1. Module 12
Math 075

2. Let’s Review
What do boxplots, histograms, and dot plots have in common?
What do circle graphs and bar graphs have in common?
What do box plots, histograms, dot plots, and scatterplots have in common?
How are box plots, dot plots, histograms different then scatterplots?
Which histogram will have a greater mean?
Skewed left
Skewed right
Symmeteric

3. Let’s review some more

4. Let’s review some more

5. What do you know about scatterplots?
More review
What do you know
about scatterplots?

6. What would we expect the sodium level to be in a hot dog that has 170 calories?

7. Least Squares Regression
BETTER model to summarize overall pattern by drawing a line on scatterplot
Not any line; we want a best-fit line over scatterplot
Least Squares Regression Line (LSRL) or Regression Line

8. Least-Squares Regression Line

9. Let’s do some predicting by using the LSRL...
About how much would a home cost if it were:
2,000 square feet?
2,600 square feet?
1,600 square feet?
Categorical data embedded; sometimes scatterplots include this ‘extra’ information.

10. Let’s do some predicting by using the LSRL...
About how large would a home be if it were worth:
$450,000?
$350,000?
$220,000?
Also, let’s discuss where the x and y axes start...
Categorical data embedded; sometimes scatterplots include this ‘extra’ information.

11. y-intercept Slope What does this all mean? x-intercept
Algebra Review Time
Slope
y-intercept
What does this
all mean?
x-intercept

12. Least Squares Regression equation to predict values
LSRL Model: 𝑦 =𝑎+𝑏𝑥
𝑦 is predicted value of response variable
a is y-intercept of LSRL
b is slope of LSRL; slope is predicted (expected) rate of change
x is explanatory variable

13. Least Squares Regression equation
Typical to be asked to interpret slope & y-intercept of the equation of the LSRL, in context
Caution: Interpret the slope of the equation of LSRL as the predicted or average change or expected change in the response variable given a unit change in the explanatory variable
NOT change in y for a unit change in x; LSRL is a model; models are not perfect

14. LSRL: Our Data
Go back to our data (age & # states visited; height and weight data from Math 075; calories & fat cereal data).
Create scatter plot; then put LSRL on our scatter plot; also determine the equation of the LSRL
Stat Crunch: stat, regression, simple linear, x variable, y variable, graphs, fitted line plot

15. LSRL: Our Data Look at graph of our LSRL for our data
Look at our LSRL equation for our data
Our line fits scatterplot well (best fit) but not perfectly
Make some predictions… do we use our graph or our equation? Which is easier? Which is better? More on this in a minute...
Interpret our y-intercept; does it make sense? Interpretation of our slope?

16. Another example… value of a truck

17. Truck example…
Suppose we were given the LSRL equation for our truck data as 𝒑𝒓𝒊𝒄𝒆 =𝟑𝟖,𝟐𝟓𝟕− 𝟎.𝟏𝟔𝟐𝟗(𝒎𝒊𝒍𝒆𝒔 𝒅𝒓𝒊𝒗𝒆𝒏)
We want to find a more precise estimation of the value if we have driven 100,000 miles. Use the LSRL equation.
Using graph, estimate price if we have driven 40,000 miles. Then use the above LSRL equation to calculate the predicted value of the truck.

18. Ages & Heights…
Age (years)
Height (inches) 18 1 28 4 40 5 42 8 49

19. Let’s review for a moment…
Input data into Stat Crunch
Create scatterplot and describe scatterplot (what do we include in a description?)
Calculate r (different from slope; why?), equation of LSRL; interpret equation of LSRL in context; does y-intercept make sense? Create a graph of LSRL
Based on the graph of the LSRL or the equation of the LSRL (you choose), make a prediction as to the height of a person at age 35.

20. Friends don’t let friends extrapolate!
LSRL: Our Data
Extrapolation: Use of a regression line (or equation of a regression line) for prediction outside the range of values of the explanatory variable, x, used to obtain the line/equation of the line.
Such predictions are often not accurate.
Friends don’t let friends extrapolate!

21. Detour…Take it back Tuesday
What is r? What is r’s range? What does it describe?

23. Detour…Take it back Tuesday
r (or correlation) is a numerical measure of how linear scatter plot is r (or correlation) tells us the direction of the scatterplot r (or correlation) ranges from -1 to 1 r (or correlation) describes the scatterplot only (not LSRL)