Statistics Class 8
2/13/2013
Quiz 18
Listed below are brain sizes (in 𝑐𝑚3 ) and
Wechsler IQ scores of subjects. Is there
sufficient evidence to conclude that there is a
linear correlation between brain size and IQ
Score? Does it appear that people with larger
brains are more intelligent?
Brain 965
1029
1030
1285
1049
1077
1037
1068
1176
1105
IQ
85
86
102
103
97
124
125
102
114
90
Regression
Given a collection of sample data, the regression equation
𝑦 = 𝑏𝑜 + 𝑏1 𝑥
Algebraically describes the relationship between the two
variables x and y. The graph of the regression equation is
called the regression line (or line of best fit, or least-squares
line).
x is called the predictor variable, or Independent variable.
𝑦 is called the response variable, or dependent variable
Regression
Given a collection of sample data, the regression equation
𝑦 = 𝑏𝑜 + 𝑏1 𝑥
Algebraically describes the relationship between the two
variables x and y. The graph of the regression equation is
called the regression line (or line of best fit, or least-squares
line).
x is called the explanatory variable, or independent variable.
𝑦 is called the response variable, or dependent variable
The regression equation should remind you of
𝑦 = 𝑚𝑥 + 𝑏
Regression
Requirements
1. The sample of paired (x,y) data is a random sample of
quantitative data.
2. Visual examination of the scatterplot shows that the
points approximate a straight line pattern.
3. Outliers can have a strong effect on the regression, so
remove any outliers that are known to be errors. Consider
the effects of any outliers that are not known errors
Regression
Formulas
• 𝑏1 =
𝑛
𝑥𝑦 −( 𝑥)( 𝑦)
𝑛 𝑥 2 −( 𝑥)2
• 𝑏0 =
( 𝑦)
𝑛
𝑥 2 −( 𝑥)( 𝑥𝑦)
𝑥 2 −( 𝑥)2
𝑠𝑦
• 𝑏1 = 𝑟 , where 𝑠𝑦 is the standard deviation of the y values
𝑠𝑥
and 𝑠𝑥 is the standard deviation of the x’s
• 𝑏0 = 𝑦 − 𝑏1 𝑥
Regression
Formulas
• 𝑏1 =
𝑛
𝑥𝑦 −( 𝑥)( 𝑦)
𝑛 𝑥 2 −( 𝑥)2
• 𝑏0 =
( 𝑦)
𝑛
𝑥 2 −( 𝑥)( 𝑥𝑦)
𝑥 2 −( 𝑥)2
𝑠𝑦
• 𝑏1 = 𝑟 , where 𝑠𝑦 is the standard deviation of the y values
𝑠𝑥
and 𝑠𝑥 is the standard deviation of the x’s
• 𝑏0 = 𝑦 − 𝑏1 𝑥
Rounding
Round to 3 significant digits.
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
We are using the Pizza/Subway data from last time. Where
pizza is the x variable (explanatory) and the subway fare is
the y variable (response).
1. Put the data into the Calculator.
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
We are using the Pizza/Subway data from last time. Where
pizza is the x variable (explanatory) and the subway fare is
the y variable (response).
1. Put the data into the Calculator.
2. Hit the [STAT] button, then [→], then [4] or select
LinReg(ax+b).
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
We are using the Pizza/Subway data from last time. Where
pizza is the x variable (explanatory) and the subway fare is
the y variable (response).
1. Put the data into the Calculator.
2. Hit the [STAT] button, then [→], then [4] or select
LinReg(ax+b).
3. Skip down to Calculate and hit [ENTER].
For us a is 𝒃𝟏 and b is 𝒃𝟎 .
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
1. Put the data into the Calculator.
2. Hit the [STAT] button, then [→], then [4] or select
LinReg(ax+b).
3. Skip down to Calculate and hit [ENTER].
For us a is 𝒃𝟏 and b is 𝒃𝟎 .
4. Make a scatter plot.
Regression
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
0.15
0.35
1.00
1.35
1.50
2.00
1. Put the data into the Calculator.
2. Hit the [STAT] button, then [→], then [4] or select
LinReg(ax+b).
3. Skip down to Calculate and hit [ENTER].
For us a is 𝒃𝟏 and b is 𝒃𝟎 .
4. Make a scatter plot.
5. Press [Y=] with the flashing box in the the 𝑌1 spot, hit the
[VARS] button then [5] then [→] twice so that you are in
the EQ menu. Press [1] or select REGEQ.
6. Press [GRAPH].
Regression
Using the Regression Equation for Predictions
1. Use the regression equation only if the graph of the
regression line on the scatterplot fits the data reasonably
well.
Regression
Using the Regression Equation for Predictions
1. Use the regression equation only if the graph of the
regression line on the scatterplot fits the data reasonably
well.
2. Use the regression equation only if there is a linear
correlation.
Regression
Using the Regression Equation for Predictions
1. Use the regression equation only if the graph of the
regression line on the scatterplot fits the data reasonably
well.
2. Use the regression equation only if there is a linear
correlation.
3. Use the regression equation for predictions that do not go
very far past the sample data.
Regression
Using the Regression Equation for Predictions
1. Use the regression equation only if the graph of the
regression line on the scatterplot fits the data reasonably
well.
2. Use the regression equation only if there is a linear
correlation.
3. Use the regression equation for predictions that do not go
very far past the sample data.
4. If the regression equation does not appear to be useful for
making predictions the best predicted value of a variable
is its point estimate, which is its sample mean.
Regression
Using the Regression Equation for Predictions
1. Use the regression equation only if the graph of the
regression line on the scatterplot fits the data reasonably
well.
2. Use the regression equation only if there is a linear
correlation.
3. Use the regression equation for predictions that do not go
very far past the sample data.
4. If the regression equation does not appear to be useful for
making predictions the best predicted value of a variable
is its point estimate, which is its sample mean.
Try!
The paired value of the Consumer Price Index
(CPI) and the cost of subway fare are listed
below. Find the best predicted cost of Subway
fare when the Consumer Price index is 182.5
CPI
30.2
48.3
112.3
162.2
191.9
197.8
Subway
Fare
0.15
0.35
1.00
1.35
1.50
2.00
Try Some more!
Find the best predicted temperature (in °F) at a
time when cricket chirps 3000 times in one
minute. What is wrong with this predicted
value?
Chirps
882
1188 1104
864
1200 1032
960
900
Temp
(°F)
69.7
93.3
76.3
88.6
71.6
79.6
84.3
82.6
HOMEWORK!!
• 10-3: 1-9 odd, 13-17 odd, 23, 25, 27.