The sampling distribution of a sample mean

About this lesson

Statistics (AP-aligned) › Sampling distributions

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The sampling distribution of a sample mean

Warm-up

Three quick questions from earlier lessons.

Discover

Heights have SD 10 cm. Is the SD of the mean of 25 random heights also 10 cm?

  1. No, it is 2 cm
  2. Yes

Key idea

Mean and spread of x̄.

For random samples of size nn from a population with mean μ\mu and SD σ\sigma: μxˉ=μ\mu_{\bar{x}} = \mu and σxˉ=σn\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}. If the population is normal, xˉ\bar{x} is exactly normal.

Worked example

μ=170\mu = 170, σ=10\sigma = 10, n=25n = 25. How unusual is xˉ=174\bar{x} = 174?

σxˉ=2\sigma_{\bar{x}} = 2
10/5.
z=2z = 2
(174 − 170)/2.
Unusual (about 2.5% are higher)
Beyond 2 SDs.

Your turn, guided

Your turn: σ=12\sigma = 12, n=16n = 16. Find σxˉ\sigma_{\bar{x}}.

33
12/4.

Practice ★

  1. A population has standard deviation σ = 9. Find the SD of the sample mean for samples of size 49.
  2. A population has standard deviation σ = 29. Find the SD of the sample mean for samples of size 4.
  3. A population has standard deviation σ = 22. Find the SD of the sample mean for samples of size 64.

Watch out!

Practice ★★

  1. Samples of size 26 give a certain SD of x̄. By what factor must the sample size be multiplied to halve it?
  1. A population has mean 76 and SD 5. A random sample of 4 has mean 77. Find the z-score of this sample mean.
  2. A population has mean 62 and SD 14. A random sample of 16 has mean 69. Find the z-score of this sample mean.

Exit ticket

Three questions to check you have got it.

  1. A population has standard deviation σ = 9. Find the SD of the sample mean for samples of size 49.
  2. A population has standard deviation σ = 29. Find the SD of the sample mean for samples of size 4.
  3. A population has standard deviation σ = 22. Find the SD of the sample mean for samples of size 64.

Summary card

Sampling distribution of x̄

μxˉ=μ\mu_{\bar{x}} = \mu, σxˉ=σ/n\sigma_{\bar{x}} = \sigma/\sqrt{n}.

Watch out: divide by n\sqrt{n}, not nn.

End of the lesson

Next lesson: The central limit theorem Back to the chapter

Answers: The sampling distribution of a sample mean

A

  1. 97\frac{9}{7}
  2. 292\frac{29}{2}
  3. 114\frac{11}{4}

B

  1. 44

C

  1. 25\frac{2}{5}
  2. 22

D

  1. 97\frac{9}{7}
  2. 292\frac{29}{2}
  3. 114\frac{11}{4}

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