Statistics (university) · Part 05
Sampling and estimation
Learn about a whole group from a sample and say how sure you are.
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- Population and sampleA population is the whole group you want to know about. A sample is the part you actually measure. A number describing the population is a parameter. A number from the sample is a statistic.
- Law of large numbersAs a sample gets bigger, the sample mean gets closer to the true mean. Early results can wobble a lot, but the average settles down.
- Sampling distribution and standard errorA sampling distribution shows how a statistic, such as the sample mean, varies from sample to sample. Its standard deviation is the standard error: SE = sigma / square root of n.
- Central limit theoremThe central limit theorem says that sample means from almost any population are close to normal when n is large enough (often 30 or more). The mean of the sample means is mu and their SD is sigma / square root of n.
- Point estimate and biasA point estimate is a single number guessing a parameter, such as the sample mean for the population mean. An estimator is unbiased if its average over many samples equals the true parameter.
- Confidence interval for a meanA confidence interval is a range likely to contain the true parameter. With known sigma, a 95 percent interval is the sample mean plus or minus 1.96 x sigma / square root of n.
- t distributionWhen sigma is unknown, replace it by the sample SD s and use the t distribution instead of the normal. It has heavier tails, shaped by degrees of freedom n minus 1. The interval is mean plus or minus t* x s / square root of n.
- Confidence interval for a proportionFor a sample proportion p-hat from n trials, the standard error is the square root of p-hat(1 - p-hat) / n. A 95 percent interval is p-hat plus or minus 1.96 x SE. It needs enough successes and failures, at least 10 of each.