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Explore creating confidence intervals and prediction intervals for data predictions. Learn about fitting lines, estimating responses, and checking regression conditions. Understand the importance of conditions in making accurate predictions in statistics.
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P. STATISTICSLESSON 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 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 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.
