Practice Sampling Variability in Statistics

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

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Quick Recap

Sampling variability is the natural sample-to-sample difference that appears when we take repeated random samples from the same population. Even good random samples do not all produce identical statistics.

If you take two honest random samples, they can still disagree a little. That disagreement is not necessarily bias or a mistake; it is part of how sampling works.

Showing a random 20 of 50 problems.

Example 1

challenge
A pollster reports a result with no margin of error, claiming their single random sample of 500 gives the precise population value. Explain the conceptual error and what they should report.

Example 2

medium
Two researchers each randomly sample 100100 adults and find different mean incomes. They should attribute the gap to ____ before suspecting fraud.

Example 3

easy
A population has mean μ=100\mu = 100. Three random samples give means 9898, 101101, and 103103. What explains the spread of these sample means?

Example 4

challenge
Sample A (n=25) and sample B (n=100) come from the same population with σ=15\sigma=15. By what factor is sample A's typical variability larger than sample B's?

Example 5

challenge
A factory's bolts have μ=5.00\mu = 5.00 cm, σ=0.10\sigma = 0.10 cm. A quality engineer rejects a shift if xˉ\bar{x} from n=25n=25 bolts differs from 5.005.00 by more than 0.040.04 cm. What is the rejection probability for an in-control shift?

Example 6

challenge
A poll of n=1000n=1000 adults estimates support at 51%51\%. The margin of error (at 95%95\%) is about ±3.1%\pm 3.1\%. Why can't the poll conclude that 'a majority supports' the measure?

Example 7

medium
You take 1000 random samples of size 40 and plot all 1000 sample means. The plot's spread reflects what?

Example 8

hard
A class simulates taking 10001000 random samples of n=40n=40 and records each sample's mean. What does the spread of those 10001000 means estimate?

Example 9

medium
A teacher says: 'Class A averaged 82 and Class B averaged 80 on a random quiz sample, so A is the stronger class.' Why is caution warranted?

Example 10

medium
A teacher takes many random samples of n=36n=36 test scores from a population with μ=75\mu=75 and σ=18\sigma=18. Sketch in words the sampling distribution of xˉ\bar{x}.

Example 11

easy
Polling 30 students gives a mean study time of 4.2 hours; another 30 gives 3.9 hours. What kind of difference is this most likely?

Example 12

easy
A scientist remeasures the same object 5 times with a scale and gets slightly different readings. Is this sampling variability?

Example 13

easy
A radio host claims one sample of 500500 adults gives the 'exact' national approval rating. What is the flaw?

Example 14

medium
A coin is fair (p=0.5p=0.5). Out of 100100 flips, you get 5858 heads. Is this strong evidence the coin is biased?

Example 15

challenge
A population has σ=20\sigma=20. A researcher wants the typical sampling variability of the mean to be at most 2. What minimum sample size is needed?

Example 16

easy
Which sample produces less sampling variability for the sample mean: n=25n = 25 or n=400n = 400?

Example 17

medium
In one random sample of n=50n=50, the sample mean is xˉ=12\bar{x}=12. Why is it wrong to claim 'the population mean is exactly 1212'?

Example 18

medium
A population has μ=60\mu=60, σ=10\sigma=10. Compare the standard error of xˉ\bar{x} for n=25n=25 vs n=100n=100.

Example 19

easy
True or false: sampling variability disappears completely if the sample is taken truly at random.

Example 20

hard
A population has σ=15\sigma = 15. You want the standard error of xˉ\bar{x} to be at most 1.51.5. What is the minimum nn?
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