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Comparing Two Groups’ Means or Proportions Independent Samples t-tests Review Confidence Interval for a Mean Slap a sampling distribution* over a sample mean to determine a range in which the population mean has a particular probability of being—such as 95% CI. If our sample is one of the middle 95%, we know that the mean of the population is within the CI. 2.5% -1.96z Slap a sampling distribution* over a guess of the population mean to determine if the sample has a very low probability of having come from a population where the guess is true—such as α-level = .05. If our sample mean is in the outer 5%, we know to reject the guess, our sample has a low chance of having come from a population with the mean we guessed. Y-bar? µ? 2.5% Significance Test for a Mean +1.96z Y-bar 95% CI: Y-bar +/- 1.96 *(s.e.) Y-bar? 2.5% 2.5% -1.96z +1.96z µo=guess z or t = (Y-bar - µo)/ s.e. *sampling distribution: the way statistics for samples of a certain size would stack up or be distributed after all possible samples are collected Review Let’s collect some data on educational aspirations and produce a 95% confidence interval to tell us where the population parameter likely falls and then let’s do a test of significance where we guess that average aspiration will be 16 years. I collected a sample of 625 kids who reported their educational aspirations where 12 = high school, 16 equals 4 years of college and so forth. The average for the sample was 15 years with a standard deviation of 2 years. 95% confidence interval 95% CI = Sample Mean +/- z * s.e. 1. Find the standard error of the sampling distribution: s / n = 2/√625 = 2/25 = 0.08 2. Build the width of the Interval. 95% corresponds with a z of +/- 1.96. +/- z * s.e = 1.96 * 0.08 = 0.157 3. Insert the mean to build the interval: 95% CI = Sample Mean +/- z * s.e = 15 +/- 0.157 The interval: 14.84 to 15.16 We are 95% confident that the population mean falls between these values. (What does this say about my guess???) Review Let’s collect some data on educational aspirations and produce a 95% confidence interval to tell us where the population parameter likely falls and then let’s do a test of significance where we guess that average aspiration will be 16 years. I collected a sample of 625 kids who reported their educational aspirations where 12 = high school, 16 equals 4 years of college and so forth. The average for the sample was 15 years with a standard deviation of 2 years. Significance Test z or t = (Y-bar - µo)/ s.e. 1. 2. 3. 4. 5. 6. 7. Decide -level ( = .05) and nature of test (two-tailed) Set critical z or t: (+/- 1.96) Make guess or null hypothesis, Ho: = 16 Ha: 16 Collect and analyze data Calculate Z or t: z/t = Y-bar - o (s.e. = s/√n = 2/√625 = 2/25 = .08) s.e. z/t = (15 – 16)/.08 = -1/.08 = -12.5 Make a decision about the null hypothesis (reject the null: -12.5 < -1.96) Find the P-value (look up 12.5 in z or t table). P < .0001 It is extremely unlikely that our sample came from a population where the mean is 16. Comparing Two Groups We’re going to move forward to more sophisticated statistics, building on what we have learned about confidence intervals and significance tests. Sociologists look for relationships between concepts in the social world. For example: Does one’s sex affect income? Focus on the relationship between the concepts: Sex and Income Does one’s race affect educational attainment? Focus on the relationship between the concepts: Race and Educational Attainment 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. 