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Chapter 6Introduction to Inferential Statistics Sampling and the Sampling Distribution

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Outline The logic and terminology of inferential statistics Random sampling The sampling distribution

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Logic And Terminology Problem: The populations we wish to study are almost always so large that we are unable to gather information from every case.

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Logic And Terminology (cont.) Solution: We choose a sample -- a carefully chosen subset of the population – and use information gathered from the cases in the sample to generalize to the population.

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Basic Logic And Terminology Statistics are mathematical characteristics of samples. Parameters are mathematical characteristics of populations. Statistics are used to estimate parameters. PARAMETER STATISTIC

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Samples: Must be representative of the population. Representative: The sample has the same characteristics as the population. How can we ensure samples are representative? Samples drawn according to the rule of EPSEM (every case in the population has the same chance of being selected for the sample) are likely to be representative.

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Random Sampling Techniques Simple Random Sampling (SRS) Systematic Random Sampling Stratified Random Sampling Cluster Sampling

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Suppose we select a random sample of 500 from a university student body and find that 74% of our sample has worked during the semester…. Population = All 20,000 students. Sample = The 500 students selected and interviewed. Statistic =74% (% of sample that held a job during the semester). Parameter = % of all students in the population who held a job.

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The Sampling Distribution We can use the sampling distribution to calculate our population parameter based on our sample statistic. The single most important concept in inferential statistics. Definition: The distribution of a statistic for all possible samples of a given size (N). The sampling distribution is a theoretical concept.

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The Sampling Distribution Every application of inferential statistics involves 3 different distributions. Information from the sample is linked to the population via the sampling distribution. Population Sampling Distribution Sample

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The Sampling Distribution: Properties 1. Normal in shape. 2. Has a mean equal to the population mean. μ x =μ 3. Has a standard deviation (standard error) equal to the population standard deviation divided by the square root of N. σ x = σ/√N

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First Theorem Tells us the shape of the sampling distribution and defines its mean and standard deviation. If we begin with a trait that is normally distributed across a population (IQ, height) and take an infinite number of equally sized random samples from that population, the sampling distribution of sample means will be normal.

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Central Limit Theorem For any trait or variable, even those that are not normally distributed in the population, as sample size grows larger, the sampling distribution of sample means will become normal in shape. The importance of the Central Limit Theorem is that it removes the constraint of normality in the population.

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The Sampling Distribution The Sampling Distribution is normal so we can use Appendix A to find areas. We do not know the value of the population mean (μ) but the mean of the S.D. is the same value as μ. We do not know the value of the pop. Stnd. Dev. (σ) but the Stnd. Dev. of the S.D. is equal to σ divided by the square root of N.

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Three Distributions Shape Central Tendency Dispersion SampleVaries _X_Xs Sampling Distribution Normalμx=μμx=μσ x = σ/√N PopulationVaries μ σ

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