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Soc202 Chapter 11.docx

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University of Toronto St. George
Scott Schieman

Chapter 11 Bivariate Relationships: t-Test for Comparing the Means of Two Groups Introduction: Bivariate Analysis -Prediction provides the connection between statistical analysis and probability theory -In studying psychological depression, if we can determine what triggers depression, we can identify “pops. at risk” and institute preventative measures…example being research shows people who experience multiple life crises are at greater risk of developing severe depression -Bivariate (2-variable) analysis involves searching for statistical relationships between 2 variables -A Statistical relationship between 2 variables asserts that “the measurements of one variable tend to consistently fluctuate with the measurements of the other, making one variable a good predictor of the other” -3 common approaches to measuring statistical relationships: 1) Difference of means testing…comparing means of an interval/ratio variable among the categories or groups of nominal/ordinal variable 2)Counting the frequencies of joint occurrences of attributes of two nominal variables 3) Measuring the correlation between 2 interval/ratio variables Difference of Means Test -use this approach when hypothesizing between dependant intervals/ratios and independent dichotomous nominal/ordinal variables -to test for a difference among 3 or more means, use a procedure called analysis of variances (ANOVA)…group means are compared indirectly by finding how much of the variance in the interval/ratio variable is explained by group membership -example: if we have an interval/ratio variable called “favourableness to team concept” for hospital care for hospital care (a measurement scale that consists of 40 questionnaire items), it discerns whether a hospital’s medics believe in sharing authority for patient care…the ANOVA test would compare the mean favourableness to team concept scores for these 4 groups to see whether physicians are less favourable -ANOVA test: compares means among the different groups or categories of a nominal/ordinal variable Joint Occurrences -another way to view relationship between 2 variables pertains to 2 nominal variables with a focus on joint occurrence of attributes -attribute: a quality of a subject that is conveyed in the category of names of a nominal/ordinal variable -example: for gender, the attributes can be male or female -joint occurrence of attributes involves 2 variables for a single individual, with pairings of attributes of the 2 variables (Mary is a white female… she has the joint occurrence of white and female) -example of a research question about the relationship between 2 nominal/ordinal variables: are women more likely than men to support government-subsidized child care services? Correlation -a relationship between 2 interval/ratio variables indicates that scores on one variable tend to correlate with scores on the other -example: correlation between the variables frequent experience of stressful life events and psychological depression -for 2 ordinal level variables, we use a “rank-order correlation” test, which sees whether those study subjects who rank high on one measure also tend to rank high on another -in all research questions on the relationship between 2 variables we hypothesize a relationship between and independent and dependant variable (main interest on dependant) example: do stressful life circumstances cause depression? Do poverty conditions contribute to crime? 2-Group Difference of Mean Tests (t-Test) for Independent Samples -2-group difference of means tests compares the means of an interval/ratio variable for 2 groups or categories of a nominal;/ordinal variable -example: test if college test scores are higher for students at City Universities rather than State Universities -the variable on which the mean is computed must be an interval/ratio variable, in this 2-group means test, this variable is typically the dependant variable -the 2 comparison groups of the independent variable may come from separate populations -the same test can be used if the 2 groups are categories of a dichotomous nominal/ordinal variable from a single pop. (such as the variable gender, with categories male and female)… we compare the separate pops. of men and women within the total population -with a 2 group difference of means test, it is not necessary to have a target balue for a parameter of X for the entire population or its subgroups -for the hypothesis tests though, we must be able to predict a parameter and describe the sampling distribution around it -the 2 group difference of means is a t-test, focused on the computed difference between 2 sample means X1-X2, where X1 and X2 are the means of groups, respectively -in describing the sampling distribution, we can predict that the difference between any 2 randomly drawn sample means is zero, give or take sampling error. -the third criterion (the independence of groups) distinguishes this statistical test from one in which the same group of subjects is being compared on 2 variables or at 2 different times on 1 variable. th -the 4 criterion is called the assumption of equal variances (or equal standard deviations)…if pop. Variances are not equal, adjustments must be made to the statistical test The Standard Error and Sampling Distribution for the T-Test of the Difference between 2 Means -for a 2group difference of means test, imagine bean counting, repeatedly sampling from 2 with the same means for some interval/ration variable X, for each sample, we compute the mean of X and subtract these 2 sample means to get the difference of means: X1-X2 (with the numbers as subscripts) -when X1> X2, the difference is positive - when X1< X2, the difference is negative -when X1=X2, the difference is 0 -the sampling population distribution centers on a difference of zero between the 2 pop. Means (the difference between parameters) -where equal variances are assumes, the standard error of the difference of means is computed by averaging the 2 variables, called a pool variance estimate of the standard error, with the formula as seen on p. 375 of the textbook -t-test statistic is computed with the equation on p. 376 -this test statistic is designed to answer questions about parameters, the null hypothesis for this test will always be that the 2 population means are equal (equation on p.376) Six Steps of Statistical Inference -(using the symbol H for Hypothesis) 1) State the H0and H A and stipulate test direction 2) Describe the sampling Distribution 3) State the level of significance and test direction and specify the critical test score 4) Observe the actual sample outcomes and compute the test effects, the test statistic, and p-value 5
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