1. Understanding CI for Means Ayona Chatterjee Math 2063 University of West Georgia

2. Why do we need CI? We want to estimate the mean of the population, since we cannot measure the whole population, we use sample mean to estimate the population mean. Assume we have a small city with 20 houses and this is our whole population. We want to find an estimate for the average house price in this city.

3. House Prices House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295 The house prices for all the houses in the population are given in thousands of dollars. Here the population mean is 305.1 In real life the population size is so large that we cannot find the true mean. There are about 90 million homes in USA. In 2007 the average house price in USA was 299700.

4. House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295 Suppose we can only sample 10 houses from the population to find an estimate for the population mean. Sample 1 mean: 301

5. Suppose we can only sample 8 houses from the population to find an estimate for the population mean. Sample 1 mean: 301 Sample 2 mean: 337.5 House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295

6. Suppose we can only sample 8 houses from the population to find an estimate for the population mean. Sample 1 mean: 301 Sample 2 mean: 337.5 Sample 3 mean: 284.2 House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295

7. Suppose we can only sample 8 houses from the population to find an estimate for the population mean. Sample 1 mean: 301 Sample 2 mean: 337.5 Sample 3 mean: 284.2 Sample 4 mean: 300.2 House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295

8. Suppose we can only sample 8 houses from the population to find an estimate for the population mean. Sample 1 mean: 301 Sample 2 mean: 337.5 Sample 3 mean: 284.2 Sample 4 mean: 300.2 Sample 5: mean: 342.7 House prices 180275 300395 400210 450285 300280 260320 180400 250450 267355 250295

9. Note We have now five DIFFERENT estimates for the same population mean. Are all of the correct? How large or how small can the sample estimate get? It would be helpful to have an interval such that we can conclude that the population mean will lie within this interval with some confidence.

10. Confidence Interval A confidence interval is a range of values used to estimate the true value of a population parameter. A CI estimate of a parameter consists of an interval of numbers generated by a point estimate, together with an associated confidence level.

11. Confidence Level The confidence level is the probability 1-α that is the proportion of times the CI will actually contain the population parameter. Most common confidence levels are 90%, 95% and 99%. A 95% confidence interval means that in the long run, the proportion of intervals that will contain the parameter will equal 95%.