I am studying statistics using R and a book "Understandable Statistics", by
Brase and Brase. The book has two
worked examples for calculating a confidence interval around a predicted
value from a linear model. The answers
to the two examples in the book differ from those I get from R. The
regression line, the standard error, and the
predicted value in
R and the book all agree for the examples. Hence I gather that R and the
book use different formula to calculate
the confidence interval. Could someone explain why the difference exists,
and which function(s) in R I might use
to get the answers in the book, and (perhaps) an explanation as to which
method to use in various situations).
The example:
x<-c(10,20,30,40,50,60,70)
y<-c(17,21,25,28,33,40,49)
dat <- data.frame(temp=x,amnt=y)
temp amnt
1 10 17
2 20 21
3 30 25
4 40 28
5 50 33
6 60 40
7 70 49
being a table of temperatures (temp) and the corresponding amounts of copper
sulfate that disolve in 100g of water
at that temperature.
The regression line:
mod <- lm(amnt ~ temp,dat)
summary(mod)
Call:
lm(formula = amnt ~ temp, data = dat)
Residuals:
1 2 3 4 5 6 7
1.7857 0.7143 -0.3571 -2.4286 -2.5000 -0.5714 3.3571
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.14286 1.98463 5.111 0.00374 **
temp 0.50714 0.04438 11.428 8.98e-05 ***
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: 2.348 on 5 degrees of freedom
Multiple R-Squared: 0.9631, Adjusted R-squared: 0.9558
F-statistic: 130.6 on 1 and 5 DF, p-value: 8.985e-05
The .95 confidence interval for a temperature of 45 degrees:
foo<-predict(mod,data.frame(temp=45),level=.95,interval="confidence",se.fit=
T)
$fit
fit lwr upr
[1,] 32.96429 30.61253 35.31604
$se.fit
[1] 0.9148715
$df
[1] 5
$residual.scale
[1] 2.348252
The book gives the confidence interval as 26.5 <= y <= 39.5. The book
defines the confidence interval calculation thus:
yp - E <= y <= yp + E
Where
E = tc*sC *sqrt(1 + 1/n + (x-xBar)^2/SSx)
yp is the predicted value from the regression line
tc is the value from Student's t distribution for a confidence
level, c, using n-2 degrees of freedom,
sC is the standard error of estimate
SSx is Sum(x^2)-[Sum(x)]^2/n
n is the number of data pairs.
So that even though the model, predicted value, standard error all agree, R
gives a much smaller confidence
interval than the book does.
Thanks for any advice/help.
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