Module 12 Math 075

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

Let’s review some more

Let’s review some more

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

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

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

Least-Squares Regression Line

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.

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.

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

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

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

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

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?

Another example… value of a truck

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.

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

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.

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!

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

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)