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© 2002 Thomson / South-Western Slide 8-1 Chapter 8 Estimation with Single Samples.

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Presentation on theme: "© 2002 Thomson / South-Western Slide 8-1 Chapter 8 Estimation with Single Samples."— Presentation transcript:

1 © 2002 Thomson / South-Western Slide 8-1 Chapter 8 Estimation with Single Samples

2 © 2002 Thomson / South-Western Slide 8-2 Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean for large sample sizes. Estimate a population mean from a sample mean for small sample sizes. Estimate a population proportion from a sample proportion. Estimate the minimum sample size necessary to achieve given statistical goals.

3 © 2002 Thomson / South-Western Slide 8-3 Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample Interval Estimate -- a range of values calculated from a sample statistic(s) and standardized statistics, such as the Z. –Selection of the standardized statistic is determined by the sampling distribution. –Selection of critical values of the standardized statistic is determined by the desired level of confidence.

4 © 2002 Thomson / South-Western Slide 8-4 Confidence Interval to Estimate  when n is Large Point estimate Interval Estimate

5 © 2002 Thomson / South-Western Slide 8-5 Distribution of Sample Means for (1-  )% Confidence  X  Z 0

6 © 2002 Thomson / South-Western Slide 8-6 Z Scores for Confidence Intervals in Relation to   X Z 0

7 © 2002 Thomson / South-Western Slide 8-7 Distribution of Sample Means for (1-  )% Confidence  X Z 0

8 © 2002 Thomson / South-Western Slide 8-8 Probability Interpretation of the Level of Confidence

9 © 2002 Thomson / South-Western Slide 8-9 Distribution of Sample Means for 95% Confidence .4750 X 95%.025 Z 1.96-1.960

10 © 2002 Thomson / South-Western Slide 8-10 Example: 95% Confidence Interval for 

11 © 2002 Thomson / South-Western Slide 8-11 95% Confidence Intervals for   X 95% X X X X X X

12 © 2002 Thomson / South-Western Slide 8-12 95% Confidence Intervals for   X 95% X X X X X X Is our interval, 3.98  4.54, in the red?

13 © 2002 Thomson / South-Western Slide 8-13 Demonstration Problem 8.1

14 © 2002 Thomson / South-Western Slide 8-14 Demonstration Problem 8.2

15 © 2002 Thomson / South-Western Slide 8-15 Confidence Interval to Estimate  when n is Large and  is Unknown

16 © 2002 Thomson / South-Western Slide 8-16 Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level Z Value 1.645 1.96 2.33 2.575

17 © 2002 Thomson / South-Western Slide 8-17 Estimating the Mean of a Normal Population: Small n and Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. The sample size is small, n < 30. Z distribution is not appropriate for these conditions t distribution is appropriate

18 © 2002 Thomson / South-Western Slide 8-18 The t Distribution A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a Z t formula

19 © 2002 Thomson / South-Western Slide 8-19 Comparison of Selected t Distributions to the Standard Normal -3-20123 Standard Normal t (d.f. = 25) t (d.f. = 5) t (d.f. = 1)

20 © 2002 Thomson / South-Western Slide 8-20 Table of Critical Values of t df t 0.100 t 0.050 t 0.025 t 0.010 t 0.005 13.0786.31412.70631.82163.656 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 231.3191.7142.0692.5002.807 24 1.318 1.711 2.0642.4922.797 251.3161.7082.0602.4852.787 291.3111.6992.0452.4622.756 301.3101.6972.0422.4572.750 401.3031.6842.0212.4232.704 601.2961.6712.0002.3902.660 1201.2891.6581.9802.3582.617 1.2821.6451.9602.3272.576  tt  

21 © 2002 Thomson / South-Western Slide 8-21 Confidence Intervals for  of a Normal Population: Small n and Unknown 

22 © 2002 Thomson / South-Western Slide 8-22 Solution for Demonstration Problem 8.3

23 © 2002 Thomson / South-Western Slide 8-23 Solution for Demonstration Problem 8.3

24 © 2002 Thomson / South-Western Slide 8-24 Confidence Interval to Estimate the Population Proportion

25 © 2002 Thomson / South-Western Slide 8-25 Solution for Demonstration Problem 8.5

26 © 2002 Thomson / South-Western Slide 8-26 Determining Sample Size when Estimating  Z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated 

27 © 2002 Thomson / South-Western Slide 8-27 Example: Sample Size when Estimating 

28 © 2002 Thomson / South-Western Slide 8-28 Solution for Demonstration Problem 8.6

29 © 2002 Thomson / South-Western Slide 8-29 Determining Sample Size when Estimating P Z formula Error of Estimation (tolerable error) Estimated Sample Size

30 © 2002 Thomson / South-Western Slide 8-30 Solution for Demonstration Problem 8.7

31 © 2002 Thomson / South-Western Slide 8-31 Determining Sample Size when Estimating P with No Prior Information P n 0 50 100 150 200 250 300 350 400 00.10.20.30.40.50.60.70.80.91 Z = 1.96 E = 0.05 P 0.5 0.4 0.3 0.2 0.1 PQ 0.25 0.24 0.21 0.16 0.09

32 © 2002 Thomson / South-Western Slide 8-32 Solution for Demonstration Problem 8.8

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