Chapter 8 Estimation with Single Samples © 2002 Thomson / South-Western Slide 8-1 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. © 2002 Thomson / South-Western Slide 8-2 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. © 2002 Thomson / South-Western Slide 8-3 Confidence Interval to Estimate when n is Large • Point estimate • Interval Estimate X X X Z n or X Z © 2002 Thomson / South-Western n n X Z n Slide 8-4 Distribution of Sample Means for (1-)% Confidence 2 2 X Z Z 2 © 2002 Thomson / South-Western 0 Z 2 Slide 8-5 Z Scores for Confidence Intervals in Relation to 2 .5 .5 2 2 2 X Z Z 2 © 2002 Thomson / South-Western 0 Z 2 Slide 8-6 Distribution of Sample Means for (1-)% Confidence 2 1 1 2 2 2 X Z Z 2 © 2002 Thomson / South-Western 0 Z 2 Slide 8-7 Probability Interpretation of the Level of Confidence P r ob[ X Z 2 © 2002 Thomson / South-Western n X Z 2 ] 1 n Slide 8-8 Distribution of Sample Means for 95% Confidence .025 .025 95% .4750 .4750 X Z -1.96 © 2002 Thomson / South-Western 0 1.96 Slide 8-9 Example: 95% Confidence Interval for X 4 .26 , 1.1, and n 60 . X Z 4 .26 1.96 n 1.1 60 X Z n 4 .26 1.96 1.1 60 4 .26 0 .28 4 .26 0 .28 3.98 4 .54 © 2002 Thomson / South-Western Slide 8-10 95% Confidence Intervals for 95% X X X X X X X © 2002 Thomson / South-Western Slide 8-11 95% Confidence Intervals for Is our interval, 95% 3.98 4.54, in the red? X X X X X X X © 2002 Thomson / South-Western Slide 8-12 Demonstration Problem 8.1 X 10 .455 , 7 .7 , and n 44 . 90% confidence Z 1.645 X Z n 7 .7 1 0 .4 5 5 1.6 4 5 44 X Z n 1 0 .4 5 5 1.6 4 5 7 .7 44 1 0 .4 5 5 1.9 1 1 0 .4 5 5 1.9 1 8 .5 4 5 1 2 .3 6 5 P r ob[ 8.545 12 .365 ] 0.90 © 2002 Thomson / South-Western Slide 8-13 Demonstration Problem 8.2 X 34 .3 , 8 , N = 800 and n 50. 98% confidence Z 2 .33 X Z N n N 1 n 34.3 2.33 8 50 800 50 800 1 X Z n 34.3 2.33 N n N 1 8 50 800 50 800 1 34.3 2.55 34.3 2.55 31.75 36.85 © 2002 Thomson / South-Western Slide 8-14 Confidence Interval to Estimate when n is Large and is Unknown X Z 2 S n or X Z 2 © 2002 Thomson / South-Western S n X Z 2 S n Slide 8-15 Z Values for Some of the More Common Levels of Confidence Confidence Level © 2002 Thomson / South-Western Z Value 90% 1.645 95% 1.96 98% 2.33 99% 2.575 Slide 8-16 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 © 2002 Thomson / South-Western Slide 8-17 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 X t S n © 2002 Thomson / South-Western Slide 8-18 Comparison of Selected t Distributions to the Standard Normal Standard Normal t (d.f. = 25) t (d.f. = 5) t (d.f. = 1) -3 -2 © 2002 Thomson / South-Western -1 0 1 2 3 Slide 8-19 Table of Critical Values of t t0.100 t0.050 t0.025 t0.010 t0.005 df 1 2 3 4 5 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.706 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.747 3.365 63.656 9.925 5.841 4.604 4.032 1.714 25 1.319 1.318 1.316 1.708 2.069 2.064 2.060 2.500 2.492 2.485 2.807 2.797 2.787 29 30 