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

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?

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.

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

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.

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

Organize work using a table Sample 1 Sample 2 Difference 1 x1 y1 d1=x1 - y1 2 x2 y2 d2=x2 - y2 3 x3 y3 d3=x3 - y3 … ..l. …. ….. n xn yn dn=xn -yn

Organize work using a table Sample 1 Sample 2 Difference 1 x1 y1 d1=x1 - y1 2 x2 y2 d2=x2 - y2 3 x3 y3 d3=x3 - y3 … ..l. …. ….. n xn yn dn=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.

Notation for Two Dependent Samples

The (1-a)*100% confidence interval for the mean difference md is Confidence Interval for the Mean Difference (Dependent Samples: Paired Data ) The (1-a)*100% confidence interval for the mean difference md is

Test Statistic for the Mean Difference (Dependent Samples) For n<30 the appropriate test statistic for testing the mean difference between paired samples is  with n-1 degrees of freedom. For n>30 then we use ‘z’ 

We now turn to the more challenging case of independent samples

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<30). In the small sample case, we must assume that both populations are normal and have equal variances.

Example Suppose we wish to compare two brands of 9-volt batteries, Brand 1 and Brand 2. Specifically, we would like to compare the mean life for the population of batteries of Brand 1, 1, and the mean life for the population of batteries of Brand 2, 2. To obtain a meaningful comparison we shall estimate the difference of the two population means by picking samples from the two populations.

For Brand 1 a sample of size 64 was chosen. From the data a point estimate for 1, would be 7.13. From the data a point estimate for 2 would be 7.78. It would therefore be natural for us to take as a point estimate for (1-2) to be -0.65 hours.

Point Estimator (Independent Samples) The estimate is the best point estimator of (1-2). Having found a point estimate, our next goal is to determine a confidence interval for it.

Point Estimator (Independent Samples) To construct a confidence interval for (1-2) we need to know the distribution of its point estimator. The distribution of is normal with mean (1-2) and standard deviation where n1 is the size of sample 1, n2 is the size of sample 2.

Confidence Interval for Difference in two Means Confidence Interval for Difference in two Means (Large samples or known variance)

Example: Life span of Batteries Let a = .05 so we are looking for the 95% confidence interval for the mean difference. What conclusion can you draw from the above?

Example: Life span of Batteries Let a = .05 so we are looking for the 95% confidence interval for the mean difference. We are 95 percent certain that the difference is negative. Thus, we are 95% certain that

Test Statistic for Two Means: Independent and large samples

Example: Life span of Batteries Hypothesis Testing. I claim that the two brands of batteries do not have the same life span. Using a 5% level of significance, test this claim.

Example: Life span of Batteries Hypothesis Sample Data Test Statistic

Example: Life span of Batteries Critical Region Decision The test statistic lies in the critical region, therefore we reject H0. The samples provide sufficient evidence to claim that the Batteries do indeed have different life spans.

Exercise Show that we would have rejected the null hypothesis even if we had used level of significance .008 (instead of .05. Thus… We conclude that the mean battery lives ARE different (p = .008)

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.

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 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