# VCU STAT 210 - Lecture14 (80 pages)

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## Lecture14

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- Pages:
- 80
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 14 Scatterplots and Correlation September 27 2017 Practice Problems Pages 130 through 137 Relevant problems V 1 V 2 a b c V 3 a V 4 a and b V 5 V 6 a V 8 a V 9 a V 10 a and V 11 Recommended problems V 1 V 5 V 8 a V 9 a V 10 a and V 11 Additional Reading and Examples Read pages 109 and 110 Top Hat Motivating Example Watching television also often means watching or dealing with commercials and of interest is to describe the relationship between the number of hours of television watched per day and the number of commercials watched Relationships Suppose we have two variables and our goal is to describe the relationship between the two variables Causal Relationship The question does one variable cause changes or explain changes in the other variable This would imply a causal relationship Example in young children as they get older they gain weight and grow taller Hence changes in age cause explain changes in weight and height Association Often two variables are associated and yet one variable does not cause changes in the other variable Example students taking the same math and chemistry classes those students who did well on the first chemistry test also tended to do well on the first math test Quantitative Variables Suppose we have two quantitative variables X and Y We want to explain the causal relationship between X and Y by writing Y as a linear function of X This linear function will then be used to predict values of Y for specified values of X Independent Variable X is called the independent or explanatory variable which is a measurement variable that has no restraints placed on it and attempts to explain the observed outcomes of Y Example X age of child Dependent Variable Y is called the dependent or response variable and is the measurement variable that measures an outcome of a process that is the effect or consequence of the independent variable Example Y weight of child As a child ages they gain weight hence the process is the aging process and growing older causes weight gain Example 23 1 X number of TV ads run Y number of cars sold or Top Hat 2 X number of cars sold Y number of TV ads run Example 23 1 X number of TV ads run Y number of cars sold Lurking Variables A variable which has an important effect on the relationship between the independent X and dependent Y variables but which is not included in the list of variables being studied is called a lurking variable When a lurking variable exists we see an association between the two variables but we cannot say that one variable is causing changes in the other Lurking Variables Example X score on first chemistry test Y score on first math test Scoring well on the first chemistry test does not cause you to score well on the first math test but it is often associated with a good score on the first math test There are potential lurking variables such as the overall ability of the student or the amount of time and effort spent studying that would explain the relationship that we see Lurking Variables When a lurking variable exists then we say that the effect that X is having on Y is confounded with the effect of the lurking variable So it appears that X is causing changes in Y but really the lurking variable is involved in the relationship and hence confounds the results Describing Relationships We now turn our attention to describing the relationship between the independent variable X and the dependent variable Y A complete description of this relationship includes specifying the direction form and strength of the relationship and to accurately describe these three things we need both a graph and a numerical descriptor In the two sections that follow we learn about the scatterplot a graph and the correlation coefficient a numerical descriptor Scatterplot Graphical procedure for displaying the relationship between two quantitative variables Label X along the horizontal axis Label Y along the vertical axis Plot each X Y observation on the plot Example 24 X number of TV ads run Y number of cars sold Example 24 0 5 10 15 20 25 30 35 Number of ads run Example 24 Number of cars sold 40 36 32 28 24 20 16 12 8 4 0 5 10 15 20 25 30 35 Number of ads run Example 24 Number of cars sold 40 36 32 28 24 20 16 12 8 4 0 5 10 15 20 25 30 35 Number of ads run Example 24 Number of cars sold 40 36 32 28 24 20 16 12 8 4 0 5 10 15 20 25 30 35 Number of ads run Example 24 Number of cars sold 40 36 32 28 24 20 16 12 8 4 0 5 10 15 20 25 30 35 Number of ads run Example 24 Number of cars sold 40 36 32 28 24 20 16 12 8 4 0 5 10 15 20 25 30 35 Number of ads run Description of Relationship To completely describe the relationship between two variables one must specify the direction form and strength of the relationship Direction Type of association between X and Y 1 Two variables are positively associated if small values of X are associated with small values of Y and if large values of X are associated with large values of Y There is an upward trend from left to right Positive Association Y grade on test X amount of time studying Direction Type of association between X and Y 1 Two variables are positively associated if small values of X are associated with small values of Y and if large values of X are associated with large values of Y 2 Two variables are negatively associated if small values of X are associated with large values of Y and large values of X are associated with small values of Y There is a downward trend from left to right Negative Association Y number of questions missed X amount of time studying Form Describe the type of trend between X and Y 1 Linear points fall close to a straight line Linear Association Y X Form Describe the type of trend between X and Y 1 Linear points fall close to a straight line 2 Quadratic points follow a parabolic pattern either in the shape of a U or an upside down U Quadratic Association Y X Form Describe the type of trend between X and Y 1 Linear points fall close to a straight line 2 Quadratic points follow a parabolic pattern 3 Exponential points follow a curved pattern either as exponential growth upward or exponential decay downward Exponential Growth Y X Strength Measures the amount of scatter around the general linear trend The closer the points fall to a straight line the stronger the linear relationship between the two variables Weak Association Y X Moderate Association Y X Strong Association Y X Example …

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