Estimation
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Estimation is the process of using sample data to draw inferences about the population. A point estimate provides a single value, while an interval estimate provides a range of values expressing uncertainty. Good estimates are unbiased, meaning the expected value equals the true value, and precise, meaning the estimate is close to the true value across samples. The 95% confidence interval for a mean is calculated as the sample mean plus or minus 1.96 standard deviations, providing a 95% probability the interval contains the true mean. Similar principles apply to estimating proportions, differences between means/proportions, and small samples which use the t-distribution instead of the normal.
Estimation
- 1. Chapter 4: Estimation Estimation is the process of using sample data to draw inferences about the population Sample information Population parameters Inferences
- 2. Point and interval estimates Point estimate – a single value the temperature tomorrow will be 23 ° Interval estimate – a range of values, expressing the degree of uncertainty the temperature tomorrow will be between 21 ° and 25°
- 3. 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
- 4. Bias and precision – a possible trade-off True value
- 5. 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
- 6. The 95% confidence interval (Eqn. 3.17) Hence the 95% probability interval is Rearranging this gives the 95% confidence interval
- 7. The 95% probability interval
- 8. The 95% confidence interval
- 9. Example: estimating average wealth Sample data: = 130 (in £000) s 2 = 50,000 n = 100 Estimate , the population mean
- 10. Point estimate: 130 (use the sample mean) Interval estimate – use Example: estimating average wealth (continued)
- 11. What is a confidence interval? One sample out of 20 (5%) does not contain the true mean, 15.
- 12. Estimating a proportion Similar principles The sample proportion provides an unbiased point estimate The 95% CI is obtained by adding and subtracting 1.96 standard errors In this case we use
- 13. Example: unemployment Of a sample of 200 men, 15 are unemployed. What can we say about the true proportion of unemployed men? Sample data p = 15/200 = 0.075 n = 200
- 14. Point estimate: 0.075 (7.5%) Interval estimate: Example: unemployment (continued)
- 15. Estimating the difference of two means A survey of holidaymakers found that on average women spent 3 hours per day sunbathing, 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 sunbathing habits.
- 16. 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
- 17. Point estimate – use For the standard error, use Hence the point estimate is 3 – 2 = 1 hour Calculating point estimate CE: AQ: We have introduced the caption “Calculating point estimate”. Please confirm if it is ok.
- 18. For the confidence interval we have i.e. between 0.3 and 1.7 extra hours of sunbathing by women. Confidence intervals CE: AQ: We have introduced the caption “Confidence intervals”. Please confirm if it is ok.
- 19. Using different confidence levels The 95% confidence level is a convention The 99% confidence interval is calculated by adding and subtracting 2.57 standard errors (instead of 1.96) to the point estimate. The higher level of confidence implies a wider interval.
- 20. Estimating the difference between two proportions Similar to before – point estimate plus and minus 1.96 standard errors
- 21. Estimation with small samples: using the t distribution If: The sample size is small (<25 or so), and The true variance 2 is unknown Then the t distribution should be used instead of the standard Normal.
- 22. Example: beer expenditure A sample of 20 students finds an average expenditure on beer per week of £12 with standard deviation £8. Find the 95% CI estimate of the true level of expenditure of students. Sample data:
- 23. The 95% CI is given by The t value of t 19 = 2.093 is used instead of z =1.96 Example: beer expenditure (continued)
- 24. Summary The sample mean and proportion provide unbiased estimates of the true values The 95% confidence interval expresses our degree of uncertainty about the estimate The point estimate ± 1.96 standard errors provides the 95% interval in large samples
Editor's Notes
- #2: The sample information is known, the population parameters are not, in general.