type ii error
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/1.jpg)
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.
![Page 2: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/2.jpg)
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.
![Page 3: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/3.jpg)
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
![Page 4: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/4.jpg)
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
![Page 5: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/5.jpg)
Type II Error
P. 371-374Non-graded homework:P. 374, #46, 48
![Page 6: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/6.jpg)
Chapter 14
Simple Linear Regression Model
![Page 7: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/7.jpg)
Regression
Used to estimate how much one variable changes with a change in another variable.
Carl Friedrich Gaus
![Page 8: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/8.jpg)
Regression
Dependent variable – The variable whose behavior we are trying to predict.
Independent variable – The variable used to predict the dependent variable.
![Page 9: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/9.jpg)
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
![Page 10: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/10.jpg)
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
![Page 11: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/11.jpg)
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
![Page 12: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/12.jpg)
Regression
Simple Linear Regression Modely = b0 + b1x + e
Simple Linear Regression Equationy = b0 + b1x
Estimated Simple Linear Regression Equationxbby 10ˆ
![Page 13: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/13.jpg)
Least Squares Criterion 2ˆ ii yymin
xbyb
xx
yyxxb
i
ii
10
21
:equation Intercept
:equation Slope
![Page 14: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/14.jpg)
Excel Regression Output
CoefficientsIntercept 45.88
X Variable 1 -0.66
xyxbby66.088.45ˆ
ˆ 10
![Page 15: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/15.jpg)
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
![Page 16: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/16.jpg)
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
![Page 17: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/17.jpg)
Computing b0 and b1, Example
Car Age (years) Price ($000)1 1 152 3 143 3 114 4 125 9 8
![Page 18: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/18.jpg)
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
![Page 19: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/19.jpg)
Coefficient of Determination
The portion of the variation in the data explained by the regression model
![Page 20: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/20.jpg)
Total Sum of Squares
The measure of the total variation in the data.
2 yySST i
![Page 21: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/21.jpg)
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
![Page 22: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/22.jpg)
Sum of Squares Due to Regression
The measure of the variation explained by the regression line.
2ˆ yySSR i
![Page 23: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/23.jpg)
Sum of Squares Due to Error
The measure of the variation left unexplained by the regression line.
2ˆ ii yySSE
![Page 24: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/24.jpg)
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
![Page 25: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/25.jpg)
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
![Page 26: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/26.jpg)
Coefficient of Determinination
The share of the variation explained by the regression line.
r2 = SSR/SST
![Page 27: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/27.jpg)
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
![Page 28: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/28.jpg)
Sample Correlation Coefficient
954.09099.1
2
xy
xy
r
rr 1b of sign
![Page 29: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/29.jpg)
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
![Page 30: Type II Error](https://reader036.vdocuments.us/reader036/viewer/2022081515/56816665550346895dd9f95f/html5/thumbnails/30.jpg)
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