Part I. Sampling Distributions And Confidence Intervals

7-1Chapter 4Part I. Sampling Distributionsand Confidence Intervals1

7-2Section 1.Sampling Distribution2

7-3Using Statistics Statistical Inference: Predict and forecast values ofpopulation parameters. Test hypotheses about valuesof population parameters. Make decisions.On basis of sample statisticsderived from limited andincomplete sampleinformationMake generalizationsabout thecharacteristics of apopulation.On the basis ofobservations of asample, a part of apopulation3

7-4Sample Statistics as Estimators of Population Parameters A sample statistic is anumerical measure of asummary characteristicof a sample.A population parameteris a numerical measure ofa summary characteristicof a population. An estimator of a population parameter is a sample statistic used to estimate or predict the populationparameter.An estimate of a parameter is a particular numericalvalue of a sample statistic obtained through sampling.A point estimate is a single value used as an estimateof a population parameter.4

7-5Estimators The sample mean, X , is the most commonestimator of the population mean, The sample variance, s2, is the most commonestimator of the population variance, 2. The sample standard deviation, s, is the mostcommon estimator of the population standarddeviation, . The sample proportion, p̂, is the most commonestimator of the population proportion, p.5

7-6A Population Distribution, a Sample from aPopulation, and the Population and Sample MeansPopulation mean ( )Frequency distributionof the populationXXXXXXXXXXXXXXXXXXSample pointsSample mean X( )6

7-7Other Sampling Methods Stratified sampling: in stratified sampling, the population ispartitioned into two or more subpopulation called strata, andfrom each stratum a desired sample size is selected atrandom. Cluster sampling: in cluster sampling, a random sample ofthe strata is selected and then samples from these selectedstrata are obtained. Systemic sampling: in systemic sampling, we start at arandom point in the sampling frame, and from this pointselected every kth, say, value in the frame to formulate thesample.7

7-8Sampling Distributions The sampling distribution of a statistic is the probability distributionof all possible values the statistic may assume, when computed fromrandom samples of the same size, drawn from a specifiedpopulation. The sampling distribution of X is the probability distribution of allpossible values the random variable X may assume when a sampleof size n is taken from a specified population.8

7-9Properties of the Sampling Distribution of theSample MeanUniform Distribution (1,8)0.2P(X)Comparing the population distributionand the sampling distribution of themean: The sampling distribution is morebell-shaped and symmetric. Both have the same center. The sampling distribution of themean is more compact, with asmaller variance.0.10.012345678XSampling Distribution of the MeanE( X ) X XVar( X ) X20.10P(X) 0.05n0.001.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0X9

7-10Sampling from a Normal PopulationWhen sampling from a normal population with mean and standard deviation , thesample mean, X, has a normal sampling distribution:This means that, as the sample sizeincreases, the sampling distributionof the sample mean remainscentered on the population mean,but becomes more compactlydistributed around that populationmeann2)Sampling Distribution of the Sample Mean0.4Sampling Distribution: n 160.3f(X)X N ( , Sampling Distribution: n 40.2Sampling Distribution: n 20.1Normal population0.0 10

7-11The Central Limit Theoremn 50.250.20P(X)0.150.100.050.00Xn 20P(X)0.2For “large enough” n: X N ( , / n)20.10.0XLarge n0.4f(X)0.30.20.10.0- XWhen sampling from a population withmean and finite standard deviation , thesampling distribution of the sample meanwill tend to a normal distribution with mean and standard deviationas the samplensize becomes large (n 30).11

7-12Student’s t DistributionIf the population standard deviation, , is unknown, replace withthe sample standard deviation, s. If the population is normal, theresulting statistic:t X s/ nhas a t distribution with (n - 1) degrees of freedom. The t is a family of bell-shaped andsymmetric distributions, one for eachnumber of degree of freedom.The expected value of t is 0.The t distribution approaches astandard normal as the number ofdegrees of freedom increases.Standard normalt, df 20t, df 10

