A.A. P. STATISTICSP. STATISTICSLESSON 14 – 2LESSON 14 – 2

( DAY 2)( DAY 2)

PREDICTIONS AND CONDITIONSPREDICTIONS AND CONDITIONS

ESSENTIAL QUESTIONS: What ESSENTIAL QUESTIONS: What are the conditions that must be in are the conditions that must be in

place in order to make place in order to make predictions?predictions?

Objective:Objective:•To create confidence intervals and To create confidence intervals and prediction intervals for a single prediction intervals for a single observation.observation.

Predictions and conditionsPredictions and conditions

One of the most common reasons to fit a One of the most common reasons to fit a line to data is to predict the response to line to data is to predict the response to a particular value of the explanatory a particular value of the explanatory variable.variable.

y = -0.0127 + 0.0180(5) = 0.077y = -0.0127 + 0.0180(5) = 0.077Do you want to predict the BAC of one Do you want to predict the BAC of one

individual student who drink 5 beers and individual student who drink 5 beers and all students who drink 5 beers.all students who drink 5 beers.

^

Predictions and Conditions cont.Predictions and Conditions cont.

The margin of error is different for the two The margin of error is different for the two kinds of prediction. Individual students kinds of prediction. Individual students who drink 5 beers don’t all have the who drink 5 beers don’t all have the same BAC. So we need a larger same BAC. So we need a larger margin of error to pin down one margin of error to pin down one student’s who have 5 beers.student’s who have 5 beers.

Prediction and confidence Prediction and confidence intervalsintervals

To estimate the mean response, we use a To estimate the mean response, we use a confidence interval. It is an ordinary confidence interval. It is an ordinary interval for the parameterinterval for the parameter

μμyy = = αα + + ββx*x*

The regression model says that The regression model says that μμy y is the is the mean of response y when x has the mean of response y when x has the value x*. It is a fixed number whose value x*. It is a fixed number whose value we don’t know. value we don’t know.

Prediction intervalPrediction interval

To estimate an individual response y, we To estimate an individual response y, we use a prediction interval. A prediction use a prediction interval. A prediction interval estimates a single random interval estimates a single random response y rather than a parameter like response y rather than a parameter like μμyy. The response y is not a fixed . The response y is not a fixed

number. If we took more observations number. If we took more observations with x = x*, we would get different with x = x*, we would get different responses responses

Confidence intervals for Confidence intervals for regression responseregression response

A level C confidence interval for the mean A level C confidence interval for the mean response response μμ when x takes the value x* is when x takes the value x* is

y ± t* SEy ± t* SEμμ

The standard error SE is The standard error SE is SE = s √ 1/n + (x* - x)SE = s √ 1/n + (x* - x)22/ ∑(x - x)/ ∑(x - x)22

The sum runs over all the observations on The sum runs over all the observations on the explanatory variable x. the explanatory variable x.

^

Prediction intervals for regression Prediction intervals for regression responseresponse

A level C prediction interval for a single A level C prediction interval for a single observation on y when x takes the value x* is observation on y when x takes the value x* is

y = t* SEy = t* SEyy

The standard error for prediction SEThe standard error for prediction SEy y isis

SESEyy = s√ 1 + 1/n + (x* - x) = s√ 1 + 1/n + (x* - x)22/ ∑(x - x)/ ∑(x - x)22

In both recipes, t* is the upper (1-C)/2 critical In both recipes, t* is the upper (1-C)/2 critical value of the t distribution with value of the t distribution with

n – 2 degrees of freedom.n – 2 degrees of freedom.

Example 14.7Example 14.7

Predicting Blood AlcoholPredicting Blood Alcohol

Page 798Page 798

Look at minitab.Look at minitab.

Checking the regression Checking the regression conditionsconditions

If the scatterplot doesn’t show a roughly linear If the scatterplot doesn’t show a roughly linear pattern, the fitted line may be almost useless.pattern, the fitted line may be almost useless.

• The observations are independent.The observations are independent.

In particular, repeated observations on In particular, repeated observations on the same individual are not allowed.the same individual are not allowed.

• The true relationship is linear.The true relationship is linear.

we almost never see a perfect straight-we almost never see a perfect straight-line relationship in our data. line relationship in our data.

Checking the regression Checking the regression conditions continuedconditions continued

• The standard deviation of the response about The standard deviation of the response about the true line is the same everywhere. Look at the true line is the same everywhere. Look at the scatterplot again. The scatter of the data the scatterplot again. The scatter of the data points about the line should be roughly the points about the line should be roughly the same over the entire range of the data.same over the entire range of the data.

• The response varies normally about the true The response varies normally about the true regression line. We can’t observe the true regression line. We can’t observe the true regression line. We can observe the least-regression line. We can observe the least-squares line and the residual, which show the squares line and the residual, which show the variation of the response about the fitted line.variation of the response about the fitted line.