Transcript P. STATISTICS LESSON 14 – 2 ( DAY 2)

A. P. STATISTICS
LESSON 14 – 2
( DAY 2)
PREDICTIONS AND CONDITIONS
ESSENTIAL QUESTIONS: What
are the conditions that must be in
place in order to make
predictions?
Objective:
•To create confidence intervals and
prediction intervals for a single
observation.
Predictions and conditions
One of the most common reasons to fit a
line to data is to predict the response to
a particular value of the explanatory
variable.
y^ = -0.0127 + 0.0180(5) = 0.077
Do you want to predict the BAC of one
individual student who drink 5 beers and
all students who drink 5 beers.
Predictions and Conditions cont.
The margin of error is different for the two
kinds of prediction. Individual students
who drink 5 beers don’t all have the
same BAC. So we need a larger
margin of error to pin down one
student’s who have 5 beers.
Prediction and confidence
intervals
To estimate the mean response, we use a
confidence interval. It is an ordinary
interval for the parameter
μy = α + βx*
The regression model says that μy is the
mean of response y when x has the
value x*. It is a fixed number whose
value we don’t know.
Prediction interval
To estimate an individual response y, we
use a prediction interval. A prediction
interval estimates a single random
response y rather than a parameter like
μy. The response y is not a fixed
number. If we took more observations
with x = x*, we would get different
responses
Confidence intervals for
regression response
A level C confidence interval for the mean
response μ when x takes the value x* is
y^ ± t* SEμ
The standard error SE is
SE = s √ 1/n + (x* - x)2/ ∑(x - x)2
The sum runs over all the observations on
the explanatory variable x.
Prediction intervals for regression
response
A level C prediction interval for a single
observation on y when x takes the value x* is
y = t* SEy
The standard error for prediction SEy is
SEy = s√ 1 + 1/n + (x* - x)2/ ∑(x - x)2
In both recipes, t* is the upper (1-C)/2 critical
value of the t distribution with
n – 2 degrees of freedom.
Example 14.7
Predicting Blood Alcohol
Page 798
Look at minitab.
Checking the regression
conditions
If the scatterplot doesn’t show a roughly linear
pattern, the fitted line may be almost useless.
• The observations are independent.
In particular, repeated observations on the
same individual are not allowed.
• The true relationship is linear.
we almost never see a perfect straightline relationship in our data.
Checking the regression
conditions continued
• The standard deviation of the response about
the true line is the same everywhere. Look at
the scatterplot again. The scatter of the data
points about the line should be roughly the
same over the entire range of the data.
• The response varies normally about the true
regression line. We can’t observe the true
regression line. We can observe the leastsquares line and the residual, which show the
variation of the response about the fitted line.