7-13The Sampling Distribution of the SampleProportion, p n 2, p 0.30 .4P(X)The sample proportion is the percentage ofsuccesses in n binomial trials. It is thenumber of successes, X, divided by thenumber of trials, n.0 .50 .30 .20 .10 .0012Xn 10,p 0.3Xn0.2P(X)Sample proportion: p 0.30.10.002345678910Xn 15, p 0.30.2P(X)As the sample size, n, increases, the sampling approaches a normaldistribution of pdistribution with mean p and standarddeviationp (1 p )n10.10.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 1 2 3 4 5 6 7 8 9 10 11 12 13 141515 15 15 15 15 15 15 15 15 1515 15 15 15 1515X p13

7-14Estimators and Their PropertiesAn estimator of a population parameter is a sample statistic used toestimate the parameter. The most commonly-used estimator of the:Population ParameterSample StatisticMean ( )is theMean ( X )Variance ( 2)is theVariance (s2)Standard Deviation ( )is theStandard Deviation (s)Proportion (p)is theProportion ( p ) Desirable properties of estimators cy14

7-15Types of Estimators Point Estimate A single-valued estimate. A single element chosen from a sampling distribution. Conveys little information about the actual value of thepopulation parameter, about the accuracy of the estimate. Confidence Interval or Interval Estimate An interval or range of values believed to include theunknown population parameter. Associated with the interval is a measure of theconfidence we have that the interval does indeed containthe parameter of interest.15

7-16Section 2. Confidence Intervals forPopulation Means (Z-CI and t-CI)16

7-17Confidence Interval for when is knownIf the population distribution is normal, the sampling distribution ofthe mean is normal. If the sample is sufficiently large ( 30), regardless of the shape of thepopulation distribution, the sampling distribution is normal (CentralLimit Theorem).In eithercase :Standard Normal Distribution 95%:Interval P 1 . 96 X 1 . 96 0 . 95nn 0.4f(z)0.30.2or0.1 P X 1 . 96 X 1 . 96 0 . 95nn 0.0-4-3-2-101234z17

7-18Confidence Interval for when is knownBefore sampling, there is a 0.95probability that the interval 1.96 nwill include the sample mean (and 5% that it will not).Conversely, after sampling, approximately 95% of such intervalsx 1.96 nwill include the population mean (and 5% of them will not).That is, x 1.96 is a 95% confidence interval for .n18

7-19A 95% Interval around the Population MeanSampling Distribution of the Mean0.495%f(x)0.3Approximately 95% of sample meanscan be expected to fall within theinterval 1.96 , 1.96 .0.20.12.5%0.0 1.96 2.5% 196.n xnSo 5% can be expected to fall outsidethe interval 1.96 , 1.96 . xx2.5% fall belowthe interval xxn nnn xx2.5% fall abovethe intervalxxx95% fall withinthe interval19

7-2095% Intervals around the Sample MeanSampling Distribution of the MeanApproximately 95% of the intervals0.4f(x)x 1 .9695%0.30.20.12.5%2.5%0.0 1.96 196.nx nxx naround the sample mean can be expectedto include the actual value of thepopulation mean, . (When the samplemean falls within the 95% interval aroundthe population mean.)x xx x*xxxxxx xx *20

7-21The 95% Confidence Interval for A 95% confidence interval for μ when σ is known and sampling isdone from a normal population, or a large sample is used:x 1 .96 n The quantity 1.96is often called the margin of error or thesampling error. nA 95% confidence interval:For example, if: n 25 20x 122x 1.96 2025n 122 (1.96)( 4 ) 122 7 .84 114 .16,129.84 122 1.9621

7-22A (1- )100% Confidence Interval for We define z as the z value that cuts off a right-tail area of under the standard22normal curve. (1- ) is called the confidence coefficient. is called the errorprobability, and (1- )100% is called the confidence level. P z z 2 P z z 2 P z z z (1 ) 2 2S tand ard Norm al Distrib ution0.4(1 )f(z)0.30.20.1 22(1- )100% Confidence Interval:0.0-5-4-3-2-1 z 20Z1z 22345x z 2 n22

