A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.
Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.
Showing a random 20 of 76 problems.
Example 1
medium
To halve the width of a confidence interval (same confidence, same σ), how must n change?
Example 2
medium
If a 95% CI for μ excludes 0, what does a two-sided H0:μ=0 test at α=0.05 conclude?
Example 3
hard
A sample of 100 students has a mean test score of xˉ=72 with population standard deviation σ=10. Construct a 95% confidence interval for the population mean.
Example 4
medium
n=25, xˉ=72, s=10. Find the 95% t-CI for μ. Use t∗=2.064.
Example 5
hard
A 95% CI for p is [0.40,0.50]. What is the approximate sample size if the CI uses z∗=1.96 and p^=0.45?
Example 6
hard
True or false: a 99% CI for μ has ≥99% probability of containing the sample mean xˉ.
Example 7
medium
A 95% CI for a mean is entirely above 0, say [2, 8]. What does this imply about the parameter?
Example 8
hard
xˉ=68, s=12, n=9. Construct a 95% t-CI for μ. Use t∗=2.306 (df =8).
Example 9
easy
Which is wider: a 90% confidence interval or a 99% confidence interval from the same data?
Example 10
easy
A sample proportion is p^=0.40 with margin of error 0.05. State the 95% confidence interval.
Example 11
hard
A claim says 'the true mean is in [42,50] with probability 95%.' Under the frequentist interpretation, what is wrong with the wording?
Example 12
medium
If n quadruples (everything else fixed), the CI width changes by what factor?
Example 13
challenge
A sample of n=400 has p^=0.5. Build a 95% CI for the proportion using z∗=2 and SE =p^(1−p^)/n.
Example 14
medium
A study reports a 95% CI of [2.0,4.0] kg for average weight loss. Translate this into estimate-and-ME form.
Example 15
easy
As sample size grows, what happens to the width of a confidence interval (other things equal)?
Example 16
easy
Fill in: a wider confidence interval reflects ____ uncertainty about the parameter.
Example 17
easy
Increasing the confidence level from 90% to 99% makes the CI ___.
Example 18
medium
A sample of n=100 has mean xˉ=80 with σ=10. Build a 95% z-interval using z∗=1.96.Sampling distribution of x̄ (n = 100, σ = 10, SE = 1); 95% CI = [78.04, 81.96]
Example 19
medium
Two-sample 95% CI for μ1−μ2: [−1.5,4.5]. Can we reject H0:μ1=μ2 at α=0.05?
Example 20
hard
A 95% CI for one mean is [5,15] and an independent 95% CI for another mean is [10,20]. Can you conclude the means differ at α=0.05 just because the intervals overlap?