims exact normality rather than approximate normality.
A factory measures the amount of time (in seconds) it takes to assemble a part. The population distribution is moderately right-skewed due to occasional machine slowdowns. Quality control repeatedly takes random samples of $n=64$ assembly times and computes the sample mean $\bar{x}$ for each sample, forming the sampling distribution of $\bar{x}$. Why is the sampling distribution approximately normal?
Because any random sample mean is approximately normal regardless of how small $n$ is
Because increasing $n$ makes the population distribution itself become normal
Because the sample mean always has the same shape as the population distribution
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
Because the sampling distribution of $\bar{x}$ is normal only if the population has no skewness at all
Explanation
The focus in this AP Statistics question is the Central Limit Theorem and its implications for the sampling distribution of the sample mean. The population of assembly times is moderately right-skewed, but with n=64, the CLT ensures