Response Variable: measures the outcome of a study (aka Dependent Variable) Explanatory Variable: helps explain or influences the change in the response variable (aka Independent Variable) Bivariate Data Not every bivariate data has a response and explanatory variable. Sometimes you are simply exploring a relationship between the variables.

Which is response and which is explanatory? OR Just exploring relationship? - The amount of time spent studying for a stats exam and the grade on the exam. - Inches of rainfall in the growing season and the yield of corn in bushels per acre. - A student’s grade in statistics and their grade in Spanish. - A family’s income and the years of education of the eldest child Classifying one variable as explanatory and the other as response DOES NOT imply CAUSATION. There may also be other variables lurking in the background that are influencing the relationship.

Scatterplot a graph that displays the relationship between bivariate, quantitative data. Each point represents an individual from the data. When interpreting a scatterplot, you must look at both the overall pattern of the points as well as points that deviated from the pattern.

Interpreting Scatterplots a) direction Positive AssociationNegative Association b) form – curvature and clusters b) form – curvature and clusters c) strength – how closely the points follow a clear form d) outliers – individual deviations from the overall pattern

If it is important to differentiate the bivariate data by individuals, a categorical component can be added to the scatterplot by using different colors or symbols

Scatterplot on the calculator Place the explanatory data in L1 Place the response data is L2 2 nd Y= button (Stat Plot) Choose the scatterplot….1 st graph X list = L1 Y List = L2 Zoom 9

Correlation: the direction and strength of a linear relationship between bivariate, quantitative variables Eyes are not always a good judge of this… We need a numerical method to determine the correlation.

Correlation (symbolized as r) is calculated by the following formula: Where z x is the z-score of the explanatory variable and z y is the z-score of the corresponding response variable

Correlation on the calculator Be sure Diagnostic is turned on… Go to Catalog and scroll down to Diagnostic On, hit enter Place the explanatory data in L1 Place the response data is L2 Go to STAT  CALC  #8 and hit enter the r value will be in that list of values

-It does not matter which is x and y for the formula -If you change the units of measure on either or both variables, the correlation will NOT change - For any correlation, -If r > 0, there is a positive association -If r < 0, there is a negative association -Correlation describes LINEAR relationships ONLY (not curved) -Correlation (r) is affected by outliers and is NOT a complete summary of bivariate data

The closer the correlation coefficient is to either 1 or -1, the more linear the scatterplot with appear.

Least Squares Regressions Correlation measures the direction and strength of a LINEAR relationship. A line that summarizes the overall pattern of the relationship is called a regression line. Only applicable to bivariate data with explanatory and response variables. The line MODELS the form of the data and can be used to PREDICT response values (y) given an explanatory value (x).

General Equation of any Regression Line ab y = a + bx slope the approximate amount by which y changes when x increases by one unit y-intercept the value of y when x = 0 …if appropriate for the data

Non exercise activity (NEA) vs. Fat gain Fat gain = a + b(NEA) extrapolation Predicting values BEYOND the scale is called extrapolation…these predictions are not always accurate This data measure calories burned in everyday activity (non-exercise) compared to the fat gained (in kilograms)

MINIMIZING LEAST SQUARES REGRESSION LINELSRL A good regression line best models the data by MINIMIZING the vertical distance of each point from the line. This line is called the LEAST SQUARES REGRESSION LINE (or LSRL) This is symbolized as where and the line passes through

Using Calculator to find the LSRL: STAT  CALC  #8 LinReg (a +bx) List x, List y, Y 1 This will store the LSRL for graphing