Maths › Further Statistics 2 › Estimators, standard error and confidence intervals
Estimators, standard error and confidence intervals
A sample statistic is a random variable with a distribution of its own. Knowing that distribution turns a single estimate into an interval, and gives an honest account of how much the estimate can be trusted.
Builds on The Central Limit Theorem and The normal distribution.
IN THIS TOPIC
- Explain what an unbiased estimator is and check whether one is.
- Calculate a standard error and say what it measures.
- Construct and interpret a confidence interval for a normal mean.
WHAT YOU PROBABLY THINK
A 95% confidence interval has a 95% chance of containing the sample mean.
Estimators and their quality
An estimator is a rule for turning a sample into a guess at a population parameter. It is unbiased if its expected value equals the parameter: the sample mean is unbiased for the population mean, and dividing by n − 1 rather than n makes the sample variance unbiased for σ².
Between two unbiased estimators, prefer the one with the smaller variance: it lands near the target more often. The standard error of the sample mean, σ/√n, is exactly that variance expressed as a standard deviation, and it shrinks like the square root of the sample size, so quadrupling the data halves the uncertainty.
WORKED EXAMPLE
Comparing two estimators
A population has mean μ. Two estimators are proposed from a sample of 3: the mean of all three values, and the first value alone. Compare them.
Both are unbiased: E(sample mean) = μ and E(X1) = μ.
Their variances are σ²/3 and σ². The mean of all three is three times more tightly clustered.
So both are honest, but only one is efficient: unbiasedness alone does not make an estimator good.
Turning an estimate into an interval
If the population variance is known, the sample mean is normal with standard error σ/√n, so a fixed proportion of samples land within a fixed number of standard errors of μ. Reversing that gives a confidence interval:
The interpretation needs care. The population mean is a fixed number, not a random one, so it either is or is not in your interval. What is random is the interval: 95% of the intervals built this way from repeated samples would contain μ. The opening claim gets it backwards, and also names the wrong quantity, since the sample mean is always at the centre of its own interval.
YOUR TURN
Building the interval
A sample of 25 from a normal population with σ = 10 has mean 104. Find 95% and 99% confidence intervals for μ, and comment.
Show the working
The standard error is 10/√25 = 2.
95%: 104 ± 1.96 × 2 = 104 ± 3.92, giving (100.08, 107.92).
99%: 104 ± 2.5758 × 2 = 104 ± 5.15, giving (98.85, 109.15).
More confidence costs width. Since 100 lies inside both, a claim that μ = 100 would survive a two-tailed test at either level.
THE EXAM BIT
- Calculate the standard error as a separate line; it earns a mark and prevents slips later.
- Quote the z value you used, and use 2.5758 rather than 2.58 when the question asks for accuracy.
- Interpret the interval as a statement about the procedure, not about the probability that μ lies inside.
- Link the interval to a test: a value outside the interval is rejected by the matching two-tailed test.
CHECK YOURSELF
A sample of 100 from a population with σ = 20 has mean 55. Find a 95% confidence interval for μ.
Show a hint
Standard error first, then 1.96 of them either side.
Show the answer
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An estimator is unbiased when its expected value is the parameter; among unbiased ones, prefer the smaller variance.
The standard error σ/√n sets the width: a 95% interval is the sample mean ± 1.96 standard errors, and 95% of such intervals capture μ.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Explain what an unbiased estimator is and check whether one is.
- Calculate a standard error and say what it measures.
- Construct and interpret a confidence interval for a normal mean.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.