Aim: How do we use data to make predictions – (linear regression)?

Aim: Line of Best Fit Course: Alg. 2 & Trig.

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Aim: How do we use data to make predictions – (linear regression)?

Do Now: A candle is 6 inches tall after burning for 1 hour. After 3 hours, it is 5 ½ in. tall. Write a linear equation to model the height y of the candle after burning x hours.

2 1

2 1

y ym

x x

2 1 2 1y y m x x

point-slope form.

6 5.5 1

1 3 4m

16 1

4y x

1 16

4 4y x

(1, 6)(3, 5.5)

slope is the rate of change


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Do Now Extension

Do Now: A candle is 6 inches tall after burning for 1 hour. After 3 hours, it is 5 ½ in. tall.

(1, 6) (3, 5.5)

1 16

4 4y x

In how many hours will the candle be 4 inches tall?

1 14 6

4 4x

y - height of candle x - hours.

1 12

4 4x

1 14 2

4 4x

x = 9 hours

(4, 9)


Interpreting Data

The cost of attending college is steadily increasing. The chart shows the average tuition and fees for a full-time resident student at a public four-year college. Estimate the average college cost in the academic year beginning in 2007 if tuition and fees continue at this rate.

Year Tuition & Fees

Year Tuition & Fees

1990-91 2159 1995-96 2811

1991-92 2410 1996-97 2975

1992-93 2349 1997-98 3111

1993-94 2537 1998-99 3243

1994-95 2681


Finding Best Fit Equation

Average Tuition and Fees3500

1700

1900

2100

2300

2500

2700

2900

3100

3300

1990 1991 1992 1993 1994 1995 1996 19971998

(1992, 2349)

(1997, 3111)12

12

xx

yym

4.152m

1212 xxmyy

8.231,3014.152 xy

•Select 2 points that represent the data

•Determine slope

•Calculate equation of best fit line

19924.1522349 22 xy


Scatter Plots & Correlation

A Scatter Plot is a graph that relates two different sets of data by plotting the data as ordered pairs.

Positive Correlation•y tends to increase as x

increases•slope is positive

Negative Correlation•y tends to decrease as x

increases•slope is negative

No Correlation


Correlation Co-efficient

Data that are linear in nature will have varying degrees of goodness of fit to the lines of fit.

The correlation coefficient r describes the nature of data.

The closer the fit of the data to the line, the closer r gets to + 1 or -1

0 < r < 0.5 positive/weak

0.75 < r < 1 strongly positive

-0.5 < r < 0 moderately

negative


Making Predictions

Projected cost in 2006-2007

8.231,3014.152 xy

8.231,301)2006(4.152 y

6.4482y

Predict cost for 2006-2007 academic year by substituting 2006 for x in best fit

equation.

Prediction: cost for tuition and fees will be $4482.60

Trend line – approximates the relationships between data sets of a scatter plot.


Least Squares Regression Line

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xayn

band

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yxyxna

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The least square regression line, y = ax + b, for the point (x1, y1), (x2, y2), (x3, y3), ….. (xn, yn) is given by

How can we find an equation that fits the data more closely if the correlation is not very high?

Thank you TI 83+!!


Using Graphing Calculator to Determine Equation of Best Fit.

Enter years into L1

Enter dollars into L2

STAT 1

STAT 1

Ensure coordinate pairs correspond

Enter data:



ENTERSTAT 4

Calculate the regression line:

2ndGraph results: Y = ENTER

ZOOM 9

ENTER

8.231,3014.152 xy



ENTERSTAT 4

Graph the regression line:

VARS 1 ENTER

ENTER ENTER GRAPH TRACE


6 pt. Regents Question

The availability of leaded gasoline in New York State is decreasing, as shown in the accompanying table.

Determine a linear relationship for x (years) versus y (gallons available), based on the data given. The data should be entered using the year and the gallons available (in thousands), such as (1984, 150)

YEAR 1984 1988 1992 1996 2000

Gallons Available

(in thousands)150 124 104 76 50




If this relationship continues, determine the number of gallons of leaded gasoline available in New York State in the year 2005.

YEAR 1984 1988 1992 1996 2000

Gallons Available

(in thousands)150 124 104 76 50




If this relationship continues, during what year will leaded gasoline first become unavailable in New York State?

YEAR 1984 1988 1992 1996 2000

Gallons Available

(in thousands)150 124 104 76 50


Model Problem

The table contains the fat grams and calories in various fast-food chicken sandwiches.

a. Find the equation of the regression b. Predict the number of calories in a sandwich

with 20 grams of fat.

Sandwich fat (grams)

calories Sandwich fat (grams)

calories

A breaded

28 536 G breaded

9 300

B grilled

20 430 H ch. salad

5 320

C ch. salad

33 680 I breaded

26 530

D broiled

29 550 J breaded

18 440

E breaded

43 710 Kgrilled

8 310


Documents

Aim: How do we use data to make predictions – (linear regression)?