Statistical inference: sampling, estimation and hypothesis testing
Statistical inference: sampling, estimation and hypothesis testing
Inference is examined as a confidence interval or a hypothesis test, usually finishing with a decision expressed in the context of the problem. The arithmetic is routine; the interpretation is where marks are gained or lost.
Core rules
- Sampling distribution of the mean: for a sample of size n from a population with mean mu and standard deviation sigma, the mean of X-bar is mu and its standard error is sigma/sqrt(n). For n at least 30 the central limit theorem gives X-bar approximately normal whatever the population; for a normal population with unknown sigma, use the t distribution with n - 1 degrees of freedom.
- Proportion: the estimator p-hat = x/n has mean p and standard error sqrt(p(1-p)/n).
- Confidence interval for the mean with known sigma: x-bar plus or minus z times sigma/sqrt(n), where z = 1.96 for 95 percent, 1.645 for 90 percent and 2.576 for 99 percent. For a proportion: p-hat plus or minus z times sqrt(p-hat(1-p-hat)/n).
- Margin of error E = z sigma/sqrt(n), so the required sample size is (z sigma/E)^2, always rounded up.
- Hypothesis test: state H0 and H1, fix the significance level, compute the statistic z = (x-bar - mu0)/(sigma/sqrt(n)) or z = (p-hat - p0)/sqrt(p0(1-p0)/n), and compare it with the critical value (plus or minus 1.96 two-tailed at 5 percent, 1.645 one-tailed) or compare the p-value with the level. Rejecting H0 at level alpha is equivalent to mu0 lying outside the corresponding confidence interval.
- Errors: Type I is rejecting a true H0, with probability alpha; Type II is failing to reject a false H0.
Worked example
In a sample of 400 voters, 232 support a proposal. Test at 5 percent whether the population proportion differs from 0.5.
H0: p = 0.5 against a two-tailed H1. Here p-hat = 0.58 and the standard error under H0 is sqrt(0.25/400) = 0.025, so z = 0.08/0.025 = 3.2. Since 3.2 exceeds 1.96, H0 is rejected: there is significant evidence that the true proportion is not 0.5. The p-value is about 0.0014, well below 0.05.
Traps that cost marks
Dividing by s instead of s/sqrt(n); using 1.96 when the test is one-tailed; writing that H0 is accepted or proved instead of not rejected; reading the p-value as the probability that H0 is true; and rounding a required sample size down instead of up.