Introduction to estimation: 2 cases Estimation is the process of using sample data to draw inferences about the population Sample information Population parameters Inferences The sample information is known, the population parameters are not, in general.

Point and interval estimates Point estimate – a single value the temperature tomorrow will be 2° Interval estimate – a range of values, expressing the degree of uncertainty the temperature tomorrow will be between -2° and +5° It is useful to have some idea of the uncertainty. Most people are far too confident about what they (think they) know.

Criteria for good estimates Unbiased – correct on average the expected value of the estimate is equal to the true value Precise – small sampling variance the estimate is close to the true value for all possible samples An unbiased estimator is correct on average, but not every time. It can give a bad estimate, via an unrepresentative sample. A precise estimator tends to give a similar answer over repeated samples. It’s a theoretical property – in practice we usually only ever take one sample.

Bias and precision – a possible trade-off True value The Bill Gates example illustrates this nicely. Leaving him out of the sample (if he were randomly picked) would bias the estimator, but leaving him in gives a completely unrepresentative result.

Estimating a mean (large samples) Point estimate – use the sample mean (unbiased) Interval estimate – sample mean ± ‘something’ What is the something? Go back to the distribution of It is helpful to focus on what is the random variable and its distribution. Once that is derived, the rest is easy.

The 95% confidence interval Hence the 95% probability interval is Rearranging this gives the 95% confidence interval Note that this is just an algebraic manipulation, but the meaning of the two intervals is quite different.

Example: estimating average wealth Sample data: = 130 K s2 = 50,000 n = 100 Estimate m, the population mean

Example: estimating average wealth (continued) Point estimate: 130 (use the sample mean) Interval estimate – use The interval estimate formula can be memorised as the point estimate plus and minus two standard errors. This is a general rule for the 95% interval in large samples.

Estimating the difference of two means A survey found that on average women spent 3 hours per day shopping, men spent 2 hours. The sample sizes were 36 in each case and the standard deviations were 1.1 hours and 1.2 hours respectively. Estimate the true difference between men and women in shopping habits.

Same principles as before… Obtain a point estimate from the samples Add and subtract 1.96 standard errors to obtain the 95% CI We just need the appropriate formulae It’s the same principle again!

Calculating the point estimate Point estimate – use For the standard error, use Hence the point estimate is 3 – 2 = 1 hour

Confidence interval For the confidence interval we have i.e. between 0.3 and 1.7 extra hours of shopping by women.