Business Statistics A DecisionMaking Approach, 6th …：業(yè)務(wù)統(tǒng)計(jì)決策方法第六…

1、business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-1 statistical inference confidence intervals and hypothesis testing business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-2 module goals basic questions: n what is a statistic? n what is t

2、he distribution of a statistic? n whats the difference between the distribution of a statistic and the distribution of x, the underlying r.v.? n why do we care ? business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-3 module goals busy version after completing this modul

3、e, you should be able to: nexplain what a statistic is, why we use it(them), and why it has a distribution ndistinguish between a point estimate and a confidence interval estimate, and construct and interpret a confidence interval estimate for a single population mean using both the z and t distribu

4、tions nform and interpret a confidence interval estimate for a single population proportion nperform hypothesis testing business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-4 confidence intervals content of this chapter nconfidence intervals for the population mean, nwh

5、en population standard deviation is known nwhen population standard deviation is unknown ndetermining the required sample size nconfidence intervals for the population proportion, p business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-5 statistics nx is r.v., nt is stat

6、istic, nt is an estimator of a parameter, e.g., m na statistic t has a distribution, e.g. nif know then you can make probability statements about the parameter youre rtying to estimate. ( ),( )( ; , ,etc.) xxx xfxfxfxm 1 1 ()e.g., n i i tt xtxx n m x xf ( ) t ft business statistics: a decision-makin

7、g approach, 6e 2005 prentice-hall, inc.chap 7-6 point and interval estimates na point estimate is a single number, na confidence interval provides additional information about variability point estimate lower confidence limit upper confidence limit width of confidence interval business statistics: a

8、 decision-making approach, 6e 2005 prentice-hall, inc.chap 7-7 point estimates we can estimate a population parameter with a sample statistic (a point estimate) mean proportion p (or p) x or pp business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-8 confidence intervals

9、ngeneral idea: nhow much uncertainty is associated with a point estimate of a population parameter? nan interval estimate provides more information about a population characteristic than does a point estimate nsuch interval estimates are called confidence intervals me business statistics: a decision

10、-making approach, 6e 2005 prentice-hall, inc.chap 7-9 confidence interval estimate nan interval gives a range of values: ntakes into consideration variation in sample statistics from sample to sample nbased on observation from 1 sample ngives information about closeness to unknown population paramet

11、ers nstated in terms of level of confidence nnever 100% sure business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-10 estimation process (mean, , is unknown) population random sample mean x = 50 sample i am 95% confident that is between 40 100% do not. sampling distribut

12、ion of the mean n zx /2 n zx /2 x x1 x2 /2/2 1 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-19 margin of error nmargin of error (e): the amount added and subtracted to the point estimate to form the confidence interval n zx /2 n ze /2 example: margin of error fo

13、r estimating , known: business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-20 factors affecting margin of error ndata variation, : e as nsample size, n : e as n nlevel of confidence, 1 - : e if 1 - n ze /2 business statistics: a decision-making approach, 6e 2005 prentic

14、e-hall, inc.chap 7-21 example na sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. we know from past testing that the population standard deviation is .35 ohms. ndetermine a 95% confidence interval for the true mean resistance of the population. business statis

15、tics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-22 2.4068 .1.9932 .2068 2.20 )11(.35/ 1.96 2.20 n z x /2 example na sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. we know from past testing that the population standard deviation is .35 ohm

16、s. nsolution: (continued) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-23 interpretation nwe are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms nalthough the true mean may or may not be in this interval, 95% of intervals formed in

17、this manner will contain the true mean nan incorrect interpretation is that there is 95% probability that this interval contains the true population mean. (this interval either does or does not contain the true mean, there is no probability for a single interval) business statistics: a decision-maki

18、ng approach, 6e 2005 prentice-hall, inc.chap 7-24 confidence intervals population mean unknown confidence intervals population proportion known business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-25 nif the population standard deviation is unknown, we can substitute th

19、e sample standard deviation, s nthis introduces extra uncertainty, since s is variable from sample to sample nso we use the t distribution instead of the normal distribution confidence interval for ( unknown) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-26 nassu

20、mptions npopulation standard deviation is unknown npopulation is normally distributed nif population is not normal, use large sample nuse students t distribution nconfidence interval estimate confidence interval for ( unknown) n s tx /2 (continued) business statistics: a decision-making approach, 6e

21、 2005 prentice-hall, inc.chap 7-27 students t distribution nthe t is a family of distributions nthe t value depends on degrees of freedom (d.f.) nnumber of observations that are free to vary after sample mean has been calculated d.f. = n - 1 business statistics: a decision-making approach, 6e 2005 p

