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Populations & Samples Objectives: Students should know the difference between a population and a sample Students should be able to demonstrate populations and samples using a GATE frame Students should know the difference between a parameter and a statistic Students should know the main purpose for estimation and hypothesis testing Students should know how to calculate standard error

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GATE Frame: Populations & Samples Population Sample/ Participants A population is any entire collection of people, animals, plants or objects which demonstrate a phenomenon of interest. A sample is a subset of the population; the group of participants from which data is collected. Eligible In most situations, studying an entire population is not possible, so data is collected from a sample and used to estimate the phenomenon in the population.

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Parameters & Statistics A population value is called a parameter. A value calculated from a sample is called a statistic. Note: A sample statistic is a point estimate of a population parameter.

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Estimating Population Parameters Confidence intervals (CI) are ranges defined by lower and upper endpoints constructed around the point estimate based on a preset level of confidence. Hypothesis Testing is used to determine probabilities of obtaining results from a sample or samples if the result is not true in the population.

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Sample Estimates of Population Parameters Sample Statistic (point estimate) Combine with measure of variability of the point estimate Population Parameter Construct a range of values with an associated probability of containing the true population value L = lower value U = upper value

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What is Standard Error? Suppose a population of 1000 people has a mean heart rate of 75 bpm (but we don’t know this). We want to estimate the HR from a sample of 100 people drawn from the population: Population N=1000 n=100 We draw our sample, and the mean HR is 72 bpm

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Standard Error If we draw another sample, the mean will probably be a little different from 72, and if we draw lots of samples we will probably get lots of estimates of the population mean: Population N=1000 n=100

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Standard Error Population N=1000 The mean of the means of all possible samples of size 100 would exactly equal the population mean: All possible samples of size n=100 The standard deviation of the means of all possible samples is the standard error of the mean

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Sample Representativeness The sample means will follow a normal distribution, and: 95% of the sample means will be between the population mean and ±1.96 standard errors.

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In addition, if we constructed 95% confidence intervals around each individual sample mean: 95% of the intervals will contain the true population mean. Population Mean Sample Means

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Why is This Important and Useful? We rarely have the opportunity to draw repeated samples from a population, and usually only have one sample to make an inference about the population parameter: The standard error can be estimated from a single sample, by dividing the sample standard deviation by the square root of the sample size: Note: You will need to calculate the standard error in this course.

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Standard Error and Confidence Intervals The sample SE can then be used to construct an interval around the sample statistic with a specified level of confidence of containing the true population value: The interval is called a confidence interval The Most Commonly Used Confidence Intervals: 90% = sample statistic SE 95% = sample statistic SE 99% = sample statistic SE

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