http://www.ck12.org Chapter 9. Regression and CorrelationYˆ=the predicted score
tcv=critical value oftford f(n− 2 )
sY=standard error of the predicted score
Example:
Develop a 95% confidence interval for the predicted scores from a student that scores a 4 on the short physical fitness
exam(X= 4 ).
Solution:
We calculate the standard error of the predicted value using the formula:sYˆ=sY∗X√
1 +
1
n+
(X−X ̄)^2
SSx= 0. 56
√
1 +
1
24
+
( 4 − 4. 92 )^2
123. 83
= 0. 57
Using the general formula for the confidence interval, we find thatCI=Yˆ±(tcvsY)
CI 95 = 3. 76 ±( 2. 074 )( 0. 57 )
CI 95 = 3. 76 ± 1. 18
CI 95 = ( 2. 58 , 4. 94 )
2. 58 <CI 95 < 4. 94 )Therefore, we can say that we are 95% confident that given a students’ short physical fitness test score(X)of 4, the
interval from 2.58 to 4.94 will contain the students’ score for the longer physical fitness test.Regression Assumptions
We make several assumptions under a linear regression model including:1.At each value of X,there is a distribution of Y. These distributions have a mean centered around the predicted
value and a standard error that is calculated using the sum of squares.
2.The best regression model is a straight line. Using a regression model to predict scores only works if the
regression line is a straight line. If this relationship is non linear, we could either transform the data (i.e.,
a logarithmic transformation) or try one of the other regression equations that are available with Excel or a
graphing calculator.
3.Homoscedasticity. The standard deviations, or the variances, of each of these distributions for each of the
predicted values is equal.
4.Independence of observation.For each give value ofX, the values ofYare independent of each other.Lesson Summary
- When we estimate a linear regression model, we want to ensure that the regression coefficient in the population
(β)does not equal zero. To do this, we perform ahypothesis testwhere we set the regression coefficient equal
to zero and test forsignificance.