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Aim: How do we use data to make predictions – (linear regression)?. (1, 6). (3, 5.5). point-slope form. 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. - PowerPoint PPT Presentation
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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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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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)
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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.
Aim: Line of Best Fit Course: Alg. 2 & Trig.
Least Squares Regression Line
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xayn
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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+!!
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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:
Aim: Line of Best Fit Course: Alg. 2 & Trig.
Using Graphing Calculator to Determine Equation of Best Fit.
ENTERSTAT 4
Calculate the regression line:
2ndGraph results: Y = ENTER
ZOOM 9
ENTER
8.231,3014.152 xy
Aim: Line of Best Fit Course: Alg. 2 & Trig.
Using Graphing Calculator to Determine Equation of Best Fit.
ENTERSTAT 4
Graph the regression line:
VARS 1 ENTER
ENTER ENTER GRAPH TRACE
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
6 pt. Regents Question
The availability of leaded gasoline in New York State is decreasing, as shown in the accompanying table.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
6 pt. Regents Question
The availability of leaded gasoline in New York State is decreasing, as shown in the accompanying table.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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
Aim: Line of Best Fit Course: Alg. 2 & Trig.
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