7-23Critical Values of z and Levels of 00.100Stand ard N o rm al Distrib utio nz 0.4(1 )22.5762.3261.9601.6451.2820.3f(z)(1 ) 0.20.1 220.0-5-4-3-2-1 z 20Z1z 2342235

7-24Level of confidence and width of the confidence intervalWhen sampling from the same population, using a fixed sample size,the higher the confidence level, the wider the confidence interval.St an d ar d N or m al Di s tri b uti o n0.40.40.30.3f(z)f(z)St an d ar d N o r m al Di s tri b uti o 5Z80% Confidence Interval:x 1.280 n95% Confidence Interval:x 196. n24

7-25The Sample Size and the Width of the Confidence IntervalWhen sampling from the same population, using a fixed confidence level, thelarger the sample size, n, the narrower the confidence interval.S a m p lin g D is trib utio n o f th e M e anS a m p lin g D is trib utio n o f th e M e an0 .40 .90 .80 .70 .3f(x)f(x)0 .60 .20 .50 .40 .30 .10 .20 .10 .00 .0x95% Confidence Interval: n 20x95% Confidence Interval: n 40Note: The width of a confidence interval can be reduced only at the price of:a lower level of confidence, or a larger sample.25

7-26Example 1Population consists of the Fortune 500 Companies (Fortune WebSite), as ranked by Revenues. You are trying to to find out theaverage Revenues for the companies on the list.The population standard deviation is 15,056.37. A random sampleof 30 companies obtains a sample mean of 10,672.87. Give a 95%and 90% confidence interval for the average Revenues26

7-27Chi-square DistributionThe random sample X 1, X 2 , , X n , is from a normal distribution N( , 2 ),X is the sample mean and S 2 is the sample variance, then( n 1) S 2 2 2 n 1 where 2 n 1 is the chi - square distribution with degrees of freedom (n - 1).Property of Chi - square distribution W 2 r Gamma(r/ 2 , 1 / 2) :1. E(W) r, Var(W) 2 r2. If Z N(0, 1), then Z 2 2 1 .3. Additive Property : Independent Chi - square r.v. Wi 2 ri then W1 W2 Wm 2 r1 rm 27

7-28t distribution Assume Z N(0, 1), W 2 r , Z and W are independent, then The statisticX T S /n2 t n 1 Z t r W /rdegrees of freedom (n-1)Standard Normalt (df 13)Bell-ShapedSymmetric‘Fatter’Tailst (df 5)0Zt28

7-29Student’s t TableLet: n 3df n - 1 2 .10 /2 .05Upper Tail 651.6382.353 /2 .050t Values2.920tFind t values:1. α 0.10, n 202. α 0.01, n 83. α 0.025, n 1029

7-30Confidence intervals for when is unknown (t distribution)A (1- )100% confidence interval for when is not known(assuming a normally distributed population):x t n 1 2snwhere t n 1 is the value of the t distribution with n-1 degrees of2 freedom that cuts off a tail area of 2 to its right.Example 2:A stock market analyst wants to estimate the average return on a certainstock. A random sample of 15 days yields an average (annualized)return of x 10.37% and a standard deviation of s 3.5%. Assuming anormal population of returns, give a 95% confidence interval for theaverage return on this stock.30

7-31Section 3.Confidence Interval for Proportions31

7-32Large-Sample Confidence Intervalsfor the Population Proportion, pA large - sample (1- )100% confidence interval for the population proportion, p :ˆˆpˆ z α pqn2where the sample proportion, p̂, is equal to the number of successes in the sample, x,divided by the number of trials (the sample size), n, and q̂ 1- p̂.For estimating p, a sample is considered large enough when np 5 and n(1- p) 532