AP Statistics Ch 3 Aim 1: Scatter Diagrams

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1 Page 1 of4 Univariate nata Bivariate Data data involving only one variable such as test scores, height, etc. data involving the relationships between two variables, such as test scores and time studying, height and weight, etc. Assodation Between Variables Two variables measured on the same individuals are associated if some values of one variable tend to occur more often with certain values of the second variable. Frequently with bivariate data, one of the variables can explain variation in the other or can even cause changes in the other variable. A response variable (the dependent variable) measures an outcome of a study. An explanatory variable (the independent variable) explains or causes changes in the response variable. Example 1: In each of the following situations, is it more reasonable to simply explore the relationship between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? In the latter case, which is the explanatory variable and which is the response variable? A. The smoking habits of an individual and the incidence of lung cancer B. The weight and height of a person C. The amount of yearly rainfall and the yield of a crop D. A student's scores on the SAT math exam and the SAT verbal exam E. The occupational class of a father and son F. Reading level of a student in an elementary school and the shoe size of the student Solution: The existence of a relationship does not imply a strong relationship and the relationship may not appear true for all observations. A. explanatory variable: smoking habits response variable: incidence of lung-eancer B. There is a relationship between height and weight, but there is not a causal relationship. C. explanatory variable: amount of yearly rainfall response variable: yield of a crop D.There is a relationship between a student's scores on the SATmath exam and the SATverbal exam, but it is not a causal relationship. Students who do relatively well on the SAT math exam tend to do relatively well on the SATverbal exam, and vice versa. E. explanatory variable: occupational class of a father response variable: occupational class of a son (The relationship may be very weak, but it is a causal relationship.) F. There is a relationship between the reading level of a student in elementary school and the shoe size of the student, but it is not a causal relationship. Both variables are associated with age, and students' who are older tend to have larger feet and higher reading levels. copyright

2 Page 2 of4 Notes: 1. Not all explanatory-response relationships involve direct causation, but the relationship could still be used for prediction of values. 2. It is easiest to identify explanatory and response variables when the values of one variable are set in order to see how it affects the values of the other variable. A scatter diagram or scatter plot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Always plot the explanatory variable, if there is one, on the horizontal axis of a scatter plot. As a reminder, we usually call the explanatory variable x and the response variable y. If there is no explanatory-response distinctfon, either variable can go on the horizontal axis. When describing the variables in a scatter plot, the following terminology is frequently used: the graph of the response variable against the explanatory variable the graph of the response variable versus the explanatory variable When interpreting a scatter diagram, consider the following features: Direction: Positive or Negative? Form: Linear or Curvilinear? Strength: Strong, Moderately Strong, or Weak? Direction: Positive or Negative Association? Two variables are positively associated when above-average values of one variable tend to accompany above-average values of the other variable and below-average values tend to occur together. The graph appears to go up from left to right.. Two variables are negatively associated when above-average values of one variable accompany belowaverage values of the other variable, and vice versa. The graph appears to go down from left to right. Form: Linear or Curvilinear? A linear form is much easier to analyze. If the form is not clearly curvilinear, describe the form as linear. Strength: Strong, Moderately Strong, or Weak? The strength of a relationship in a scatter plot is determined by how closely the points follow a clear form. If the scatter diagram can be used to predict values accurately, the relationship is strong. The strength of a relationship can be determined mathematically. copyright

3 Page 3 of4 Deseribe relationships using the fodowingsentence structure! "There is a STRENGTH DIRECTION FORM relationship between STATE THE VARIABLES" Example 4: A teacher wants to determine the effect of watching television on a student's grade -poist average. In determining a student' sgrade point average, the following scale was used: A = 4, B = 3, C= 2,D = 1, and F =0. The following data was collected from a simple random sample of 10 students from the loth grade in her school: Homs watchinf!televisionper week Grade point average ~ A. Make a scatter plot of grade point average against hours watching television per week B. Describe the direction, form, and strength of the relationship. C. Identify any outliers and identify lurking (confounding) variables which might help to explain why the outliers are present. Solution: A. "Hours watching television per week" is the explanatory variable and is placed on the horizontal axis. "Grade point average" is the response variable and is placed on the vertical axis. Axes must be labeled or ltis a 25% deduction on the AP exam! The points areplotted in the same way that the point ix.v) isplottechn the coordinate plane CD ~ 3 ~ 2.5 as c: 2 oq. 1.5 CD '" 1 l! " 0.5 o o ' Hours watching TV per week B. There is a strong negative linear relationship between hours watching N per week and grade point average. copyright

4 Page 4 of4 (The association is negative since the pattern goes down from left to right. As the hours watching TV per week increases, the grade point average tends to decrease. The strength is strong since the scatter plot appears to be useful for prediction purposes. For example, if the student watched 5 hours of TV per week, the predicted value for the student's grade point average would be approximately 2.6.) C. The point representing 1.5 hours watching TV per week and a grade point average of 1.6is on outlier. A possible confounding variable may be that the student has a job and has little time to watch TV or to study. Another possible confounding variable may. be the student's academic ability. The point representing 8.1 hours watching TV per week and a grade point average of 3.6 is also an outlier. A possible confounding variable may be that the student watches educational television. Another possible confounding variable may be the student's academic ability. Homework 27: Sullivan's Statistics: Informed Decisions Using Data Read pages pg ,2,3.4, llabd

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