Chapter 3-Examining Relationships

Scatterplots and CorrelationLeast-squares Regression

October 26, 2015.Objectives:1. Interpreting Scatterplots in terms of

direction, form, and strength.

• Activity: Tootie Fruities.1. Each person grabs one hand full of Tootie

Fruities and does a quantitative analysis of the that event (how much did you grab?). Discuss an exact procedure on how to grab the cereal.

2. Make a histogram of the data and discuss the shape center and spread of the data.

3. Measure another variable and create a scatter plot.

4. Discuss the following of the scatter plot:• Overall pattern.• Are there any striking deviations.• Direction.• Form.• Strength.• Any outliers.

Beer and blood alcohol

• Data.Student Beers BAC Student Beers BAC

1 5 .1 9 3 .02

2 2 .03 10 5 .05

3 9 .19 11 4 .07

4 8 .12 12 6 .1

5 3 .04 13 5 .085

6 7 .095 14 7 .09

7 3 .07 15 1 .01

8 5 .06 16 4 .05

October 27, 2015Objectives:1. Construct and interpret a scatterplot for a set of bivariate data. 2. Compute and interpret the correlation r between two variables.

3. Identify the properties of correlation r.

Describing a Scatterplot

Direction: Positive association, Negative association.

Form: Linear, Curved, clusters.

Strength: Weak, Moderate, Strong.

Outliers: Influential or not influential.

Correlation r:The correlation measures the direction and strength of the linear relationship between two quantitative variables.Correlation = r

1

1 x yr Z Zn

Correlation facts:1. Correlation makes no distinction between

explanatory and response variables.2. Because r uses the standardized values of the

observations, r does not change when we change the units of measure.

3. Positive r = positive association Negative r = negative association4. Correlation is always between -1 and 1.

Meet the Archaeopteryx.

1. Make a scatterplot. Do you think that all 5 specimens come from the same species?

2. Find the correlation step by step. Find the standardized score of each data point and use the formula to calculate r.

Femur 38 56 59 64 74Humerus 41 63 70 72 84

Oct. 29, 2015• Objectives:

1. Explain the meaning of a least squares regression line.

2. Given a bivariate data set, construct and interpret a regression line.

3. Explain what extrapolation means.

Regression Line

A regression line is a line that describes how a response variable y changes as an explanatory variable x changes. This line is used to predict response variables y given an explanatory variable x.

y a bx

Extrapolation

• Extrapolation is the use of a regression line for prediction outside the range of values of the explanatory variable x used to obtain the line. Such predictions are usually unreliable.

1. Use your calculator to construct a scatterplot of these data using 1997 as year 1 and 1998 as year 2 2. Find the equation of the least-squares line on your calculator. Record the equation below. Be sure to define any variables used.3. Interpret the slope of the least-squares line in context.4. How many students would you predict took

the AP Statistics exam in 2006? Show your method.

October 30, 2015Objectives:1. Define and calculate the residual of a point in

a scatterplot.2. Interpret a Residual plot as a means of

assessing the linearity of a scatterplot.

A residual is the difference between an observed value of the response variable and the value predicted by the regression line.

Activity:Enter this data on your calculator and calculate the residuals of each of the three points.

Residual plot.A residual plot is a scatterplot of the regression residual against the explanatory variable. Residual plots help us assess how well a regression line fits the data.

November 3, 2015Objectives:1. Be able to read computer print outs and

identify the coefficients of the LSRL.2. Given the following information:

Students will be able to calculate the LSRL., , s , , and rx xx y s

1. What is the equation of the LSRL?2. Interpret the slope and y intercept in the

context of this setting.3. Use the regression line to predict the amount

of natural gas that the Sanchez family will use in a month that averages 20 degrees.

Equation of the Least-Squares Regression LineGiven the data of the explanatory variable and the response variable of n individuals:

The LSRL is:

The slope is:

The line passes through:

, , s , , and rx yx y s

y a bx y

x

sb r

s

, x y

• Enter the following data in your calculator and find , , s , , and rx yx y s

Nov. 5, 2015Objectives:1. Review the Coefficient of determination2. Practice finding the LSRL with only

3. Analyze outliers and influential observations within a scatterplot.

2r

, , s , , and rx yx y s

Coefficient of determination.

The coefficient of determination is the fraction of the variation in the values of y that is explained by the LSRL of y on x. Exp.60.6% of the variation of fat gain (y-var.) can be accounted for by the linear relationship (linear model) with NEA (x-var.) .

2r

Correlation

1. Correlation and regression describe only linear relationships.

2. Extrapolation often produces unreliable predictions.

3. Correlation is not resistant.4. Correlation indicates direction, positive or

negative.5. Correlation ignores distinction between

explanatory and response variables.

6. The r value is not affected by change in units of measure.

7. The correlation is between 8. Correlation r measures the strength and

direction between two quantitative variables. (how close the points are in a scatterplot to the LSRL)

1 1r

Outlier:An outlier is an observation that lies outside the overall pattern of the other observations.

Influential Observation:An observation is influential if removing it would significantly change the statistical measure such as: the slope and y-intercept of the LSRL.