3.1 9.19.2017

Explanatory and Response variables Explanatory variables Also known as “predictor variable” or “x-variable” or “independent variable” Response variables Also known as “outcome variable” or “y-variable” or “dependent variable” An explanatory variable may help to explain/predict/influence changes in the response variable Note: many studies aim to show that changes in the explanatory variable cause changes in the response variable. We will delve into this distinction more in chapter 4

Examples We believe that the GPA of AP Stats students can help us to predict their grade in the course Explanatory variable: Response variable: We believe that taller trees are more likely to be struck by lightning Predictor variable: Outcome variable:

Examples We believe that the GPA of AP Stats students can help us to predict their grade in the course Explanatory variable: GPA Response variable: Grade in AP Stats We believe that taller trees are more likely to be struck by lightning Predictor variable: Tree height Outcome variable: Probability of being struck by lightning

Scatterplots Plots values for two quantitative variables together One on each axis Note: must be values from the same individual Cannot plot the scores of 30 AP Stats students on the x-axis with the height of 30 bears on the y-axis Explanatory variable typically goes on x-axis Response variable typically goes on the y-axis

Example

Interpretation

Interpretation Strength (subjective) Direction (usually obvious) Form Strong Moderately Strong Weak Direction (usually obvious) Positive Negative No association Form Linear Non-linear

Direction (slope)

Strength

Interpretation

Interpretation There is a moderately strong, negative, curved relationship between the percent of students in a state who take the SAT and the average SAT math score in the state

Provide a brief description of the relationship between murder rate and number of ice creams sold

Provide a brief description of the relationship between murder rate and number of ice creams sold There is a moderately strong positive linear relationship between the murder rate and the number of ice creams sold

Correlation We can measure the strength and direction of this relationship Usually better than just estimating by looking at it We use the letter r to represent correlation (also known as the correlation coefficient) r is always between -1 and 1 Positive values represent a positive association Negative values represent a negative association Closer to 0 indicate a weaker correlation; closer to 1 or -1 indicate a stronger relationship r= 1 or r= -1 indicate a perfect linear relationship

Calculating r

Toy Example—calculate r GPA AP Stats Grade Student #1 3.317 94.37 Student #2 2.500 76.06 Student #3 4.024 100.73 Student #4 4.238 97.11

Toy Example—calculate r GPA AP Stats Grade Student #1 3.317 94.37 Student #2 2.500 76.06 Student #3 4.024 100.73 Student #4 4.238 97.11 Mean: 3.52 st. dev: .786 mean: 92.068 st. dev: 10.985 𝑟= 1 3 [ −.258 .210 + −1.298 −1.457 + .641 .789 + .913 .459 ] 𝑟= 1 3 [−.054+1.891+.506+.419] 𝑟= 1 3 [2.762] 𝑟=.92

Correlation Cautions It is more important to understand what the correlation means than to be able to calculate it by hand We will learn how to do it on our calculator next class Correlation makes no distinction between explanatory and response variables So it is the correlation for the relationship between the two, not for the effect that one has on the other r does not change when you change the units of measurement If the correlation between weight in pounds and height in feet is .74, then the correlation between weight in grams and height in inches will also be .74 r has no units—it is just a number

More Correlation Cautions Requires that both variables be quantitative So you cannot calculate the correlation coefficient between blue eyes and brown hair Correlation only measures the strength of a linear relationship, not a curved relationship If the data looks like it is curved, correlation not super meaningful Correlation is strongly affected by outliers Correlation is not a complete summary So if a question asks you to describe the relationship between two variables, saying “r=.45” is not sufficient

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