2-Sample Independent Test of Hypotheses Part 1

2-Sample Independent Test of Hypotheses ______________________________________ 1) Distinguish between the statistical question underlying CIs, 1-sample hypothesis tests and 2-sample hypothesis tests. 2) Describe the procedures for conducting independent 2sample hypothesis tests when n is large.  calculation of standard error  calculation/interpretation of CI 3) Describe the procedures for conducting independent 2sample hypothesis tests when n is small.  calculation of pooled estimate of variance  calculation of standard error  calculation/interpretation of CI 4) Compare the results of tests conducted using the small and large-sample methods. Important Questions ______________________________________ Confidence Interval: What is the range within which we can be (1 - )% sure that that  falls? 1-Sample Tests of Hypotheses: What is the likelihood that the sample we have collected was drawn from a population with  = ___? 2-Sample Tests of Hypotheses: What is the likelihood that two samples we have collected were drawn from populations with the same value for ? What does STATISTICAL SIGNIFICANCE mean? ______________________________________ The procedures we use for conducting hypothesis tests evaluate how much error we can expect by chance, which is a function of the variability in the data. 1. Assume all values of x =  (expected value). 2. Error = deviations from the expected value. 3. We reject the null if the observed error is SIGNIFICANTLY larger than what we would expect by chance. ______________________________________ 1-sample test:  If the difference between M and 0 is significantly larger than we would expect by chance, we conclude that there is a significant difference between  and 0. 2-sample test:  If the difference between M1 and M2 is significantly larger than we would expect by chance, we conclude that there is a significant difference between 1 and 2. ______________________________________ Significant does not necessarily imply "meaningful" Procedure for 2-sample Hypothesis Testing ______________________________________ Step 1: Decided whether to conduct a one or a two-tailed test. We’ll start with two-tailed tests. Steps 2 and 3: Set up your null and alternative hypotheses Ho: 1 - 2 = D0 or 1 = 2 Ha: 1 - 2  D0 or 1  2 Step 4: Choose alpha Step 5: Set up a RR by determining zcrit or tcrit Step 6: Calculate zobs or tobs Step 7: Make a decision regarding the null.  Does the observed value of your test statistic fall in the rejection region? Step 8: Interpret what your decision regarding the null means in terms of your original research question. Two Types of 2-sample Tests ______________________________________ Paired test – two conditions are comprised of the same elements; same observation is measured under different testing conditions. EX: Independent test – two conditions are comprised of different elements/observations; o often called unpaired EX: My wildest dreams come true ______________________________________ I won the Mass State Lottery (true statisticians never enter lotteries or sweepstakes, and probably shouldn’t go to Las Vegas though some still do). I have decided to invest the proceeds in Charlie's Tavern. I want to know whether people drink more on the weekends or weekdays. So I collect data from 36 patrons on the weekend and 36 patrons during the week. Below is the information from the two samples. Is there enough evidence to conclude that drinking on the weekend is different than drinking during the week? α = .05. Weekend Weekday Mwe = 3.5 drinks / night Mwd = 2.1 drinks / night σwe = 1.4 σwd = 0.8 Step 1: Let’s conduct a two-tailed test. Steps 2 and 3: Set up your null and alternative hypotheses Ho: we - wd = 0 or we = wd Ha: we - wd  0 or we  wd Step 4: α = .05. Step 5: zcrit = ±1.96 Step 6: Calculate zobs or tobs Sampling Distribution for 2-sample test (unpaired) ______________________________________ We can construct a sampling distribution for the difference between two sample means just like we can construct the sampling distribution for the mean drawn from a single population (e.g., CIs and one-sample hypothesis tests). 1. Calculate the mean for every possible sample of size n from population #1. 2. Calculate the mean for every possible sample of size n from population #2. 3. Calculate the difference between every possible sample from Pop1 and Pop2. ______________________________________ Q: What is the mean of the sampling distribution going to be? µM1-M2 Q: What is the standard deviation of the sampling distribution going to be? σM1-M2 = 12  22  n1 n 2 95% CI for Difference between means ___________________________________________ (M1-M2)  z/2 (σM1-M2) ___________________________________________ σM1-M2 = 12  22 =  n1 n 2 = 1.96 .64  36 36 1.4 .8  36 36 2 2 = .0544  .0178 = .0722 = .27 ___________________________________________ 95% CI = = 1.4 1.4   1.96 (.27) .53 Notice anything interesting about this CI? [.87 - 1.93] Back to hypothesis testing ______________________________________ M1  M 2  D0 zobs = Step 6:  M M 1 (vs. z = ( xM - 0) / M) zobs = 3.5  2.1  0 .27 = 1.4 – 0 / .27 = 1.4 / .27 = 5.19 zcrit =  1.96 Step 7: Decision regarding the null? Step 8: Interpretation? 2 2-Sample Hypotheses Tests when n is large, σ is known ______________________________________ Step 1: One- vs. Two-tailed test Step 2: Specify the NULL hypothesis (HO)  1 - 2 = D0 Step 3: Specify the ALTERNATIVE hypothesis (Ha)  1 - 2  D0 Step 4: Designate the rejection region by selecting . Step 5: Determine the critical value of your test statistic Step 6: Use sample statistics to calculate observed value of your test statistic. 2 2 x x D   1 2 zobs =  x  x  =   n1 n 2 Step 7: Compare observed value with critical value: o If test statistic falls in RR, we reject the null. o Otherwise, we fail to reject the null. 1 2 0 1 2 x1  x 2 Step 8: Interpret your decision regarding the null in terms of your original research question. Two-Sample Tests: Anthropology Example ______________________________________ An anthropologist wants to collect data to determine whether the two different cultural groups that occupy an isolated Pacific Island grow to be different heights. The results of his samples of the heights of adult females are as follows. Group A Group B n = 120 n = 150 M = 62.7 M = 61.8 σ = 2.5 σ = 2.62 Do these samples constitute enough evidence to reject the null hypothesis that the heights of the two groups are the same? Set alpha to .05. Step 1: Run a two-tailed test. Step 2: Ho: a = b Step 3: Ha: a  b Steps 4 and 5: α = .05; zcrit = ±1.96 Step 6: zobs = 62.7 – 61.8 – 0 2.5 2.62  120 150 2 2 Anthropology example: more calculations ______________________________________________ Step 6: .9 .052  .046 zobs = = .9 / .31 = 2.88 Step 7: Decision regarding the null? Step 8: Interpretation? ______________________________________________ Observed p-value zobs = 2.88 p-value = 2(Tailzobs) = 2(.002) = .004