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Least Squares Regression Lines

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Least Squares Regression Lines

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Least Squares Regression Lines

Text: Chapter 3.3

Unit 4: Notes page 58

x – variable: is the independent or explanatory variable

y- variable: is the dependent or response variable

Use x to predict y

b – is the slope

it is the approximate amount by which y increases when x increases by 1 unit

a – is the y-intercept

it is the approximate height of the line when x = 0

in some situations, the y-intercept has no meaning

- (y-hat) means the predictedy

Be sure to put the hat on the y

The line that gives the best fit to the data set

The line that minimizes the sum of the squares of the deviations from the line

Sum of the squares = 61.25

(-4)2 + (4.5)2 + (-5)2 = 61.25

(3,10)

y =.5(6) + 4 = 7

2 – 7 = -5

4.5

y =.5(0) + 4 = 4

0 – 4 = -4

-5

y =.5(3) + 4 = 5.5

10 – 5.5 = 4.5

-4

(6,2)

(0,0)

(0,0)

(3,10)

6

Find y - y

-3

(6,2)

-3

(0,0)

Sum of the squares = 54

What is the sum of the deviations from the line?

Will it always be zero?

Use a calculator to find the line of best fit

STAT EDIT L1, L2

STAT CALC

4 LinReg (ax+b)

The line that minimizes the sum of the squares of the deviations from the line is the LSRL.

Slope:

For each unit increase in x, there is an approximateincrease/decrease of b in y.

Correlation coefficient:

There is a direction, strength, and linear association between x and y.

The ages (in months) and heights (in inches) of seven children are given.

x1624426075102120

y24303540485660

Find the LSRL.

Interpret the slope and correlation coefficient in the context of the problem.

Ans: r = .994,

Correlation coefficient:

There is a strong, positive, linear association between the age and height of children.

Slope:

For an increase in age of one month, there is an approximateincrease of .34 inches in heights of children.

The ages (in months) and heights (in inches) of seven children are given.

x1624426075102120

y24303540485660

Predict the height of a child who is 4.5 years old. (4.5 yrs = 54 months)

Predict the height of someone who is 20 years old. (240 months)

The LSRL should not be used to predict y for values of xoutsidethe data set.

It is unknown whether the pattern observed in the scatterplot continues outside this range.

The ages (in months) and heights (in inches) of seven children are given.

The LSRL is

Can this equation be used to estimate the age of a child who is 50 inches tall?

Calculate: LinReg L2,L1

For these data, this is the best equation to predict y from x.

Do you get the same LSRL?

However, statisticians will always use this equation to predict x from y

The ages (in months) and heights (in inches) of seven children are given.

x1624426075102120

y24303540485660

Calculate x & y.

62.71, 41.86

Plot the point (x, y) on the LSRL.

Will this point always be on the LSRL?

YES!

The correlation coefficient and the LSRL are both non-resistant measures.

The following statistics are found for the variables posted speed limit and the average number of accidents.

Find: the LSRL & predict the number of accidents for a posted speed limit of 50 mph.

(Hint: Find b1, then b0, then LSRL)

Predict the number of accidents for a posted speed limit of 50 mph.

Homework:

- Packet page 64, “Linear Regression Activity”
- Packet page 68