What does the central limit theorem say about the mean x̄ of a large random sample from a population with mean μ and standard deviation σ?

It is nearly normal, with mean μ and standard deviation σ/(√(n))

It is nearly normal, with mean μ and standard deviation σ

It is exactly equal to μ, whatever the sample

Question 26

MediumOriginal question

What does the central limit theorem say about the mean $\bar{x}$ of a large random sample from a population with mean $\mu$ and standard deviation $\sigma$ ?

It is nearly normal, with mean

\mu

and standard deviation

\sigma

Correct answerYour answerIt is exactly equal to

\mu

, whatever the sampleCorrect answerYour answerIt always has the same shape as the population it comes fromCorrect answerYour answerIt is nearly normal, with mean

\mu

and standard deviation

\frac{\sigma}{\sqrt{n}}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

For a large sample, the sample mean is close to normally distributed, whatever the shape of the population.
Its distribution is centred at $\mu$ , with spread $\frac{\sigma}{\sqrt{n}}$ , smaller than the population's.

Worth noting

Averages vary less than single values, so their spread is $\frac{\sigma}{\sqrt{n}}$ , not $\sigma$ .

Concept: The central limit theorem
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsParameters, statistics and the central limit theorem

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Ten questions from Sampling, Inference & the t-Test, at your own pace.

What does the central limit theorem say about the mean xˉ\bar{x}xˉ of a large random sample from a population with mean μ\muμ and standard deviation σ\sigmaσ?

Explanation

What does the central limit theorem say about the mean $\bar{x}$ of a large random sample from a population with mean $\mu$ and standard deviation $\sigma$ ?