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.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 = (5) = 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 μ y = α + βx* The regression model says that μ y 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 μ y. 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* SE μ 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* SE y The standard error for prediction SE y is SE y = 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.The observations are independent. In particular, repeated observations on the same individual are not allowed. The true relationship is linear.The true relationship is linear. we almost never see a perfect straight- line 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 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 least- squares line and the residual, which show the variation of the response about the fitted line.The response varies normally about the true regression line. We can’t observe the true regression line. We can observe the least- squares line and the residual, which show the variation of the response about the fitted line.