1. Chapter 14 Single-Population Estimation

2. Population Statistics Population Statistics:  , usually unknown Using Sample Statistics to estimate population statistics: –Point estimates –Confidence interval

3. Confidence Interval Wrong Meaning: Probability within the confidence interval is..% True meaning: The interval will contain the population statistics..% times

4. Confidence Interval for Population Mean Confidence Interval If  is known, If  is unknown,

5. Common Confidence Levels Confidence Level 1-  /21-  /2 90%.90.10.05.95 95%.95.05.025.975 99%.99.01.005.995 z 0.9001.282 0.9501.645 0.9751.960 0.9902.326 0.9952.576 0.100-1.282 0.050-1.645 0.025-1.960 0.010-2.326 0.005-2.576 1-   /2 -z z 1-   -z

6. Example 14.2  =350, n=40, 90% CI 95% CI 99% CI

7. Confidence Interval for Population Mean – Sample size Can be (1-  ) confident that the error will not exceed a specified amount E when sample size is Round up to an integer

8. Confidence Interval for Population Mean One-sided Confidence Bounds If  is known, If  is unknown,

9. t Distribution Let x 1, x 2,.. x n be a random sample from a normal distribution with unknown  and  2, the random variable has a t distribution with n-1 degrees of freedom. Probability Density Function, with k degrees of freedom, Mean Variance

10. t Distribution www.boost.org/.../graphs/students_t_pdf.png

11. t Distribution df90%95%99% 22.9204.3039.925 52.0152.5714.032 101.8122.2283.169 301.6972.0422.750 501.6762.0092.678 1001.6601.9842.626 10001.6461.9622.581 z1.6451.9602.576 Confidence Level: 1-  Two-sided CI

12. Example 14.4 95% CI, s=5.79, n=11 95% CI, n=80 95% CI,  =5.79 Excel: tinv( , df)

13. Confidence Interval for Population Variance Confidence Interval –Where: n-1 is the degree of freedom, and 1-  is the confidence level One-sided Confidence Bounds

14.  2 Distribution Let x 1, x 2,.. x n be a random sample from a normal distribution with  and  2, and let s 2 be the sample variance, then the random variable (n-1)s 2 /  2 has  2 distribution with n-1 degrees of freedom. Probability Density Function, with k degrees of freedom, Mean Variance Mode = k-2 (when k  3)

15.  2 Distribution fr.academic.ru/pictures/frwiki/67/Chi-square_..

16. Example 14.11 n=24, s 2 =.47 99% Confidence Interval Excel: chiinv( , df)

17. Confidence Interval for Population Proportion Confidence Interval One-sided Confidence Bound

18. Example 14.7 n=350, x=212 95% Confidence Interval

19. Confidence Interval for Population Proportion – Sample size Can be (1-  ) confident that the error will not exceed a specified amount E when sample size is Round up to an integer

20. Confidence Interval for Population Proportion – when p is small Wilson Estimator Confidence Interval

21. Example 14.8 n=350, x=212 95% Confidence Interval with Wilson Estimator

22. Example 14.9 N=50, x=1 99% Confidence Interval with Wilson Estimator