Type II Error

The probability of making a Type II error is denoted as b. The actual value of b is unknown, we can only calculate possible values for b.

Type II Error

Assume we are trying to test to see if the average number of gallons purchased when a driver fills up their tank has fallen. In the past it was 10 gallons and the standard deviation was 4 gallons. A sample of 100 sales is drawn. Set a at .025.

Hypothesis Test with s Known

1. H0: m > 10Ha: m < 10

2. Reject H0 if: z < -1.96Alternatively:Reject H0 if:

216.9100496.110

0

x

x

nzx sm a

Type II Error

What if m really was 9?z = (9.216-9)/.4 = .54b = P(z > .54) = .2946

What if m really was 9.5?z = (9.216-9.5)/.4 = -.71b = P(z > -.71) = .7611

What if m really was 8.5?z = (9.216-8.5)/.4 = 1.79b = P(z > 1.79) = .0367

Type II Error

P. 371-374Non-graded homework:P. 374, #46, 48

Chapter 14

Simple Linear Regression Model

Regression

Used to estimate how much one variable changes with a change in another variable.

Carl Friedrich Gaus

Regression

Dependent variable – The variable whose behavior we are trying to predict.

Independent variable – The variable used to predict the dependent variable.

Temperature and Natural Gas Usage at the

Porter Household

MonthAverage daily temperature

Thousands of cubic feet

Jun-07 66 3.6Jul-07 68 1.8

Aug-07 71 3.7Sep-07 65 2.2Oct-07 61 3.9Nov-07 42 19.3Dec-07 31 25.2Jan-08 29 23.4Feb-08 26 33.7Mar-08 31 27.7Apr-08 51 3.2

May-08 53 4.9Jun-08 67 2.3Jul-08 71 2.1

Aug-08 68 2.7Sep-08 64 2.1Oct-08 52 8.7Nov-08 40 17.3Dec-08 30 31.1Jan-09 19 30.0Feb-09 28 33.9Mar-09 37 21.7Apr-09 49 13.0

May-09 58 4.7Jun-09 65 2.7Jul-09 66 2.6

Aug-09 71 2.3Sep-09 63 2.8Oct-09 48 10.0

Jun-07

Aug-07

Oct-07

Dec-07

Feb-08

Apr-08

Jun-08

Aug-08

Oct-08

Dec-08

Feb-09

Apr-09

Jun-09

Aug-09

Oct-09

0

10

20

30

40

50

60

70

80

Temperature and Natural Gas Consumed

Average daily temperature Thousands of cubic feet

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Monthly Natural Gas Use and Temperature

Average Daily Temperature

Thou

sand

s of c

ubic

feet

Regression

Simple Linear Regression Modely = b0 + b1x + e

Simple Linear Regression Equationy = b0 + b1x

Estimated Simple Linear Regression Equationxbby 10ˆ

Least Squares Criterion 2ˆ ii yymin

xbyb

xx

yyxxb

i

ii

10

21

:equation Intercept

:equation Slope

Excel Regression Output

  CoefficientsIntercept 45.88

X Variable 1 -0.66

xyxbby66.088.45ˆ

ˆ 10

Interpreting the Output

b0 – If the average daily temperature is 0 degrees Fahrenheit the predicted gas usage is 45.88 thousand cubic feet

b1 – A 1 degree increase in the average daily temperature reduces the predicted gas usage by 0.66 thousand cubic feet over a month

Interpreting the OutputWhat is the predicted natural gas usage if the temperature is 10 degrees?45.88 – (10)(0.66) = 39.28

What if the temperature is 50 degrees?45.88 – (50)(0.66) = 12.88

What if the temperature is -10 degrees?45.88 – (-10)(0.66) = 52.48

What if the temperature is 100 degrees?45.88 – (100)(0.66) = -20.12

Computing b0 and b1, Example

Car Age (years) Price ($000)1 1 152 3 143 3 114 4 125 9 8

Computing b0 and b1, Examplex y1 15 -3 3 -9 93 14 -1 2 -2 13 11 -1 -1 1 14 12 0 0 0 09 8 5 -4 -20 25

Sum = 20 60 -30 36Mean = 4 12

b1 = -0.83b0 = 15.33

)( xxi )( yyi 2)( xxi ))(( yyxx ii

Coefficient of Determination

The portion of the variation in the data explained by the regression model

Total Sum of Squares

The measure of the total variation in the data.

2 yySST i

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Monthly Natural Gas Use and Temperature

Average Daily Temperature

Thou

sand

s of c

ubic

feet

Sum of Squares Due to Regression

The measure of the variation explained by the regression line.

2ˆ yySSR i

Sum of Squares Due to Error

The measure of the variation left unexplained by the regression line.

2ˆ ii yySSE

Total Sum of Squares

The total sum of squares equals the sum of squares due to regression plus the sum of squares due to error.

SST = SSR + SSE

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Monthly Natural Gas Use and Temperature

Average Daily Temperature

Thou

sand

s of c

ubic

feet

Unexplained

Explained

ii yy ˆ

yyi ˆ

Coefficient of Determinination

The share of the variation explained by the regression line.

r2 = SSR/SST

Excel Regression OutputRegression Statistics

Multiple R 0.953885R Square 0.909896Adjusted R Square 0.906559Standard Error 3.512402Observations 29

ANOVA

  df SS MS FSignificance 

FRegression 1 3363.7 3363.7 272.7 1.23E-15Residual 27 333.1 12.3Total 28 3696.8

3363.7/3696.8 = 0.9099

Sample Correlation Coefficient

954.09099.1

2

xy

xy

r

rr 1b of sign

Coefficient of Determinationx y SSR SSE SST1 15 14.5 6.2 0.3 93 14 12.84 0.7 1.3 43 11 12.84 0.7 3.4 14 12 12.01 0.0 0.0 09 8 7.86 17.4 0.0 16

Sum=20 Sum=60 25.0 5.0 30Mean=4 Mean=12

b1=-0.833b0=15.33

r2 = 25/30 = .833

y

Model Assumptions1. The error term e is a random variable with

an expected value of 02. The variance of e is the same for all values

of x.3. The values of e are independent4. The error term e is a normally distributed

random variable