Practice Confidence Interval in Math

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

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Quick Recap

A range of values, computed from sample data, that is likely to contain the true population parameter with a specified level of confidence.

You can't know the exact average height of all Americans, but after measuring 200 people you can say: 'I'm 95%95\% confident the true average is between 167 cm and 173 cm.' It's like casting a netβ€”wider nets catch the true value more often, but narrower nets are more useful. A 95%95\% confidence level means that if you repeated this process 100 times, about 95 of those nets would contain the true value.

Showing a random 20 of 50 problems.

Example 1

hard
Find the smallest nn for a 95%95\% CI for a proportion with E≀0.03E \le 0.03, assuming worst case p^=0.5\hat{p}=0.5, zβˆ—=1.96z^*=1.96.

Example 2

medium
A 95%95\% CI for the mean is (95.1,104.9)(95.1, 104.9). A colleague claims 'there is a 95%95\% probability the true mean lies in this interval.' What is the correct interpretation?

Example 3

easy
A 95%95\% CI for a mean is (46,54)(46, 54). What is the point estimate (the center)?

Example 4

medium
A 95%95\% CI for the difference in means is (βˆ’2,6)(-2, 6). Does this suggest a significant difference at Ξ±=0.05\alpha = 0.05?

Example 5

medium
To halve the width of a 95%95\% CI for a mean (same Οƒ\sigma, same confidence), by what factor must nn increase?

Example 6

medium
A 99%99\% CI (zβˆ—=2.576z^* = 2.576) uses xΛ‰=60\bar{x} = 60, Οƒ=10\sigma = 10, n=25n = 25. Find the interval.

Example 7

hard
A 98%98\% CI for ΞΌ\mu with xΛ‰=50\bar{x}=50, Οƒ=20\sigma=20, n=100n=100, zβˆ—=2.326z^*=2.326. Find the CI.

Example 8

easy
Should you use a z-interval or a t-interval when Οƒ\sigma is unknown and nn is small?

Example 9

medium
Compute a 95%95\% CI for ΞΌ\mu given xΛ‰=50\bar{x}=50, Οƒ=10\sigma=10, n=100n=100, zβˆ—=1.96z^*=1.96.

Example 10

easy
True or false: increasing the sample size nn (same confidence) narrows the interval.

Example 11

hard
If 100100 different 95%95\% CIs are constructed from independent samples, about how many are expected to miss the true mean?

Example 12

easy
A 95%95\% CI for ΞΌ\mu is (8,12)(8, 12). What is the width of the interval?

Example 13

hard
A 95%95\% CI for a proportion: p^=0.32\hat{p}=0.32, n=400n=400, zβˆ—=1.96z^*=1.96. Find the CI.

Example 14

medium
A 95%95\% CI for the difference of means is (1.2,4.8)(1.2, 4.8). Is the difference significantly different from zero?

Example 15

easy
A 95%95\% CI for ΞΌ\mu is (20,30)(20, 30). State the point estimate and the margin of error.

Example 16

medium
A 95%95\% CI is (10,14)(10, 14). What is the EE, and what was xˉ\bar{x}?

Example 17

easy
True or false: increasing the confidence level from 90%90\% to 99%99\% (same data) makes the interval wider.

Example 18

medium
For a proportion with p^=0.6\hat{p}=0.6, n=100n=100, build a 95%95\% CI (zβˆ—=1.96z^*=1.96).

Example 19

easy
A sample gives point estimate xˉ=50\bar{x} = 50 with margin of error 44. Write the 95%95\% confidence interval.

Example 20

easy
What zβˆ—z^* value corresponds to a 90%90\% confidence level?
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