Practice Margin of Error in Math

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

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Quick Recap

The maximum expected difference between the sample statistic and the true population parameter; it is half the width of a confidence interval.

When a poll says 'the approval rating is 52%52\% with a margin of error of Β±3%\pm 3\%,' it means the true value is likely between 49%49\% and 55%55\%. The margin of error is the 'Β±\pm' partβ€”it tells you how much wiggle room to give the estimate. Larger samples and less variability shrink the margin of error.

Showing a random 20 of 50 problems.

Example 1

hard
A study triples its sample size from n=100n=100 to n=300n=300. By what factor does margin change?

Example 2

easy
A poll reports 52%52\% approval with a margin of error of Β±3%\pm 3\%. What interval of values does this imply?

Example 3

easy
To shrink the margin of error, which of these helps: bigger sample, smaller sample, or higher confidence?

Example 4

medium
Currently E=4E = 4 at n=50n = 50. Find EE at n=200n = 200 (same zβˆ—z^* and Οƒ\sigma).

Example 5

easy
True or false: a larger sample size produces a smaller margin of error (same confidence).

Example 6

challenge
A 95% CI for a mean is (xΛ‰βˆ’4,xΛ‰+4)(\bar{x} - 4, \bar{x} + 4). If you re-run at 99% confidence with the same data, find the new margin (z95βˆ—=1.96z^*_{95}=1.96, z99βˆ—=2.576z^*_{99}=2.576).

Example 7

hard
Show that p^(1βˆ’p^)\hat{p}(1-\hat{p}) is maximized when p^=0.5\hat{p}=0.5, and find its max value.

Example 8

hard
Two candidates poll at 51%51\% and 49%49\%, each with margin Β±3%\pm 3\%. Can you declare a winner?

Example 9

medium
A margin of 4.94.9 comes from zβˆ—=1.96z^* = 1.96 and SE =2.5= 2.5. If confidence rises to 99%99\% (zβˆ—=2.576z^* = 2.576), what is the new margin?

Example 10

medium
A margin of error is zβˆ—β‹…Οƒ/nz^* \cdot \sigma/\sqrt{n}. With Οƒ=18\sigma = 18, n=36n = 36, zβˆ—=1.645z^* = 1.645 (90%90\%), find the margin.

Example 11

hard
A poll has p^=0.5\hat{p}=0.5, n=2500n=2500, zβˆ—=1.96z^*=1.96. Margin?

Example 12

medium
Currently E=4.9%E=4.9\% at n=400n=400, p^=0.5\hat{p}=0.5, 95% confidence. What is the new margin at n=1600n=1600?

Example 13

medium
You need E≀3E \leq 3 for a 95% CI on a mean with Οƒ=20\sigma = 20. Find the smallest nn.

Example 14

medium
Two candidates poll at 48%48\% and 52%52\%, each with margin Β±3%\pm 3\%. Can we declare a clear winner?

Example 15

challenge
A pollster wants E≀0.02E \leq 0.02 at 95% confidence with no prior estimate of pp. Find the smallest nn (zβˆ—=1.96z^*=1.96).

Example 16

medium
A sample has Οƒ=24\sigma = 24, n=64n = 64. Find the 95%95\% margin of error (zβˆ—=1.96z^* = 1.96).

Example 17

medium
Find the 95% margin of error for p^=0.4\hat{p} = 0.4, n=100n = 100 (zβˆ—=1.96z^* = 1.96).

Example 18

challenge
A poll uses p^=0.5\hat{p} = 0.5 for a conservative margin. Why does p^=0.5\hat{p} = 0.5 maximize the margin, and what is the maximum value of p^(1βˆ’p^)\hat{p}(1-\hat{p})?

Example 19

medium
With Οƒ=30\sigma = 30, n=100n = 100, zβˆ—=1.96z^* = 1.96, compute the 95% margin of error.

Example 20

easy
A poll of n=400n=400 voters finds p^=0.55\hat{p}=0.55 supporting a candidate. Calculate the margin of error at 95% confidence and interpret the result.
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