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. XY (X= Sex or Race) (Y= Income or Education) The statistical technique you use will depend of the level of measurement of your independent and dependent variables—the statistical test must match the variables! Levels of Measurement: Nominal, Ordinal, Interval-Ratio Comparing Two Groups The test you choose depends on level of measurement: Independent Dependent Statistical Test Dichotomous Interval-ratio Dichotomous Independent Samples t-test Nominal Ordinal Dichotomous Nominal Ordinal Dichotomous Cross Tabs Nominal Ordinal Dichotomous Interval-ratio Dichotomous ANOVA Interval-ratio Dichotomous Interval-ratio Correlation and OLS Regression Comparing Two Groups Independent Dependent Statistical Test Dichotomous Interval-ratio Dichotomous Independent Samples t-test An independent samples t-test is concerned with whether a mean or proportion is equal between two groups. For example, does sex affect income? ♀ Income µ Women’s mean ♂ Income = µ Men’s Mean ??? Comparing Two Groups Independent Samples t-tests: Earlier, our focus was on the mean. We used the mean of the sample (statistic) to infer a range for what our population mean (parameter) might be (confidence interval) or whether it was like some guess or not (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 a range for 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-bar2 – Y-bar1 For example: Average Difference in Income by Sex = Male Average Income – Female Average Income (What would it mean if men’s income minus women’s income equaled zero?) Comparing Two Groups Like the mean, if one were to take random sample after random sample from two groups and calculate and record the difference between groups each time, one would see the formation of a Sampling Distribution for D-bar that was normal and centered on the two populations’ difference. = Sampling Distribution of D-bar Z -3 -2 -1 0 1 2 3 95% Range average difference between two groups’ samples 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 our sample could have come from a population with a difference between two groups that we guessed (significance test). Comparing Two Groups So the rules and techniques we learned for means and proportions apply to the differences in groups’ means and proportions. One creates sampling distributions to create confidence intervals and do significance tests in the same ways. However, the standard error of D-bar has to be calculated slightly differently. For Means: s.e. (s.d. of the sampling distribution) = (s1)2 n1 + (s2)2 n2 For Proportions: s.e. = 1 (1 - 1) n1 2 (1 - 2) + n2 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-bar2 – Y-bar1) +/- 1.96 * (s.e.) Or = (2 – 1) +/- 1.96 * (s.e.) 99% C.I. = D-bar +/- 2.58 * (s.e.) = (Y-bar2 – Y-bar1) +/- 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 50 women, average gpa = 3.1, s.d. = 0.4 95% C.I. = Y-bar2 – Y-bar1 +/- 1.96 * s.e. 1. Find the standard error of the sampling distribution: s.e. = (.5)2/ 50 + (.4)2/50 = .005 + .003 = .008 = 0.089 2. Build the width of the Interval. 95% corresponds with a z or t of +/- 1.96. +/- z * s.e = +/- 1.96 * 0.089 = +/- 0.174 3. Insert the mean difference to build the interval: 95% C.I. = (Y-bar2 – Y-bar1) +/- 1.96 * s.e. = 3.1 - 2.9 +/- 0.174 = 0.2 +/- 0.174 The interval: 0.026 to 0.374 We are 95% confident that the difference between men’s and women’s GPAs in the population is between .026 and 0.374. If we had guessed zero difference, 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, Ho, 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 or t = +/- 1.96 3. To find if there is a difference in the population, Ho: 2 - 1 = 0 Ha: 2 - 1 0 4. Collect Data 5. Calculate z or t: z or t = (Y-bar2 – Y-bar1) – (2 - µ1) s.e. 6. Make decision about the null hypothesis (reject or fail to reject) 7. Report 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. Two-tailed significance test for -level = .05 Critical z or t = +/- 1.96 To find if there is a difference in the population, Ho: 2 - 1 = 0 Ha: 2 - 1 0 4. Collect Data Sample: 50 men, average gpa = 2.9, s.d. = 0.5 50 women, average gpa = 3.1, s.d. = 0.4 1. 2. 3. s.e. = (.5)2/ 50 + (.4)2/50 5. 6. 7. 8. = .005 + .003 = .008 = 0.089 3.1 – 2.9 – 0 = 0.2 = 2.25 0.089 0.089 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. 2.25 >1.96 Find P-value: p or (sig.) = .0122 We have a 1.2 % 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, for sure! Calculate z or t: z or t = 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. Just note the difference in calculation of the standard error for the difference. Comparing Two Groups • Now let’s do an example with SPSS, using the General Social Survey.