According to the Central Limit Theorem, the sampling distribution of x̄ is approximately normal when which condition holds?

Answer: The sample size n is sufficiently large (n ≥ 30 is a common rule). The CLT states that for sufficiently large n, the sampling distribution of x̄ is approximately normal regardless of the population shape.

A common rule of thumb says the CLT applies when which condition holds for a moderately skewed population?

Answer: n ≥ 30. For moderately skewed populations, n ≥ 30 typically suffices.

What can you say about the sampling distribution of x̄?

Answer: It is also right-skewed because n is too small for the CLT. With strongly skewed populations and small n, the CLT does not apply; x̄ is also skewed.

AP Statistics · Topic 5.3

The Central Limit Theorem Practice

Part of Sampling Distributions.(UNC-3.D)

Practice questions

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Sample questions

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Sample 1difficulty 2/5
$A statistics teacher is reviewing the conditions under which the sampling distribution of the sample mean is approximately normal.$
$According to the Central Limit Theorem, the sampling distribution of x̄ is approximately normal when which condition holds?$
- A
  $The sample size n is sufficiently large (n \geq 30 is a common rule).$
  check_circle
- B
  $The population is uniform.$
- C
  $The population standard deviation is known.$
- D
  $The sample mean equals the population mean.$
Why
$The CLT states that for sufficiently large n, the sampling distribution of x̄ is approximately normal regardless of the population shape.$
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Sample 2difficulty 2/5
$A common rule of thumb says the CLT applies when which condition holds for a moderately skewed population?$
- A
  $n \geq 30.$
  check_circle
- B
  $n \geq 100.$
- C
  $Only when σ is known.$
- D
  $n \geq 5.$
Why
$For moderately skewed populations, n \geq 30 typically suffices.$
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Sample 3difficulty 3/5
$A population is strongly right-skewed. A sample of n = 5 is taken.$
$What can you say about the sampling distribution of x̄?$
- A
  $It is approximately normal by the CLT.$
- B
  $It is uniform.$
- C
  $It is also right-skewed because n is too small for the CLT.$
  check_circle
- D
  $It is exactly normal.$
Why
$With strongly skewed populations and small n, the CLT does not apply; x̄ is also skewed.$
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Sample 4difficulty 3/5
$The figure illustrates that a skewed population can produce$
- A
  $A roughly normal sampling distribution of x̄ when n is large$
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- B
  $A bimodal sampling distribution$
- C
  $An equally skewed sampling distribution of x̄$
- D
  $A uniform sampling distribution$
Why
$CLT: with large n, sampling distribution is approximately normal.$
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Sample 5difficulty 3/5
$For a moderately skewed population, n =$
- A
  $1$
- B
  $1000+$
- C
  $5$
- D
  $30+$
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Why
$Conventional threshold; very heavily skewed populations may need larger n.$
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