Populations and SamplesBios 662Michael G. Hudgens, [email protected]://www.bios.unc.edu/∼mhudgens2007-08-21 15:35BIOS 662 1 Populations and SamplesRandom Variables• Random sample: result of independently selecting ele-ments at random from a population• Def 4.8 A random variable is a variable associated witha random samplePV Rao (p 786, 1998) A rv is a variable whose value isdetermined by the observed characteristics of an itemrandomly selected from a popluationBIOS 662 2 Populations and SamplesProbability Functions• Def 4.9 The probability mass function is a functionthat for each possible value of a discrete rv takes on theprobability of that value occurring• Def 4.10 The probability density function is a curvethat specifices, by means of the area under the curveover an inte rval, the probability that a contiunous rvfalls within the intervalBIOS 662 3 Populations and SamplesProbability Functions0 2 4 6 80.00 0.05 0.10 0.15 0.20 0.25 0.30Number of Boys in Families with Eight ChildrenProbability mass function−3 −2 −1 0 1 2 30.0 0.1 0.2 0.3 0.4xProbability density functionBIOS 662 4 Populations and SamplesCumulative Distribution Function• Def 4.9 The cumulative distribution function for a rvX isF (x) = Pr[X ≤ x]• If X is discrete,F (x) =Xy≤xpX(y)where pXis the pmf of X• If X is continuous,F (x) =Zx−∞f(y)dywhere f is the pdf of XBIOS 662 5 Populations and SamplesMean and Variance• Mean or expected value of X– If X is discrete,µ = E(X) =XyypX(y)– If X is continuous,µ = E(X) =Z∞−∞yf(y)dy• Varianceσ2= V ar(X) = E{(X − µ)2}BIOS 662 6 Populations and SamplesSkewness and Kurtosis• Skewnessα3=E{(X − µ)3}σ3• Kurtosisα4=E{(X − µ)4}σ4BIOS 662 7 Populations and SamplesParameters and Statistics• Definition: A parameter is a numerical characteristic ofa population• Definition: A statistic is a numerical characteristic of asample• Notation: Greek letters typically denote parameters;English letters denote statistics• Example:µ = population mean; σ2population variance¯Y sample mean; s2sample varianceBIOS 662 8 Populations and SamplesParameters and Statistics• Parameters are fixed constants• Statistics are random variables• Statistics have probability distributions• We will use statistics and probability theory to drawconclusions (inference) about parametersBIOS 662 9 Populations and SamplesSampling Distributions• Defintion 4.15 The probability function of a statistic iscalled the sampling distribution of the statistic.• Eg, when sampling from a population, the sample mean¯Y is a rv becuase its value depends on chance, namely,on which sample is obtained.The probability distribution of the random variable¯Yis called the sampling distribution of the mean.BIOS 662 10 Populations and SamplesSampling Distributions• Result 4.1 If a rv Y has a population mean µ and apopulation variance σ2, the sampling distribution of themean (¯Y ) has mean µ and variance σ2/n• Definition 4.16 The standard deviation of the samplingdistribution is called the standard error• Eg the standard error of¯Y is σ/√nBIOS 662 11 Populations and SamplesNormal or Gaussian