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1 Introduction to Inference Confidence Intervals William P. Wattles, Ph.D. Psychology 302

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2 Statistical Inference F Provides methods for drawing conclusions about a population from sample data. Sample (statistic) Population (parameter)

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3 The problem F Sampling Error

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4 Sampling error results from chance factors that produce a sample statistic different from the population parameter it represents.

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6 Inferential statistics F How well does the sample statistic predict the unknown population parameter? Population Sample

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7 Dealing with sampling error F Confidence intervals F Hypothesis testing

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8 Frequency Distribution F Tells what values a variable can take and how often each value occurs

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9 Sampling Distribution F Tells what values a statistic can take and how often each value occurs. F All possible samplings of a given size F Less variable than a raw score frequency distribution

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10 Confidence interval F Point versus interval estimation F confidence interval= estimate±margin of error

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11 Margin of error example F Imagine catering a function where you expect 120 students.

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12 Margin of error example F Imagine catering a function where you expect 120 students plus or minus 30 F What are the upper and lower limits?

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13 Margin of error example F Imagine catering a function where you expect 120 students plus or minus 30 F What are the upper and lower limits? F Minimum (lower limit) 90 F Maximum (upper limit) 150

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14 Obtaining confidence intervals F estimate + or - margin of error

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15 Upper and Lower limits F Bob estimates that Mary weighs 120 pounds “give or take” ten. Calculate the upper and lower limits of his estimate.

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16 Upper and Lower limits F Bob estimates that Mary weighs 120 pounds “give or take” ten. Calculate the upper and lower limits of his estimate. F Upper 130 F Lower 110

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17 Upper and Lower limits F Tom is giving a party and tells the caterer that he expects 80 friends plus or minus 20. Determine the upper and lower limits

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18 Upper and Lower limits F Tom is giving a party and tells the caterer that he expects 80 friends plus or minus 20. Determine the upper and lower limits F Upper 100 F Lower 60

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19 Upper and Lower limits F If something costs $250 plus or minus $25, what is the lower limit, the least you would expect to pay? What is the upper limit or the most you would expect to pay.

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20 Upper and Lower limits F If something costs $250 plus or minus $25, what is the lower limit, the least you would expect to pay? F Upper $275 F Lower $225

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21 The purpose of a confidence interval is to estimate an unknown parameter and an indication of: 1.of how accurate the estimate is 2.how confident we are that the result is correct.

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23 Estimating with confidence Although the sample mean is a unique number for any particular sample, if you pick a different sample, you will probably get a different sample mean. In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, . x

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24 But the sample distribution is narrower than the population distribution, by a factor of √n. Sample means, n subjects Population, x individual subjects

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25 Confidence intervals tell us two things F 1. the interval F 2. the level of confidence – C = the confidence interval – p=probability

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26 Obtaining confidence intervals F Confidence interval for a population mean

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27 Steps to upper limit 1. The Upper limit equals the Mean + Margin of error 2. Margin of error = Z times the standard error (sigma /sqrt of n) 3. Standard Error = std dev/ square root of n

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28 Determining critical Z F What is the Z for an 80% confidence interval? F We need a number that cuts off the upper 10% and the lower 10% F Table A look for.90 and.10 F Z= to cut off lower 10% F to cut off upper 10%

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30 Determining Critical values of Z F 90% F 95% F 99% F Critical Values: values that mark off a specified area under the standard normal curve.

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31 Homework

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32 Confidence intervals F Example 14.1 Page 360 F Want 95% confidence interval F σ =7.5 F Mean= 26.8 F n=654

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33 Confidence intervals F Estimate +-Margin of error F Estimate 26.8 F Margin of error.60 F Upper limit –27.4 F Lower Limit –26.2

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34 F Obtaining a confidence interval for a sample mean value gives you some idea of how far off you may expect the true population mean to be.

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35 Confidence intervals are extremely important in statistics, because whenever you report a sample mean, you need to be able to gauge how precisely it estimates the population mean.

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36 Characteristics of confidence intervals F The margin of error gets smaller when: – Z gets smaller. More confidence=larger interval. (i.e., Only 90% confident versus 95%) – sigma gets smaller. Less population variation equals less noise and more accurate prediction – n gets larger.

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37 Example from cliff notes F : Suppose that you want to find out the average weight of all players on the football team. You are select ten players at random and weigh them. F The mean weight of the sample of players is 198, so that number is your point estimate. F The population standard deviation is σ = What is a 90 percent confidence interval for the population weight, if you presume the players' weights are normally distributed?

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38 90% confidence interval F Area to the right 5% F Area between that point and the mean 45% F Z value

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39 90% Confidence Interval F Another way to express the confidence interval is as the point estimate plus or minus a margin of error; in this case, it is 198 ± 6 pounds. F

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40 Confidence Intervals F Student Study Times

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41 Confidence Intervals F Students (269) asked how many hours do you study on a typical weeknight? –sample mean 137 minutes –study times standard deviation is 65 minutes –Create a 99% confidence interval

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42 Problem 14.30

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43 Sampling Distribution Homework

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44 F Problem page 390 F Wine odors

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45 DMS odor threshold F Mean 30.4 F Std dev 7 F 95% conf interval

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46 Problem 14.27

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47 Caution page 344 F The conditions: –Perfect SRS –Population is normal –We know the population standard deviation (σ) F These conditions are unrealistic.

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48 Parametric statistics F Assume raw scores form a normal distribution F Assume the data are interval or ratio scores (measurement data) F Assume raw scores are randomly drawn F Robust refers to accuracy of procedure if one of the assumptions is violated,

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49 Random error versus bias F The margin of error in a confidence interval covers only random sampling errors.

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50 The End

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