Practice Sampling Distribution in Statistics

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The sampling distribution is the probability distribution of a statistic (such as the sample mean xˉ\bar{x}) computed from all possible random samples of a given size nn drawn from a population. It describes how that statistic varies from sample to sample.

If you took 1000 different random samples and calculated the mean of each, those 1000 means would form a distribution. That's the sampling distribution - it shows how sample statistics vary.

Showing a random 20 of 76 problems.

Example 1

medium
A sampling distribution of xˉ\bar{x} has mean 60 and SD 4. A sample gives xˉ=68\bar{x}=68. How many standard errors above the center is this?

Example 2

medium
For σ=20\sigma=20, the sampling distribution of xˉ\bar{x} should have SD =4=4. What sample size achieves this?

Example 3

medium
A population has μ=12\mu=12 and σ=6\sigma=6. What sample size makes the standard deviation of the sampling distribution of xˉ\bar{x} equal to 0.50.5?

Example 4

easy
A population has mean μ=50\mu=50. What is the mean (center) of the sampling distribution of the sample mean xˉ\bar{x}?

Example 5

easy
A population has μ=75\mu=75 and σ=10\sigma=10. For n=100n=100, the sampling distribution of xˉ\bar{x} has mean and SD equal to what?

Example 6

hard
μ=70,σ=8,n=16\mu = 70, \sigma = 8, n = 16. Use normal approximation to find P(xˉ<68)P(\bar{x} < 68).

Example 7

hard
A skewed population has μ=12,σ=4\mu = 12, \sigma = 4. For n=64n = 64, is the sampling distribution of xˉ\bar{x} approximately normal? Why?

Example 8

medium
Population has mean μ=50,σ=6\mu = 50, \sigma = 6. Find SE for n=9,36,144n = 9, 36, 144.

Example 9

medium
The sampling distribution of xˉ\bar{x} has SE =2.5= 2.5. If n=16n = 16, what is the population standard deviation σ\sigma?

Example 10

medium
What is the difference between the population distribution and the sampling distribution of xˉ\bar{x}?

Example 11

hard
A factory's tubes have μ=500\mu=500 ml and σ=10\sigma=10 ml. Quality inspectors test n=25n=25 tubes. What is the probability the sample mean lies outside the warning band 498498 to 502502?

Example 12

medium
Does increasing the sample size change the mean of the sampling distribution of xˉ\bar{x}?

Example 13

medium
Using μ=200\mu=200, SD of xˉ\bar{x} =5=5 (from n=36n=36, σ=30\sigma=30), what fraction of sample means fall above 205?

Example 14

easy
σ=30\sigma = 30, n=36n = 36. Find the standard error of xˉ\bar{x}.

Example 15

hard
A normal population has μ=20\mu=20, σ=4\sigma=4. For n=16n=16, find the 90%90\% central interval for xˉ\bar{x}.

Example 16

hard
A population has σ=20\sigma = 20. Find the standard error for sample sizes n=16n = 16 and n=64n = 64.

Example 17

hard
A population has mean μ=60\mu = 60 and standard deviation σ=12\sigma = 12. If we take samples of size n=36n = 36, what is the standard error of the sample mean?

Example 18

challenge
A population has σ=10\sigma=10. You want 95% of sample means to lie within 1 unit of μ\mu. Using ±2\pm 2 SE for 95%, find the needed sample size.

Example 19

medium
A population has μ=150\mu=150 and σ=30\sigma=30. For n=100n=100, find the probability that xˉ\bar{x} lies between 147147 and 153153.

Example 20

medium
If nn is multiplied by 44, the standard error of xˉ\bar{x} is multiplied by what factor?
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