Practice Standard Error in Statistics

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The standard error (SE) is the standard deviation of a sampling distribution, measuring how much a sample statistic (like the sample mean) typically varies from the true population parameter across repeated samples. It decreases as sample size increases.

Standard error tells you how much your sample estimate might be 'off' from the true value. Larger samples have smaller SE because they're more precise - like asking 1000 people vs 10.

Showing a random 20 of 50 problems.

Example 1

medium
Population SD is 50. What sample size makes the standard error of the mean equal to 5?

Example 2

challenge
Combine two independent sample means each with SE =3=3 into their difference. What is the standard error of the difference?

Example 3

hard
In a two-sample t-test with xห‰1=50\bar{x}_1=50, xห‰2=46\bar{x}_2=46, s1=8s_1=8, s2=6s_2=6, n1=n2=64n_1=n_2=64. Find the SE of the mean difference.

Example 4

easy
A sample of n=36n=36 has sample SD s=24s=24. Estimate the SE of the mean.

Example 5

medium
You want to shrink the SE of the mean from 55 to 11 keeping the same population. By what factor must nn grow?

Example 6

medium
A sample mean is 80 with SE =4=4. How many standard errors away is a hypothesized mean of 72?

Example 7

easy
Population SD is 1414 and n=49n=49. Find the SE of the mean.

Example 8

hard
A polling company wants the standard error of a proportion to be no more than 0.02 (2%). If a preliminary estimate suggests p^โ‰ˆ0.5\hat{p} \approx 0.5, what minimum sample size is needed? Use SE=p^(1โˆ’p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

Example 9

easy
Does the SE of the mean ever exceed the population SD?

Example 10

medium
To halve the standard error of the mean, by what factor must the sample size increase?

Example 11

medium
A sample of n=16n=16 has SD s=12s=12. Estimate the SE of the mean.

Example 12

medium
A researcher measures the reaction time of 64 participants and finds a sample standard deviation of s=40s = 40 ms. Calculate the standard error and construct an approximate 95% confidence interval if the sample mean is 250 ms.

Example 13

medium
In what limit does the SE of the mean approach zero?

Example 14

hard
Two independent samples have means xห‰1\bar{x}_1 and xห‰2\bar{x}_2 with SEs 1.21.2 and 0.90.9. Find the SE of the difference xห‰1โˆ’xห‰2\bar{x}_1 - \bar{x}_2.

Example 15

easy
A population has a standard deviation of ฯƒ=20\sigma = 20. If you take a random sample of n=100n = 100, what is the standard error of the sample mean?

Example 16

easy
Compute the SE of the mean when ฯƒ=18\sigma=18 and n=81n=81.

Example 17

challenge
Population SD is 40. A researcher needs a 95% margin of error (2โ€‰SE2\,SE) of at most 4. Find the minimum sample size.

Example 18

easy
What is the standard error if ฯƒ=0\sigma=0?

Example 19

medium
A normal population has ฯƒ=50\sigma=50. Find the sample size needed so that the SE of the mean is at most 55.

Example 20

hard
A bootstrap procedure resamples 10001000 samples and computes xห‰โˆ—\bar{x}^* each time. The SD of those 10001000 means is the bootstrap estimate of what?
โ† Back to Standard Error Worked Examples Formula Explained