Chapter 7 Testing Differences between Means

2 Testing Differences Between Means  Establish hypothesis about populations, collect sample data, and see how likely the sample results are, given the hypothesis. Example: Memory enhancement  N = 10 Method AMethod B Mean = 82Mean = 77

3 Method AMethod B Mean = 82Mean = 77  Now suppose instead that the following sets of scores produced the two sample means of 82 and 77.

4 The Null Hypothesis  No difference between means An obtained difference between two sample means does not represent a true difference between their population means  Mean of the first population = mean of the second population Retain or reject the null hypothesis Null hypothesis shown as H 0 :

5 The Research Hypothesis  Differences between groups, whether expected on theoretical or empirical grounds, often provide the rationale for research Mean of the first population does not equal the mean of the second population  If we reject the null hypothesis, we automatically accept the research hypothesis that a true population difference does exist. Different means Research hypothesis shown as H 1 :

6 Levels of Significance  To establish whether our obtained sample difference is statistically significant – the result of a real population difference and not just sampling error – it is customary to set up a level of significance Denoted by the Greek letter alpha (α)  The alpha value is the level of probability at which the null hypothesis can be rejected with confidence and the research hypothesis can be accepted with confidence.

7 Type I and II Errors Correct DecisionType I Error P (Type I Error) = alpha Type II Error P (Type II Error) = beta Correct Decision DECISION Retain Null Reject Null Null is true REALITY Null is false

8 Choosing a Level of Significance Suppose for example that a researcher were doing research on gender differences in sentence length for first time drug offenses for a random sample of males and females. What would be worse? Type I error or Type II error? Suppose that a researcher is testing the effects of marijuana smoking on SAT performance, and he compares a sample of smokers with a sample of nonsmokers. What would be worse? Type I error or Type II error?

9 What is the Difference Between P and Alpha?  The difference between P and alpha can be a bit confusing A =.001 A =.01 A =.05 A =.10 P <.001 P <.01 P <.05 P <.10

Standard Error of the Difference between Means  Standard deviation of the distribution of differences can be estimated.  The standard error of the differences between means is shown as:

Testing the Difference between Means  Why use t instead of z?  Test differences between means using t:  This is referred to as our T computed

Comparing our T value  Using Table C, we find our T critical value.  To calculate the degrees of freedom (df) when testing the difference between means we use the following formula df = N1 + N2 – 2  Alpha value is given (.05 or.01)  If T computed > T critical, reject null  If T computed < T critical, accept null 12

13 Testing the Difference between Means  Suppose that we obtained the following data for a sample of 25 liberals and 35 conservatives on the permissiveness scale.  Calculate the estimate of the standard error of the differences between means.  Then, translate the difference between sample means into a t ratio. LiberalsConservatives N 1 = 25N 2 = 35 S 1 = 12S 2 = 14

14 Continued. If necessary, find the mean and standard deviation first. Otherwise:  Step 1: Find the standard error of the difference between means.  Step 2: Compute the t ratio.  Step 3: Determine the critical value for t.  Step 4: Compare the calculated and table t values.

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16 Comparing the Same Sample Measured Twice  So far, we have discussed making comparisons between two independently drawn samples  Before-after or panel design: the case of a single sample measured at two different points in time (time 1 vs. time 2)  For example, a polling organization might interview the same 1,000 Americans both in 1995 and 2000 in order to measure their change in attitude over time.  Numerous uses for this type of test

Testing the Difference Between Means for the Same Sample Measured Twice 17

Finding the t ratio 18 Critical T: df = N – 1 α =.05 or.01 Use Table C Compare the computed T with the critical T. If |T| > critical T, reject null hypothesis. If |T| < critical T, retain null hypothesis.

19 Test of Difference between Means for Same Sample Measured Twice  Suppose that several individuals have been forced by a city government to relocate their homes to make way for highway construction.  As researchers, we are interested in determining the impact of forced residential mobility on feelings of neighborliness.  What would the null and research hypotheses state?  We interview a random sample of 6 individuals about their neighbors both before and after they are forced to move.

Their Scores RespondentBeforeAfter Stephanie21 Leon12 Carol31 Jake31 Julie12 David41 20

21 Test of Difference between Means for Same Sample Measured Twice RespondentBeforeAfterDifferenceDifference 2 Stephanie2111 Leon121 Carol3124 Jake3124 Julie121 David4139 ΣX = 14ΣX = 8 ΣD 2 = 20 N = 6

Two Sample Test of Proportions 22

Two Sample Test of Proportions Formulas 23

Two Sample Test Example A criminal justice researcher is interested in marijuana usage and driving while high of upper level undergraduates in her particular school. After taking a random sample of 300 students, she discards any surveys of students who have not smoked marijuana. She is left with the following data: Test the research hypothesis at the alpha level of.05. What do your results indicate? 24 MaleFemale Sample Size Driven High5636

25 One-Tailed Tests 1. A one-tailed test rejects the null hypothesis at only one tail of the sampling distribution. 2. It should be emphasized, however, that the only changes are in the way the hypotheses are stated and the place where the t table is entered. 3. Used when the researcher anticipates the direction of change.

26 Requirments for Testing the Differences between Means 1. A comparison between two means 2. Interval data 3. Random sampling 4. A normal distribution 5. Equal population variances