 # Comparing Two Groups’ Means or Proportions Independent Samples t-tests.

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Comparing Two Groups’ Means or Proportions Independent Samples t-tests

Comparing Two Groups Sociologists look for relationships between concepts in the social world. For example: Does one’s sex affect income? Concepts: Sex and Income Does one’s race affect educational aspirations? Concepts: Race and Educational Aspiration

Comparing Two Groups In this section of the course, you will learn ways to infer from a sample whether two concepts are related in a population. The technique you use will depend of the level of measurement of your independent and dependent variables. Independent variable (X): That which causes another variable to change when it changes. Dependent variable (Y): That which changes in response to change in another variable. X Y

Comparing Two Groups The test you choose depends on level of measurement: IndependentDependentTest DichotomousContinuous Independent Samples t-test Dichotomous NominalNominalCross TabsDichotomous NominalContinuousANOVADichotomous ContinuousContinuousBivariate Regression/Correlation Dichotomous

Comparing Two Groups Independent Samples t-tests: Earlier, our focus was on the mean. We used the mean of the sample (statistic) to infer what our population mean (parameter) might be (confidence interval) or whether it was like some guess (significance test). Now, our focus is on the difference in the mean for two groups. We will use the difference of the sample (statistic) to infer what our population difference (parameter) might be (confidence interval) or whether it is like some guess (significance test).

Comparing Two Groups The difference will be calculated as such: D-bar = Y-bar 2 – Y-bar 1 For example: Average Difference in Income by Sex = Female Average Income – Male Average Income Like the mean, if one were to take random sample after random sample and calculate and record the difference each time, one would see the formation of a Sampling Distribution for D-bar that was normal and centered on the population’s Difference.

Comparing Two Groups Like the mean, if one were to take random sample after random sample and calculate and record the difference each time, one would see the formation of a Sampling Distribution for D-bar that was normal and centered on the population’s Difference. Z -3 -2 -1 0 1 2 3 95% Range Sampling Distribution of D-bar 6 8 10 12 14 16 18

Comparing Two Groups So the rules and techniques we learned for the mean apply to the difference in the mean. One creates sampling distributions and does significance tests in the same ways. However, the standard error of D-bar has to be calculated slightly differently. For Means: (s.d. 1 ) 2 (s.d. 2 ) 2 s.e. (s.d. of the sampling distribution) = n 1 + n 2 For Proportions: One formula for C.I. and one for Sig. Tests. You may use the C.I. formula for each. CI s.e. =  1 (1 -  1 )  2 (1 -  2 ) n 1 + n 2

Comparing Two Groups It’s Power Time Again! Using just a sample, our statistics will allow us to pinpoint the difference between two groups in the population (confidence interval) or to determine whether there is a difference between two groups in the population (significance test).

Comparing Two Groups Calculating a Confidence Interval for the Difference between Two Groups’ Means By slapping the sampling distribution for the difference over our sample’s difference between groups, D-bar, we can find the values between which the population difference is likely to be. 95% C.I. = D-bar +/- 1.96 * s.e. = Y-bar 2 – Y-bar 1 +/- 1.96 * s.e. Or =  2 –  1 +/- 1.96 (s.e.) 99% C.I. = D-bar +/- 2.58 * s.e. = Y-bar 2 – Y-bar 1 +/- 2.58 * s.e. Or =  2 –  1 +/- 2.58 (s.e.)

Comparing Two Groups EXAMPLE: We want to know what the likely difference is between male and female GPAs in a population of college students with 95% confidence. Sample: 50 men, average gpa = 2.9, s.d. = 0.5 40 women, average gpa = 3.1, s.d. = 0.4 s.e. =  (.5) 2 / 50 + (.4) 2 /40 = .005 +.004 = .009 = 0.095 95% C.I. = Y-bar 2 – Y-bar 1 +/- 1.96 * s.e. = 3.1 - 2.9 +/- 1.96 * 0.095 = 0.2 +/- 0.186 = 0.014 to 0.386 We are 95% confident that the difference between men’s and women’s GPAs in the population is between.014 and 0.386. Would the difference be a significant difference?

Comparing Two Groups Conducting a Test of Significance for the Difference between Two Groups’ Means By slapping the sampling distribution for the difference over a guess of the difference between groups, H o, we can find out whether our sample could have been drawn from a population where the difference is equal to our guess. 1.Two-tailed significance test for  -level =.05 2.Critical z = +/- 1.96 3.To find if there is a difference in the population, H o :  2 -  1 = 0 H a :  2 -  1  0 4.Collect Data 5.Calculate Z: z = Y-bar 2 – Y-bar 1 -  o s.e. 6.Make decision about the null hypothesis 7.Find P-value

Comparing Two Groups EXAMPLE: We want to know whether there is a difference in male and female GPAs in a population of college students. 1.Two-tailed significance test for  -level =.05 2.Critical z = +/- 1.96 3.To find if there is a difference in the population, H o :  2 -  1 = 0 H a :  2 -  1  0 4.Collect Data Sample: 50 men, average gpa = 2.9, s.d. = 0.5 40 women, average gpa = 3.1, s.d. = 0.4 s.e. =  (.5) 2 / 50 + (.4) 2 /40 = .005 +.004 = .009 = 0.095 5.Calculate Z: z = 3.1 – 2.9 – 0 = 0.2 = 2.11 0.095 0.095 6.Make decision about the null hypothesis: Reject the null. There is enough difference between groups in our sample to say that there is a difference in the population. 7.Find P-value: p or (sig.) =.0348 8.We have a 3.5 % chance that the difference in our sample could have come from a population where there is no difference between men and women. That chance is low enough to reject the null.

Comparing Two Groups The steps outlined above for Confidence intervals And Significance tests for differences in means are the same you would use for differences in proportions. The only thing you change is the way you calculate the standard error for the difference.