22、rentice-hall, inc.chap 7-28 if the mean of these three values is 8.0, then x3 must be 9 (i.e., x3 is not free to vary) degrees of freedom (df) idea: number of observations that are free to vary after sample mean has been calculated example: suppose the mean of 3 numbers is 8.0 let x1 = 7 let x2 = 8

23、what is x3? here, n = 3, so degrees of freedom = n -1 = 3 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-29 students t distribution t 0 t (df = 5) t (df = 13) t-distributions a

24、re bell- shaped and symmetric, but have fatter tails than the normal standard normal (t with df = ) note: t z as n increases business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-30 students t table upper tail area df.25.10 .05 11.000 3.078 6.314 2 0.817 1.886 2.920 30.7

25、65 1.638 2.353 t0 2.920 the body of the table contains t values, not probabilities let: n = 3 df = n - 1 = 2 = .10 /2 =.05 /2 = .05 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-31 t distribution values with comparison to the z value confidence t t t z level (10

26、d.f.) (20 d.f.) (30 d.f.) _ .80 1.372 1.325 1.310 1.28 .90 1.812 1.725 1.697 1.64 .95 2.228 2.086 2.042 1.96 .99 3.169 2.845 2.750 2.58 note: t z as n increases business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-32 example a random sample of n = 25 has x = 50 and s =

27、8. form a 95% confidence interval for nd.f. = n 1 = 24, so the confidence interval is 2.0639tt .025,241n,/2 25 8 (2.0639)50 n s tx /2 46.698 . 53.302 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-33 approximation for large samples nsince t approaches z as the sam

28、ple size increases, an approximation is sometimes used when n 30: n s tx /2 n s zx /2 correct formula approximation for large n business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-34 determining sample size nthe required sample size can be found to reach a desired marg

29、in of error (e) and level of confidence (1 - ) nrequired sample size, known: 2 /2 2 /2 e z e z n 2 2 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-35 required sample size example if = 45, what sample size is needed to be 90% confident of being correct within 5? (

30、always round up) 219.19 5 1.645(45) e z n 22 /2 so the required sample size is n = 220 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-36 if is unknown nif unknown, can be estimated when using the required sample size formula nuse a value for that is expected to be

31、 at least as large as the true nselect a pilot sample and estimate with the sample standard deviation, s business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-37 confidence intervals population mean unknown confidence intervals population proportion known business statis

32、tics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-38 confidence intervals for the population proportion, p nan interval estimate for the population proportion ( p ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ) business statistics: a decisi

33、on-making approach, 6e 2005 prentice-hall, inc.chap 7-39 confidence intervals for the population proportion, p nrecall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation nwe will estimate this with sample data: (continued) n )p

34、(1p sp n p)p(1 p business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-40 confidence interval endpoints nupper and lower confidence limits for the population proportion are calculated with the formula nwhere nz is the standard normal value for the level of confidence des

35、ired np is the sample proportion nn is the sample size n )p(p zp /2 1 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-41 example na random sample of 100 people shows that 25 are left-handed. nform a 95% confidence interval for the true proportion of left-handers bu

36、siness statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-42 example na random sample of 100 people shows that 25 are left-handed. form a 95% confidence interval for the true proportion of left-handers. 1. 2. 3. .0433 .25(.75)/n)/np(1ps .2525/100 p p 0.3349 . . . . . 0.1651 (.

37、0433) 1.96 .25 (continued) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-43 interpretation nwe are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. nalthough this range may or may not contain the true proporti

38、on, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion. business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-44 changing the sample size nincreases in the sample size reduce the width of the confidence interval. example: nif

39、 the sample size in the above example is doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at .25, but the width shrinks to .19 .31 business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-45 finding the required sample size for pr

40、oportion problems n )p(p ze /2 1 solve for n: define the margin of error: 2 /2 e )p(pz n 1 2 p can be estimated with a pilot sample, if necessary (or conservatively use p = .50) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-46 what sample size.? nhow large a samp

41、le would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (assume a pilot sample yields p = .12) business statistics: a decision-making approach, 6e 2005 prentice-hall, inc.chap 7-47 what sample size.? solution: for 95% confidence, use z = 1.96 e = .03 p = .12, so use this to estimate p so use n = 451 450.74 (.03) .12)(.12)(1(1.96) e )p(pz n 2 2 /2 2 2 1 (continued) business statistics: a decision-making approach