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Unformatted text preview: Drawing Statistical Inferences Inferences Inference Inference General Definition Inference Reasoning Reasoning from something known to something unknown something conclusions about population parameters from sample statistics parameters Statistical Statistical Drawing Drawing Point Estimation Estimating Estimating the value of a parameter as a single point from the value of a statistic single Point Estimation Point
X = ∑X N a single value of X as an estimate of µ Provides Provides Unbiased Consistent Very Very unlikely to be correct – it’s just an approximation. Affect by sampling method, sample size, and variability Characteristics of Estimators Characteristics Unbiased The The Estimator average of estimates from an infinite number of random samples would equal the population parameter population Consistent Is Is Estimator one, that as the sample size increases, the probability that the estimate has a value close to the population parameter also increases to Unbiased Estimators Unbiased The Mean X = ∑X N Variance s2 = ∑(XX)2 N1 N1 The Consistent Estimators Consistent Mean Mean based on a sample of 25 from a population of 100 will be closer to µ , than than will a sample of 10. will Logical Proof: A mean based on a sample Logical of 100 people from a population of 100, has to equal µ !! Sampling Methods Sampling Selecting Selecting members of a sample from a population. population. How the sample is selected determines How the generalizability of the findings. the The larger the sample, the more likely the The results will be accurate results Simple Random Sample Simple Each Each member of the population has an equal likelihood of being selected. equal And, the selection of any one member is And, independent of the selection of the other members members Stratified Random Sampling Stratified
A stratum is a layer or level of some stratum characteristic of a population. characteristic Define ahead of the sampling, how many Define people will be selected from each stratum. people Each member of a population has an Each equal likelihood of being selected, but the selection of some members changes the likelihood that others will be selected. likelihood Stratified Random Sample Example Stratified Attitudes Attitudes towards Statistics  Want sample to represent the composition of the class: to Sophomores (25%) Juniors (35%) Seniors (40%) If sample size is limited to 20: 5 sophomores 7 juniors 8 seniors Sampling Distributions Sampling The basis of inferential statistics Distributions of statistics, such as the Distributions mean and variance, as opposed to the distributions of raw scores we have examined thus far examined Empirical Sampling Distribution of 9 Empirical scores each, from an infinite population with µ = 100 and σ = 12, compute, and plot the 100 sample means plot Empirical Sampling Distribution Empirical Infinite population µ = 100 σ = 12 Sampling Distribution Μ & X = 100 σ x = σ / √N = 12/ √N √9=12/3 = 4.0 √9=12/3 76 88 100 112 124 76 88 100 112 124 Standard Error of the Mean Standard The The standard deviation of the distribution of sample means: of σx = σ √N Will be smaller than the standard deviation Will of the population because averaging sample scores reduces the effect of extreme scores and decreases variability extreme Characteristics of Sampling Distributions Distributions Normally Normally Distributed (If infinite population and influence number of samples). influence Mean of the sampling distribution is equal to μ of Mean the population the Standard error (standard deviation) of the Standard sampling distribution is equal to: sampling σx = σ √N The Central Limit Theorem The If If we take an infinite number of samples (of size N) from any population containing an infinite number of terms, and an We plot the mean of these samples… The resulting sampling distribution will be The normally distributed… normally No matter what the shape of the original No distribution! distribution! Logical Example of the Central Limits Theorem Limits Imagine Imagine 200 200 Population of 600 scores 600
scores equal to value of 1 value 200 scores equal to 200 value of 2 value 200 scores equal to 200 value of 3 value
200 100 0 1 2 3 Rectangular Rectangular Distribution Distribution Logical Example of the Central Limits Theorem (continued) Limits Imagine Imagine that we draw a large number of samples of 6 scores each from the rectangular population and plot the means rectangular The The values of the means will be a continuous variable variable The most frequent value of the sample means The will be 2.00 will Sample means with values of 1.0 and 3.0 will Sample be rare be Logical Example of the Central Limits Theorem (continued) Limits
200 100 0 1 2 3 μ=2 σ = .67
Original Distribution 1 2 3 μX = 2 σX = .67/√6 = .33
Sampling Distribution Example of the Central Limits Theorem Theorem
sub 1 2 3 4 5 6 X
1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 3 1 1 1 1 2 2 1 1 1 2 2 2 1 1 1 1 2 3 1 1 2 2 2 2 1 1 1 1 3 3 1 1 1 2 2 3 1.00 1.16 1.33 1.33 1.50 1.50 1.66 1.66 1.66 Sub 1 2 3 4 5 6 X
1 2 2 2 2 2 1 1 3 2 2 2 1 1 1 3 3 2 2 2 2 2 2 2 1 1 2 2 3 3 1 1 1 3 3 3 1 2 2 2 2 3 2 2 2 2 2 3 1 2 2 2 3 3 1.83 1.83 1.83 2.00 2.00 2.00 2.00 2.16 2.16 Sub 1 2 3 4 5 6 X
1 1 2 3 3 3 2 2 2 2 3 3 1 2 2 3 3 3 1 1 3 3 3 3 2 2 2 3 3 3 1 2 2 3 3 3 2 2 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 2.16 2.33 2.33 2.33 2.50 2.50 2.66 2.83 3.00 Area/Probability of Populations and sampling distributions sampling Probability Probability of X ≥ 108 in population with μ = 100, σ = 12? z = (X μ)/ σ )/ = (108100)/12 (108100)/12 = 8/12 = .67 8/12 prob = .2514 What What is the probability of finding a sample mean sample (X) ≥ 108, based on a sample of 9 drawn from a population with μ = 100, σ = 12? Find the standard error of the mean! mean! σx = σ/√N = 12/√9 = 4.0 z = (X μ)/ σx )/ = (108100)/4.0 = 8/4 = 2.0 8/4 look up value in col c look prob = .0228 prob Interval Estimation Interval Provides Provides Uses an estimated range for μ X as a base Sx and sampling distribution Probability
a concepts Confidence intervals Confidence
range of score values expected to contain the range value of μ with a certain level of confidence 68% 95% 99% confidence interval
≤ μ ≤ X+1 sx sx ≤ μ ≤ X+1.96 sx ≤ μ ≤ X+2.58 sx X1sx confidence interval confidence interval X1.96 X2.58sx .4750 .4750 .4951 .34 .34 .4951 1 μ .68 .95 1 1.96 2.58 2.58 1.96 Total area under curve = 1.0 .50 to left of μ, .50 to right of μ .50 .50 .99 ...
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This note was uploaded on 01/16/2011 for the course PSYC 274 at USC.
 '07
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