Inferences based on TWO samples New concept: Independent versus dependent samples Comparing two...

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Inferences based on TWO samples •New concept: Independent versus dependent samples •Comparing two population means: Independent sampling •Comparing two population means: Dependent sampling

Transcript of Inferences based on TWO samples New concept: Independent versus dependent samples Comparing two...

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Inferences based on TWO samples New concept: Independent versus dependent samples Comparing two population means: Independent sampling Comparing two population means: Dependent sampling Slide 2 2 Inferences About Two Means In the previous chapter we used one sample to make inferences about a single population. Very often we are interested in comparing two populations. 1) Is the average midterm grade in Stat 201.11 higher than the average midterm grade in Stat 201.12? 2) Is the average grade in Quiz #1 higher than Quiz #2 in this section of Introductory Statistics? Slide 3 Inferences about Two Means Each sample is an example of testing a claim between two populations. However, there is a fundamental difference between 1) and 2). In # 2) the samples are not independent where as in # 1), they are. Why?1. Different people in each class. 2. Same people writing different test. Slide 4 Recognizing independent versus dependent samples 1. Is the average midterm grade in Stat 201.11 higher than the average midterm grade in Stat 201.12? Independent samples 2. Is the average grade in Quiz #1 higher than Quiz #2 in this section of Introductory Statistics? Dependent samples Slide 5 Definition. Independent and Dependent Samples Two samples are independent if the sample selected from one population is not related to the sample selected from the other population. If one sample is related to the other, the samples are dependent. With dependent samples we get two values for each person, sometimes called paired-samples. Slide 6 We consider first the case of two dependent (or paired) samples Calculations are very similar to those in the previous chapter for a CI or Test of Hypothesis involving one sample Slide 7 Organize work using a table Sample 1Sample 2Difference 1x1y1d1=x1 - y1 2x2y2d2=x2 - y2 3x3y3d3=x3 - y3 ..l.... nxnyndn=xn -yn Slide 8 Organize work using a table Sample 1Sample 2Difference 1x1y1d1=x1 - y1 2x2y2d2=x2 - y2 3x3y3d3=x3 - y3 ..l.... nxnyndn=xn -yn Can now use the methods of the previous chapter to find a confidence interval for the population mean of the difference d between x1 and x2. Slide 9 Notation for Two Dependent Samples Slide 10 Confidence Interval for the Mean Difference (Dependent Samples: Paired Data ) The (1- )*100% confidence interval for the mean difference d is Slide 11 Test Statistic for the Mean Difference (Dependent Samples) For n30 then we use z Slide 12 We now turn to the more challenging case of independent samples Slide 13 Testing Claims about the Mean Difference (Independent Samples) When making claims about the mean difference between independent samples a different procedure is used than that for dependent/paired samples. Again there are different procedures for large (n>30) samples and small samples (n Overview Comparing Two Populations: Mean (Small Dependent (paired) Samples) Asumptions: Samples are random plus eith n>=30 or the population of differences is approximately normal Mean (Large Independent Samples) Assumptions: Both samples are randomly chosen plus both sample sizes >= 30. Slide 27 NOTE In the case of SMALL independent samples, one must use the t- distribution plus additional conditions must be satisfied AND one must use what is called a pooled estimate of the variance. Slide 28 NOTE In the case of SMALL independent samples, one must use the t- distribution plus additional conditions must be satisfied AND one must use what is called a pooled estimate of the variance. You are not responsible for this material