Chapter 3 Examining Relationships

Introduction We have looked at only one-variable statistics: Quantitative & Categorical data We have looked at only one-variable statistics: Quantitative & Categorical data Now, we examine how one variable affects another. Now, we examine how one variable affects another. These variables are called RESPONSIVE and EXPLANATORY. These variables are called RESPONSIVE and EXPLANATORY.

Two Variables Response Variables – measure the outcome of a study (dependent variables). Response Variables – measure the outcome of a study (dependent variables). Explanatory Variables – explains the observed outcome (independent variables). Explanatory Variables – explains the observed outcome (independent variables). Since responsive variables depend on explanatory variables, how variables are labeled depend on how the data is used. Since responsive variables depend on explanatory variables, how variables are labeled depend on how the data is used.

Page 122: Exercise 3.1 EXPLANATORY EXPLANATORY RESPONSIVE RESPONSIVE a) Study Time Grade b) Explore the relationship … neither explanatory. c) Yearly RainfallCrop Yield d) Explore the relationship … neither explanatory. e) Occup. class of dadOccup. class of son

Page 123: Exercise 3.2 Explanatory – Age 6 Explanatory – Age 6 Responsive – Age 16 Responsive – Age 16 Both Quantitative Both Quantitative

Guide to Examine Data Plot and add numerical summaries Plot and add numerical summaries Look for overall patterns and any deviations Look for overall patterns and any deviations When the overall pattern is regular, use a mathematical model to describe what is occurring. When the overall pattern is regular, use a mathematical model to describe what is occurring.

3.1 SCATTERPLOTS

Activity

Scatterplots Review A scatterplot shows the relationship between two quantitative variables measured on the same individuals. A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Once we plot the variables, we look for a pattern called correlation. Once we plot the variables, we look for a pattern called correlation. If a correlation exists, we use a mathematical model to explain the pattern. If a correlation exists, we use a mathematical model to explain the pattern.

Scatterplots The Plot Place explanatory variables, x, (if one exists) on the horizontal axis and responsive, y, on the vertical. Place explanatory variables, x, (if one exists) on the horizontal axis and responsive, y, on the vertical. If there isn’t an explanatory-response relationship, either variable may go on the horizontal axis. If there isn’t an explanatory-response relationship, either variable may go on the horizontal axis. Read the tips on Page 128. Read the tips on Page 128.

Lions, Tigers, & Bears … Oh My! Let’s look at a study of the weight of a bear vs. the chest size of a bear. A biologist feels that the size of a bear’s chest can predict the weight of a bear. Which is the response variable? The explanatory variable? Chest size (in) Weight (lb)

Bear Data Make a scatterplot. Make a scatterplot. Save your lists. Save your lists.

Equation Diagnostics On Diagnostics On Weight = chest Weight = chest Meaning: a = slope – increase of lbs in weight for every additional inch of chest size. b = is the y-intercept – for a bear with a chest size of 0 inches, the weight would be lbs. Reality: this number has no significance. It is a numerical value to help estimate the regression line. Meaning: a = slope – increase of lbs in weight for every additional inch of chest size. b = is the y-intercept – for a bear with a chest size of 0 inches, the weight would be lbs. Reality: this number has no significance. It is a numerical value to help estimate the regression line.

Bear Data Association Does it appear a bear’s weight is explained by its chest size? Does it appear a bear’s weight is explained by its chest size? What is the shape of the dots? What is the shape of the dots? Is it a positive/negative association? Is it a positive/negative association? How strong is the relationship? How strong is the relationship? Are there any outliers? Are there any outliers?

Practice: Exercise 3.6 (a) Number of powerboats and number of manatees killed by boats. Which is the explanatory variable? (b) Make a scatterplot. What does the scatterplot show about this relationship?

PLOT a) Number of Powerboat Registrations. b) Plot Powerboat Registrations (1000) Manatees Killed

Interpreting Scatterplots Graph – look for the overall pattern and any striking deviations, outliers. Graph – look for the overall pattern and any striking deviations, outliers. Descriptions: Form, direction, and strength. Descriptions: Form, direction, and strength. –Strength is determined by how close the points follow a clear form.

Association – Positive or Negative Positive association – above-average values of one variable accompany above- average values of the other or below- average values of one yield below-average values of the other. Positive association – above-average values of one variable accompany above- average values of the other or below- average values of one yield below-average values of the other. Negative association – one variable is above-average and the other below or vice-versa. Negative association – one variable is above-average and the other below or vice-versa.

Categorical Variables When adding on categorical variables to a scatterplot, use different symbols, colors, to plot points. When adding on categorical variables to a scatterplot, use different symbols, colors, to plot points. Homework: Exercises 3.3, 3.4, 3.7, 3.9 – 3.12 (Exercise 3.13 allows you to expedite things with 3.11 with added Calc fun.) Homework: Exercises 3.3, 3.4, 3.7, 3.9 – 3.12 (Exercise 3.13 allows you to expedite things with 3.11 with added Calc fun.)