Distribution• PDF:f(x; µ, σ) =1σ√2πexp(−12x − µσ2)• CDF:F (x; µ, σ) =Zx−∞f(y; µ, σ)dy• µ mean, σ2variance• X ∼ N(µ, σ2) [beware X ∼ N(µ, σ)]BIOS 662 12 Populations and SamplesNormal Distribution−50 0 50 100 1500.00 0.01 0.02 0.03 0.04 0.05xdensityN(25, 900)N(25, 225)N(100, 81)N(100, 225)BIOS 662 13 Populations and SamplesStandard Normal Distribution• Z ∼ N(0, 1)• PDF:φ(z) =1√2πexp{−12z2}• CDF:Φ(z) =Zz−∞φ(y)dy• N(0, 1) is a standard normal distributionBIOS 662 14 Populations and SamplesStandard Normal Distribution−3 −2 −1 0 1 2 30.0 0.1 0.2 0.3 0.4ZdensityBIOS 662 15 Populations and SamplesProperties of Standard Normal Distribution• A rv w/ pdf f is symmetric about µ iff(µ + x) = f(µ − x) for all x• Z ∼ N(0, 1) is symmetric about 0φ(z) = φ(−z) for all − ∞ < z < ∞• ThusPr[Z ≤ −z] = Pr[Z ≥ z]i.e.Φ(−z) = 1 − Φ(z)BIOS 662 16 Populations and SamplesStandard Normal Distribution−3 −2 −1 0 1 2 30.0 0.1 0.2 0.3 0.4ZdensityPr[Z < −.1] = 0.4602 Pr[.1 < Z] = 0.4602BIOS 662 17 Populations and SamplesStandard Normal Distribution• R> pnorm(-.1,0,1)[1] 0.4601722> 1-pnorm(.1,0,1)[1] 0.4601722• SASdata;x = probnorm(-.1);BIOS 662 18 Populations and SamplesProperties of a Random Variable• Let X be a random variable• Suppose Y = aX + b where a and b are constants• ThenE(Y ) = aE(X) + bV ar(Y ) = a2V ar(X)• If X ∼ N(µ, σ2) and Y = aX + b, thenY ∼ N(aµ + b, (aσ)2)BIOS 662 19 Populations and SamplesConversion to Standard Normal• Suppose Y ∼ N(µ, σ2)• LetZ =Y − µσ• ThenZ ∼ N(0, 1)• In words: a normal random variable can be standardizedby subtracting its mean and dividing by its standarddeviationBIOS 662 20 Populations and SamplesComputation of Probabilities• Suppose Y ∼ N(µ, σ2)• LetZ =Y − µσ• ThenPr[a < Y < b] = Pr [a−µσ< Z <b−µσ]= Φ(b−µσ) − Φ(a−µσ)BIOS 662 21 Populations and SamplesTable 1 (p 818 text): Standard normal distribution.Let Z be a normal random variable with mean zero and variance one. For selected values ofz, three values are tabled: (1) the two-sided p-value, or Pr[|Z| ≥ z]; (2) the one-sided p-value,or Pr[Z ≥ z]; and (3) the cumulative distribution function at z, or Pr[Z ≤ z].Two One Cumu.z sided sided dist.0.00 1.0000 .5000 .50000.05 .9601 .4801 .51990.10 .9203 .4602 .53980.15 .8808 .4404 .55960.20 .8415 .4207 .57930.25 .8026 .4013 .59870.30 .7642 .3821 .61790.35 .7263 .3632 .63680.40 .6892 .3446 .65540.45 .6527 .3264 .67360.50 .6171 .3085 .6915...1.00 .3173 .1587 .84131.33 .1835 .0918 .90821.64 .1010 .0505 .94951.96 .0500 .0250 .97502.00 .0455 .0288 .97722.58 .0099 .0049 .9951BIOS 662 22 Populations and SamplesExample• Intraocular pressure (IP) is used to diagnose glaucoma• Assume IP is normally distributed with mean µ = 16mmHg and variance σ2= 9 mmHg• If pressure greater than 20 mmHg is considered abnor-mal, what proportion of the population is abnormal?Pr[X > 20] = Pr[X−163>20−163]= Pr[Z > 1.33] = 1 − Φ(1.33)= 1 − 0.9082 = 0.0918BIOS 662 23 Populations and SamplesExample (continued)• What proportion of the population has IP between 4and 18?Pr[4 < X < 18] = Pr[4−163<X−163<18−163]= Pr[−4 < Z < 2/3]= Φ(2/3) − Φ(−4)= Φ(2/3) − 1 +