1Overview of Class1. Basics of estimation2. Statistical inference3. Estimation of population parameters4. Point estimate vs. interval estimate5. Confidence interval for the population meanI. Basics of Estimation• 1. Populations and Samples• A population is the complete enumeration of all elements in a closed system. • A sample is a subset of a population. • e.g., the population of UW-Madison students (P: 42,030 according to web page; S:420 - Sampling ratio: 1:100). Key Sampling ConceptsCopyright ©2002, William M.K. Trochim, All Rights Reserved2Types of SamplesProbabilityNon-ProbabilityConveniencePurposiveSimple RandomSystematic RandomStratified RandomRandom ClusterComplex Multi-stage Random (various kinds)QuotaStratified Cluster2. Parameters and Estimates• A parameter is a characteristic of the population, such as a measure of central tendency(e.g., mean) or a measure of dispersion(e.g., variance). • An estimate is the estimated value of the true parameter from a sample. • Express measures symbolically– Denote any true population parameter by θ. – Denote a sample estimate of θ by T. T is also called a sample statistic. Examples:• Population Parameter (θ)• Population mean• Population variance• Correlation in population • Cross-tabulation in population • Sample Statistic (T)• Sample mean• Sample variance• Correlation in sample• Cross-tabulation in sampleStatistical Inference3II. Statistical Inference• 1. Introduction to Statistical InferenceStatistical inference is the methodology by which we assess the reliability of a sample estimate. For any population parameter, there could be many sample estimates. But how good are they? Are some estimates better than others? This is the subject matter of statistical inference. 2. Simple Random Sampling with Replacement• The simplest sampling method is called simple random sampling with replacement. – Random--we select sample elements by chance. – Simple--each individual in the population has the same probability of being selected. – With replacement--we put the selected element back into the population. Small issue if the sample size is small relative to the population size. • Intuitive examples – UW system students and family expectations, high school students and drug useFun Example: how do scientists estimate the number of gorillas in a closed region?• The capture-recapture method: Assuming that capture is random. • You first capture n1gorillas, mark them, and release them. • You then capture n2gorillas and see how many of them had been captured before. • Intuition: the higher the proportion in both captures, the smaller the population size.4NumericallyCaptured in Time 2 Captured in Time 1 Yes No Total Yes 10 90 100 No 90 N22 N2+ Total 100 N+2 N++ 100×100/10 = 1000SymbolicallyCaptured in Time 2Capturedin Tim e 1Yes No TotalYes N11N12N1+No N21N22N2+Total N+1N+2N++Independence means p11= p1+p+1n11/n++= (n1+/n++)(n+1/n++)n++= n1+n+1/n113. Sampling Distribution of an Estimator• We don’t observe a population. We only observe asample.• Imagine that one repeats sampling from the same population. The samples would be different from each other. All estimates for the same estimator from many, many, repetitions constitute a distribution, called the sampling distribution. • The distribution is hypothetical.5Mean GPA of UW students: sampling distribution 0510152025301.7 1.9 2.2 2.5 2.8 3.1 3.4 3.7 3.9Mean GPA in SampleNumber of samplesAnother possibility05101520253035401.7 1.9 2.2 2.5 2.8 3.1 3.4 3.7 3.9Mean GPA in SampleNumber of samplesIII. Estimation of Population Parameters• The sample mean (Ā) is a good estimator of the population mean: E(A) = μ.• Expectationof Ā: E(Ā) = μ• Varianceof Ā:V(Ā) = σ2/n, where σ2= V(A)• For a binary variable:– Ā = proportion. –V(Ā) = pq/n, q = 1-p, p is the proportion in the population.6IV. Point Estimate versus Interval Estimate• When a population characteristic is estimated by a single number, the number is called a point estimate. (Unlikely to be true.)• An interval estimate of the population characteristic θ consists of two bounds within which θ is estimated to lie:–L ≤ θ ≤ U,– where L is the lower bound, and U is the upper bound. I. Confidence Interval for the Population Mean• A confidence interval is the interval within which you believe that the true parameter lies with a fixed probability. • The probability with which the parameter lies in the interval can be assessed from the sampling distribution of the estimator. • We can place a confidence interval around the point estimate of the sample mean:Ā ± 1.96 σĀ==> Ā ± 1.96 (σ/n)1/2(Why 1.96?) – this is where you get the ± 3 percentage points statements you always hear w/re opinion pollsII. Determining Sample Size • The larger a sample size, the narrower the confidence interval, and thus the higher the precision. •V(Ā) ∝ 1/n. • If we know the precision desired, we can determine sample size.7A numerical example• e.g., for an opinion survey (yes or no), we would like to keep the "margin of error" to be 2%, or 0.02. Determine n. • Most conservative: p = q = 0.52 × (pq/n)1/2= 0.02(pq/n)1/2= 0.01.25/n = 0.0001n = 0.25/0.0001 =
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