Confidence Intervals (Dr. Monticino)

1. Confidence Intervals(Dr. Monticino)

2. Assignment Sheet • Read Chapter 21 • Assignment # 14 (Due Monday May 2nd ) • Chapter 21 • Exercise Set A: 1,2,3,7 • Exercise Set B: 1-4 • Exercise Set C: 1-8 • Exercise Set E: 1,2,3 • Review Exercises: Try them all … not to turn in. (good review for Final Exam)

3. Overview • Confidence intervals for survey sampling

4. Knowing/Not Knowing the Percentages • Up to now, have assumed that know composition of population being sampled • Example: Know the percentage of population of a particular type • Have calculated the probability that a randomly drawn sample will have a certain sample percentage • Now will draw a sample without knowing composition of population • Want to infer value of population parameter from sample statistic AND want to measure how reliable is the sample statistic

5. Confidence Intervals • A confidence interval for a parameter estimate provides a measure of the accuracy of the estimate • A c% confidence interval is a random interval, calculated from the sample, which has a c% probability of containing the population parameter • Example: 95% percent of the time a 95% confidence interval will contain the population parameter

6. Components of a Confidence Interval Calculation • Parameter statistic • A parameter statistic is the population parameter estimate obtained from the sample • Sample mean • Sample percentage • Population variance • The sample is used to estimate how much the population values vary • Population standard deviation is estimated with sample standard deviation (for large samples) • Use corrected sample standard deviation (for small samples)

7. Components of a Confidence Interval Calculation • Standard error • The standard error of the sample measures the likely amount that the sample statistic is off from the population parameter • Often use • Confidence Level • The confidence level indicates how confident you should be that population parameter lies in the confidence interval • Use the normal approximation given by the Central Limit Theorem

8. Confidence Intervals • General form of a confidence interval is (sample statistic) +/- (standard units associated with c% confidence interval)*(SE)

9. Example • A survey was conducted to determine the proportion of current UNT students who would be interested in enrolling in a web-based statistics course. In the survey of 500 students, 200 of the students expressed interest. Determine the 95% confidence interval for the percentage of students interested in a web-based course.

10. Example • Suppose now that all UNT students were surveyed and the proportion of students who were interested in a web-based math course was .28. If appropriate, calculate the 95% confidence interval.

11. Cautions and Notes • The standard deviation of the sample can be used as an estimate for the standard deviation of the population if • the sample is large enough • “Large enough” depends on many factors • the sample is obtained by simple random sampling

12. Cautions and Notes • The standard deviation says how far an element in the population differs from the population average, for a typical element • The standard error says how far the sample average differs from the population average, for a typical sample • Most methods for calculating confidence intervals assume simple random sample • These methods are not appropriate for other types of samples

13. SE without replacement =  SE with replacement Cautions and Notes • If the sample is selected from the population without replacement and the sample is large with respect to the population, then a correction factor is needed for the standard error

14. Cautions and Notes • Confidence intervals for small samples are tricky to calculate. When in doubt • Select a larger sample • Consult a statistician • If the sample data show a trend or pattern over time, then the above techniques do not apply to estimating parameter values or determining their accuracy (Dr. Monticino)