Statistical Inference and Estimation
This domain covers how a sample is used to estimate a population quantity: the properties that make an estimator trustworthy, standard errors, confidence intervals, and the sample size needed to reach a stated precision.
Properties of estimators
- Unbiased: the expected value of the estimator equals the parameter. The sample mean and s-squared are unbiased; the sample standard deviation is mildly biased downward for sigma.
- Efficient: it has the smallest variance among unbiased estimators. Consistent: it converges to the parameter as n grows.
- The standard error of the mean is sigma divided by root n, estimated by s divided by root n. Quadrupling n halves the standard error.
Building intervals
- Known sigma, or large n: x-bar plus or minus z at alpha/2 times sigma divided by root n.
- Unknown sigma, small n, roughly normal data: x-bar plus or minus t with n minus 1 degrees of freedom times s divided by root n.
- Proportion: p-hat plus or minus z times the root of p-hat(1 minus p-hat)/n, reliable when both np-hat and n(1 minus p-hat) exceed about 10.
- Required sample size for a mean: n = (z times sigma divided by margin) squared, always rounded up. With no prior estimate, use p = 0.5 for a proportion, which maximises the product.
Interpretation
- A 95% interval means that 95% of intervals constructed this way capture the parameter; it is not a 95% probability that this particular interval contains it.
- A prediction interval for a single new observation is always wider than a confidence interval for the mean.
- Narrower intervals come from larger n, smaller variability or lower confidence.
Worked example
A sample of 25 gives x-bar = 52 and s = 8. The standard error is 8/5 = 1.6, and t at 0.025 with 24 degrees of freedom is 2.064, so the margin is 3.30 and the interval is 48.70 to 55.30. To halve that margin at the same confidence you need roughly four times the sample, about 100 observations, because the margin scales with one over root n.
Traps
- Using z when sigma is unknown and n is small.
- Saying there is a 95% chance the parameter lies inside the computed interval.
- Rounding a required sample size down instead of up.
- Using s divided by root n but writing root(n minus 1) in the denominator.