Variation and Prediction Intervals Section 10.4 Variation and Prediction Intervals

Introducing … TOPIC ONE

Deviation vs. Variation Can you measure the deviation and the variation for a pair of (x, y) values? If so, what’s the difference?

Not sure? Let’s Clarify! Total Deviation: y – y (given value for y minus the average value for y) Explained Deviation: y – y (predicted value for y minus the average value for y). Unexplained Deviation: y – y (given value for y minus the predicted value for y).

Let’s Practice You are given the following: The equation of the regression line is y = 3 + 2x The mean of the y-values is 9. One of the pairs of sample data is (5, 19). Find the total deviation, explained deviation, and unexplained deviation.

If we sum the squares of deviation values we get amounts of variation. The connection The total variation is the sum of the squares of the total deviation values, the explained variation is the sum of the squares of the explained deviation values, and the unexplained variation is the sum of the squares of the unexplained deviation values. If we sum the squares of deviation values we get amounts of variation.

The connection (Total variation) = (explained variation) + (unexplained variation)

One Step Further The coefficient of determination is the amount of the variation in y that is explained by the regression line. r² = Explained variation Total variation

How do we use this now? In Section 10-2 we used paired subway and pizza costs in NY to determine the correlation coefficient r = 0.988. Find the coefficient of determination and then use this to find the percentage of total variation that can be explained by the linear relationship between the cost of a slice of pizza and the cost of subway fare.

What are two things that you wish there was a correlation between? Intermission What are two things that you wish there was a correlation between?

Introducing … TOPIC TWO

Definitions A prediction interval is an interval estimate of a predicted value y. An interval estimate of a parameter (such as the mean of all subway fares) is referred to as a confidence interval, but an interval estimate of a variable (such as the predicted subway fare) is called a prediction interval.

Definitions The standard error of estimate is a collective measure of the spread of the sample points about the regression line. It is a measure of the differences between observed sample y-value and the predicted y that are obtained from the regression equation.

Practice – By Hand Find the standard error of estimate for the paired pizza/subway fare data listed below. Cost of Pizza $0.15 $0.35 $1.00 $1.25 $1.75 $2.00 Subway Fare $1.35 $1.50

Practice – By Calc Find the standard error of estimate for the paired pizza/subway fare data listed below. Enter paired data in L1 and L2, then press STAT and select TESTS, and then choose the option LinRegTTest. For Xlist enter L1, for Ylist enter L2, use a Freq value of 1, and select ≠0. Cost of Pizza $0.15 $0.35 $1.00 $1.25 $1.75 $2.00 Subway Fare $1.35 $1.50

Now what do we use this for? Once you have found the standard error estimate, you can use it to find the prediction interval for an individual y. Prediction Interval for an Individual y Given the fixed value xo, the prediction interval for an individual y is…

How Do We Find E? The Margin of Error Formula:

Final Practice Construct a 95% confidence interval for the cost of a subway fare, given that a slice of pizza costs $2.25. Cost of Pizza $0.15 $0.35 $1.00 $1.25 $1.75 $2.00 Subway Fare $1.35 $1.50

Homework Pg. 557-559 #1, 3-6, 13, 17 Extra Practice Pg. 557-559 #14, 18