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### Ch 3 – Examining RelationshipsYMS – 3.1

### YMS – 3.2

### YMS – 3.3

### Understanding r-squared: A single point simplification

Scatterplots

Some Vocabulary

- Response Variable
- Measures an outcome of a study
- AKA dependent variable
- Explanatory Variable
- Attempts to explain the observed outcomes
- AKA independent variable
- Scatterplot
- Shows the relationship between two quantitative variables measured on the same individuals

Scatterplots

- Examining
- Look for overall pattern and any deviations
- Describe pattern with form, strength, and direction
- Drawing
- Uniformly scale the vertical and horizontal axes
- Label both axes
- Adopt a scale that uses the entire available grid
- Categorical Variables
- Add a different color/shape to distinguish between categorical variables

Classwork p125 #3.7, 3.10-3.11

Homework: #3.16, 3.22 and 3.2 Blueprint

Correlation

Correlation

- Measures the direction and strength of the linear relationship between two quantitative variables

Facts About Correlation

- Makes no distinction between explanatory and response variables
- Requires both variables be quantitative
- Does not change units when we change units of measurement
- Sign of r indicates positive or negative association
- r is inclusive from -1 to 1
- Only measures strength of linear relationships
- Is not resistant

Least-Square Regression

Regression

- Regression Line
- Describes how a response variable y changes as an explanatory variable x changes
- LSRL of y on x
- Makes the sum of the squares of the vertical distances of the data points from the line as small as possible
- Line should be as close as possible to the points in the vertical direction
- Error = Observed (Actual) – Predicted

Coefficient of determination – r2

- The fraction of the variation in the values of y that is explained by the least-squares regression of y on x
- Measures the contribution of x in predicting y
- If x is a poor predictor of y, then the sum of the squares of the deviations about the mean (SST) and the sum of the squares of deviations about the regression line (SSE) would be approximately the same.

Al Coons

Buckingham Browne & Nichols School

Cambridge, MA

Error w.r.t. mean model

Error eliminated by y-hat model

Proportion of error eliminated by Y-hat model

Error eliminated by y-hat model

=

Error w.r.t. mean model

r2 = proportion of variability accounted for by the given model (w.r.t the mean model).

Error w.r.t. mean model

Error eliminated by y-hat model

Proportion of error eliminated by Y-hat model

Error eliminated by y-hat model

=

Error w.r.t. mean model

=

~

Facts about Least-Squares Regression

- Distinction between explanatory and response variables is essential
- A change of one standard deviation in x corresponds to a change of r standard deviations in y
- LSRL always passes through the point
- The square of the correlation is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x

Classwork: Transformations and LSRL WS

Homework: #3.39 and ABS Matching to Plots Extension Question (we’ll finish the others in class)

Residuals

- observed y – predicted y or
- Positive values show that data point lies above the LSRL
- The mean of residuals is always zero

Residual Plots

- A scatterplot of the regression residuals against the explanatory variable
- Helps us assess the fit of a regression line
- Want a random pattern
- Watch for individual points with large residuals or that are extreme in the x direction

Outliers vs. Influential Observations

- Outlier
- An observation that lies outside the overall pattern of the other observations
- Influential observation
- Removing this point would markedly change the result of the calculation

Classwork: Residual Plots WS

Homework: p177 #3.52 and 3.61

Doctor’s for the PoorThis will be graded for accuracy!Ch 3 Reviewp176 #3.50-3.51, 3.56, 3.59, 3.69, 3.76-3.77

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