Confidence Intervals — Core Maths Compendium

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Understanding confidence intervals

Click each step below to reveal it — work through them in order the first time round.

Step 1 · What a confidence interval is

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A confidence interval gives a range of values, calculated from sample data, that is likely to contain the true population value (e.g. the population mean). Rather than a single estimate, it acknowledges the uncertainty involved in estimating from a sample.

Step 2 · Constructing a confidence interval

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A confidence interval for the mean is calculated as: sample mean ± (a multiplier × standard error). For a 95% confidence interval, the multiplier is approximately 1.96 (from the standard normal distribution): $\bar{x}\pm 1.96\times\frac{\sigma}{\sqrt{n}}$.

Step 3 · What ‘confidence level’ actually means

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A ‘95% confidence interval’ does not mean there's a 95% probability the true population mean lies in this particular interval. It means that if you repeated the sampling process many times and built a confidence interval each time, about 95% of those intervals would contain the true population mean.

Step 4 · Factors affecting the width of a confidence interval

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A wider confidence interval gives you more confidence it contains the true value, but is less precise. Increasing sample size narrows the interval (more precise); increasing the confidence level (e.g. 95%→99%) widens the interval (need a bigger range to be more sure).

Worked example

A sample of 64 items has a mean of 50 and the population standard deviation is known to be 8. Construct a 95% confidence interval for the population mean.

SE = 8÷√64 = 8÷8 = 1
95% CI = 50 ± 1.96×1 = 50 ± 1.96 = (48.04, 51.96).

Test yourself

Past-paper style question

A researcher takes a sample of 100 students' test scores, finding a sample mean of 62. The population standard deviation is known to be 10.

(a) Calculate the standard error of the mean.
(b) Construct a 95% confidence interval for the population mean score, using a multiplier of 1.96.
(c) Explain what this confidence interval tells us about the population mean. [5 marks]

Show the answer

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(a) SE = 10÷√100 = 10÷10 = 1.

(b) 95% CI = 62 ± 1.96×1 = 62 ± 1.96 = (60.04, 63.96).

(c) This means that if we repeated this sampling process many times, about 95% of the confidence intervals constructed this way would contain the true population mean score — we can be reasonably confident the true mean lies somewhere between about 60.04 and 63.96.

Practice

Confidence Intervals worksheet

Five short questions on confidence intervals. Work through them, then reveal the mark scheme to check.

A sample of 49 has a mean of 30, and population standard deviation 7. Calculate the standard error.
Using your answer to Q1, construct a 95% confidence interval for the population mean (using multiplier 1.96).
Explain why a 99% confidence interval is wider than a 95% confidence interval, for the same sample data.
A confidence interval is calculated as (45.2, 54.8). State the sample mean and the margin of error (half the width of the interval).
Explain why it is incorrect to say ‘there is a 95% probability that the true population mean lies within this particular confidence interval.’

Mark scheme

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SE = 7÷√49 = 7÷7 = 1.
95% CI = 30 ± 1.96×1 = (28.04, 31.96).
A 99% confidence interval needs to be wider to be more confident (99% vs 95%) that it captures the true population mean — a higher confidence level requires a larger range of plausible values, using a bigger multiplier than 1.96.
Sample mean = (45.2+54.8)÷2 = 50. Margin of error = (54.8−45.2)÷2 = 4.8.
The true population mean is a fixed (though unknown) value — it either is or isn't in a specific interval, so there's no probability about it once the interval has been calculated. The 95% refers to the long-run proportion of such intervals (across repeated sampling) that would contain the true mean, not the probability for this one specific interval.

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