1.311 1.310 1.699 1.697 2.045 2.042 2.462 2.457 2.756 2.750 40 60 120 1.303 1.296 1.289 1.282 1.684 1.671 1.658 1.645 2.021 2.000 1.980 1.960 2.423 2.390 2.358 2.327 2.704 2.660 2.617 2.576 23 24 1.711 © 2002 Thomson / South-Western t Slide 8-20 Confidence Intervals for of a Normal Population: Small n and Unknown X t S n or X t S n X t S n df n 1 © 2002 Thomson / South-Western Slide 8-21 Solution for Demonstration Problem 8.3 X 2 .1 4 , S 1.2 9 , n 1 4 , d f n 1 1 3 2 t 1 .9 9 . 0 0 5 ,1 3 2 0 .0 0 5 3.0 1 2 X t 2 .1 4 3.0 1 2 S n 1.2 9 14 X t S n 2 .1 4 3.0 1 2 1.2 9 14 2 .1 4 1.0 4 2 .1 4 1.0 4 1.1 0 3.1 8 © 2002 Thomson / South-Western Slide 8-22 Solution for Demonstration Problem 8.3 X t 2 .1 4 3.0 1 2 S n 1.2 9 14 X t S n 2 .1 4 3.0 1 2 1.2 9 14 2 .1 4 1.0 4 2 .1 4 1.0 4 1.1 0 3.1 8 P r ob [1.10 3.18 ] 0 .99 © 2002 Thomson / South-Western Slide 8-23 Confidence Interval to Estimate the Population Proportion p Z 2 p q n P p Z 2 p q n w h ere: p = sam p le p ro p o rtio n q = 1 - p P = p o p u latio n p ro p o rtio n n = sam p le siz e © 2002 Thomson / South-Western Slide 8-24 Solution for Demonstration Problem 8.5 n 2 1 2 , X 3 4 , p X n q = 1 - p 1 0 .1 6 0 .8 4 34 212 0 .1 6 9 0 % C o n fid en ce Z 1.6 4 5 p Z 0 .1 6 1.6 4 5 p q n ( 0 .1 6 )( 0 .8 4 ) 212 P p Z p q n P 0 .1 6 1.6 4 5 ( 0 .1 6 )( 0 .8 4 ) 212 0 .1 6 0 .0 4 P 0 .1 6 0 .0 4 0 .1 2 P 0 .2 0 P r ob[ 0.12 P 0.20 ] 0.90 © 2002 Thomson / South-Western Slide 8-25 Determining Sample Size when Estimating • Z formula Z X n • Error of Estimation E (tolerable error) • Estimated Sample Size X n Z 2 • Estimated © 2002 Thomson / South-Western 1 4 Z E 2 2 2 E 2 2 range Slide 8-26 Example: Sample Size when Estimating E 1, 4 90% confidence Z 1.645 Z 2 n 2 2 2 E (1.6 4 5) ( 4 ) 1 2 2 2 4 3.3 0 o r 4 4 © 2002 Thomson / South-Western Slide 8-27 Solution for Demonstration Problem 8.6 E 2 , ra n g e 2 5 9 5 % co n fid en ce Z 1.9 6 estim a ted : 1 ra n g e 2 5 6 .2 5 4 4 1 Z E (1.9 6 ) ( 6.2 5) 2 2 2 n 2 2 2 2 3 7 .5 2 o r 3 8 © 2002 Thomson / South-Western Slide 8-28 Determining Sample Size when Estimating P •Z formula Z pP P Q n • Error of Estimation (tolerable error) • Estimated Sample Size © 2002 Thomson / South-Western E pP n Z 2 PQ E 2 Slide 8-29 Solution for Demonstration Problem 8.7 E 0 .03 98% C onfidence Z 2 .33 estim ated P 0 .40 Q 1 P 0 .60 n Z 2 PQ 2 E ( 2 .33) 2 0 .40 0 .60 .003 2 1, 447 .7 or 1,448 © 2002 Thomson / South-Western Slide 8-30 Determining Sample Size when Estimating P with No Prior Information P PQ 0.5 0.25 Z = 1.96 E = 0.05 400 350 300 0.4 0.24 0.3 0.21 250 n 200 150 0.2 0.16 0.1 0.09 100 50 0 0 n Z 2 E 0.1 0.2 0.3 0.4 0.5 P 0.6 0.7 0.8 0.9 1 1 4 2 © 2002 Thomson / South-Western Slide 8-31 Solution for Demonstration Problem 8.8 E 0 .05 90% C onfidence Z 1.645 w ith no prior estim ate of P , use P 0 .50 Q 1 P 0 .50 n Z 2 PQ 2 E (1.6 4 5) 2 0 .5 0 0 .5 0 .0 5 2 2 7 0 .6 o r 2 7 1 © 2002 Thomson / South-Western Slide 8-32