Describing Bivariate Data
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1 Describing Bivariate Data Correlation Linear Regression Assessing the Fit of a Line Nonlinear Relationships & Transformations The Linear Correlation Coefficient, r Recall... Bivariate Data: data that consists of measured Can be written as (x, y) A shows the relationship between x and y (if one exists). Variable: the one manipulated/controlled Variable: the one dependent on the explanatory; the result of the control or manipulation
2 Example Scatter Plots POSITIVE RELATIONSHIPS NEGATIVE RELATIONSHIP NO RELATIONSHIP NONLINEAR RELATIONSHIP We measure how strong the linear relationship is by the Pearson's Correlation Coefficient, ρ (Sample: r ) VERY TEDIOUS!! USE CALCULATOR: Menu, Stats, Stat Calculations, 3: LinReg(mx + b) *you are looking for the "r" value
3 Let's see how well you did at the age guessing... Guessed Ages in List A Actual Ages in List B Find the correlation coefficient, r. The closer to +1, the better you did at guessing the correct ages. 5 Properties of the Linear Correlation Coefficient (r): 1. r is between 1 and +1 (inclusive) linear is perfect if r is 1 or +1 (all points on line of best fit) r = 1 r = 0 r = If r is positive, x increases and y increases OR x decreases and y decreases. 4. If r is negative, x increases while y decreases OR x decreases while y increases. 5. If r = 0, then there is no LINEAR relationship; it doesn't mean there is no relationship at all!
4 10.2 mt4.notebook strong { 0.8 moderate Weak to None { { 1 moderate { strong April 23, How strong is the relationship? A study of 528 students found r to be 0.75 when x = # hours worked after school and y = GPA. A study examined suicide rates from 1958 to 1992 and found r to be 0.97 when x = misery index and y = suicide rate. Is Foal Weight Related to Mare Weight? A study of 8 foals and their mothers produced the following data. What are the explanatory and response variables? Are the two variables related linearly? mare wgt (kg) foal wgt (kg) Calculate the linear correlation coefficient, r. +1
5 ASSIGNMENT: correlation worksheet #1 To plot a scatter plot on calculator: Type #s in list Ctrl + Doc 5: Add Data scroll down to select x axis scroll left to select y axis copy of worksheet on next slide...
6 1. For each of the following pairs of variables, determine which is explanatory x) ( and which is response (y). Then, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to 0. a. maximum daily temperature and cooling costs b. interest rate and number of loan applications c. incomes of husbands and wives when both have full time jobs d. height and IQ score e. height and shoes size f. score on math SAT section and score on verbal SAT section g. time spent on homework and time spent watching TV during the same day by elementary school children h. amount of fertilizer used per acre and crop yield 2. Is the following statement correct? If not, make it true. A correlation coefficient of 0 implies that no relationship exists between the two variables under the study. 3. The paper A Cross National Relationship between Sugar Consumption and Major Depression? concluded that there was a correlation between refined sugar consumption (calories per person per day) and annual rate of major depression (cases per 100 people) based on data from 6 countries. a. Compute the correlation coefficient, r, for this data set. Is it positive or negative? Weak, moderate, or strong? b. Based on the value of the correlation coefficient from part a, it is reasonable to conclude that increasing sugar consumption leads to higher rates of depression? Explain. c. Do you have any concerns about this study that would make you hesitant to generalize these conclusions to other countries?
7 4. Is there a relationship between test anxiety and exam performance? Data on X = score on a measure of test anxiety and Y = exam score for a sample of n = 9 students are the following. X Y Higher values for X indicate higher levels of anxiety. a. Construct a scatter plot and comment on the features of the plot. b. Does there appear to be a linear relationship between the two variables? Would you characterize it as positive or negative? Weak, moderate, or strong? c. Compute the value of the linear correlation coefficient. Is the value of r consistent with your answer to part b? d. Based on the value of the linear correlation coefficient, is it reasonable to conclude that test anxiety caused poor exam performance? Explain.
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