Linear Models We will determine and use linear models, and use correlation coefficients.

A cleaning company charges $100 for services and $5 per room. Warm-Up: A cleaning company charges $100 for services and $5 per room. Write an equation in function notation and graph What is the cost to clean 12 rooms? How many rooms can be cleaned for $230? This is a linear relationship. Perfect

Warm-Up Answers Let r = the number of rooms and C (r) = the total cost to clean r rooms. Equation C(r) = 100 + 5r C(12) = $160 230 = 100 + 5r 26 rooms

Avg. Wt. of US Girls Warm-Up: Age (yr) 0.5 1 2 3 4 6 8 Wt. (lb) 14.1 20.4 26.2 30 32.7 41.2 50.8 PLOT the points - why Estimate a line visually (your judgment) Pick two points ON the line to find model; note, the two points do NOT need to be part of the data set! A. Calculate slope B. Use y – y1 = m(x – x1) or y = mx + b

Regression Line on the Calculator 1) Enter data into STAT, 1:edit, L1 = I.V., L2 = D.V.   2) Plot on the calculator: STATPLOT, Type: scatterplot (1st one), Xlist = L1, YList = L2 ZOOM, 9:Zoomstat 3) Line of regression: STAT, →Calc, 4:LinReg (ax+b) (to put into Y=: VARS, →Y-Vars, 1:Function, 1:Y1 4) Can also hand-type the equation into Y = 5) 9:ZoomStat to see line with points

Warm-Up Use the calculator to find the line of regression for Average Wt. of US Girls Use both equations to predict the weight of a girl who is 10. What limitations do you see with this model?

Warm-Up Answers y = 4.5x + 15.01 60.01 pounds Limitations?

2 Main Reasons for Model s Interpretation – what does the slope mean, what does the y-intercept mean? Prediction – what will the weight be at 5 years old, when will the weight be 48 pounds?

Correlation: a measure that describes the strength of the relationship between 2 variables called correlation coefficient, r value from -1 to 1, inclusive closer |r| is to 1, stronger the correlation sign indicates slope of the line r = ±1 indicates perfect correlation

given r2, how can r be determined? Correlation continuum: -1 1 weak - weak + strong - strong + none weak – close to 0, little linear relation strong – close to -1 or 1, good linear relation given r2, how can r be determined?

How do I find r? Using your calculator… Go to the catalog by pressing 2nd , 0 Scroll to the “D’s” Highlight Diagnostic On, Enter The r and r2 will now appear when you find your line of regression.

Which is more accurate? interpolation: extrapolation: 2 Types of Prediction: interpolation: predicts values between known data values extrapolation: predicts values beyond known data values Which is more accurate?

Properties of a Linear Model: y =ax+ b Concavity: None…there are no turning points. Therefore, there are no inflection points. End Behavior When a > 0 Limx-∞f(x) = -∞ Limx∞f(x) = ∞ When a < 0 Limx-∞f(x) = ∞ Limx∞f(x) = -∞

Closure The table below gives the age of a cat or dog and its corresponding age in human years. Determine a line of regression, using the calculator. What is the correlation coefficient and what does it tell you? According to your equation, what happens to a dog or cats age as humans age one year? car/dog age (years) .5 1 2 4 6 8 10 14 18 21 human age (years) 15 24 32 40 48 56 72 91 106

Closure Answers y = 4.44x + 11.90 r = .998; this indicates that the data is a very strong fit to the line; also, the rate of change (slope) is positive (as x increases, so does y.) They age 4.44 years Approximately 78.5 years; Interpolation; the input value is within the known data set.