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UT Knoxville STAT 201 - Chapter 23

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Chapter 23 Chapter23 Presentation 1213 Inferences About Means Copyright 2009 Pearson Education Inc 1 Sampling Distribution of y Review If we want to estimate a population mean or test a hypothesis about a population mean we start with our understanding of the sampling distribution of y The Central Limit Theorem told us that if the sample size is large enough the sampling distribution model for y is Normal with mean and standard deviation SD y Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc n 2 A Small Problem So would a 95 confidence interval for the population mean look like this y 1 96 n Unfortunately we don t know How did we deal with a similar issue with proportions Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 3 A Small Problem cont So will this produce a 95 confidence interval for the population mean s y 1 96 n Unfortunately no When we substitute s for the sampling distribution of y is no longer normally distributed so we can t use 1 96 Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 4 A Small Problem cont In the late 1800 s and early 1900 s the race was on to understand what the sampling distribution of y was when the population standard deviation was not known Until this problem was solved one had to simply assume they knew the population standard deviation an unrealistic assumption in many cases Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 5 The Solution William Gosset William S Gosset an employee of the Guinness Brewery in Dublin Ireland worked long and hard to find out what the sampling distribution model was The sampling model that Gosset found has become known as Student s t The Student s t models form a whole family of related distributions that depend on a parameter known as degrees of freedom We often denote degrees of freedom as df and the model as tdf Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Student s t vs the Normal Distribution Student s t models are unimodal symmetric and bell shaped just like the Normal But t models with only a few degrees of freedom have much fatter tails than the Normal Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 7 The Impact on Confidence Intervals for Means Because the Student s t distribution has heavier tails than the normal distribution confidence intervals using Gosset s t model will be just a bit wider than if the Normal model applied A slightly wider interval is the price that is paid for not only estimating the mean of the population but having to estimate the standard deviation of the population as well Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 8 A Confidence Interval for Means cont One sample t interval for the mean When the conditions are met we are ready to find the confidence interval for the population mean The confidence interval is n 1 where the standard error of the mean is y t SE y s SE y n The critical value tn 1 depends on the particular confidence level C that you specify and on the number of degrees of freedom n 1 which we get from the sample size Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 9 Step by Step Example I want to find an estimate for the mean speed of vehicles driving on Triphammer Road I will take an interval a 90 confidence interval as my estimate I have data on the speeds of 23 cars sampled on April 11 2000 I will base my estimate on these data Before we apply the technique on the previous page there are some assumptions we must make To see if these assumptions are reasonable we must check three conditions Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 10 Assumptions and Conditions Independence Assumption The data values should be independent Check Condition 1 Randomization Condition Randomly sampled data are ideal Check Condition 2 10 Condition The sample should be no more than 10 of the population We check these same 2 conditions when doing confidence intervals for a population proportion Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 11 Assumptions and Conditions cont Normal Population Assumption We can never be certain that the data are from a population that follows a Normal model but we can check the following condition Check Condition 3 Nearly Normal Condition The data come from a distribution that is unimodal and symmetric Chapter23 Presentation 1213 Check this condition by making a histogram Normal probability plot or Goodness of Fit Test for the raw data in the sample Copyright 2009 Pearson Education Inc 12 Assumptions and Conditions cont Nearly Normal Condition cont The smaller the sample size n 15 or so the more closely the data should follow a Normal model For moderate sample sizes n between 15 and 40 or so the t works well as long as the data are unimodal and reasonably symmetric For sample sizes larger than 40 or 50 the t methods are safe to use unless the data are extremely skewed Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 13 Step by Step Example Cont Is it appropriate to use my sample of 23 car speeds sampled on April 11 2000 Check Condition 1 Randomization condition Technically this was not a random sample but we did select cars spread out in time on 4 11 So I still think it was representative of traffic on Triphammer Rd Check Condition 2 Less than 10 condition Is 23 cars less than 10 of all cars that travel this road Definitely it s a busy road Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 14 Step by Step Example Cont Check Condition 3 Nearly normal condition Are the data nearly normal What do you think Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 15 Step by Step Example Cont n 23 Sample mean Std Deviation y 31 0 mph s 4 25 mph SE y 4 25 23 0 886 mph For 90 confidence and df t 1 717 Margin of error ME 1 717 0 886 1 5 mph 90 confidence interval 31 0 1 5 mph or 29 5 32 5 Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 16 Step by Step Example Cont The following is the correct interpretation for our confidence interval We are 90 confident that the true mean speed of all vehicles on Triphammer Road that day was between 29 5 mph and 32 5 mph Caveat If the drivers could see the radar gun some may have slowed down This could have biased our data collection Chapter23 Presentation 1213 Copyright 2009 Pearson Education Inc 17 Using JMP to Calculate Confidence Intervals for a Population Mean Following instructions in our


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