Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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Section 10.4 Variation and Prediction Intervals

Transcript of Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

Page 1: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

Section 10.4

Variation and Prediction Intervals

Page 2: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

Introducing …

• TOPIC ONE

Page 3: 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?

Page 4: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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).

Page 5: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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.

Page 6: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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.

Page 7: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

The connection

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

Page 8: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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

Page 9: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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.

Page 10: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

Intermission

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

Page 11: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

Introducing …

• TOPIC TWO

Page 12: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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.

Page 13: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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.

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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

$0.15 $0.35 $1.00 $1.35 $1.50 $2.00

Page 15: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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

$0.15 $0.35 $1.00 $1.35 $1.50 $2.00

Page 16: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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 yGiven the fixed value xo, the prediction interval for an

individual y is…

Page 17: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

How Do We Find E?

• The Margin of Error Formula:

Page 18: Section 10.4 Variation and Prediction Intervals. Introducing … TOPIC ONE.

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

$0.15 $0.35 $1.00 $1.35 $1.50 $2.00

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Homework

• Pg. 557-559 #1, 3-6, 13, 17

• Extra Practice Pg. 557-559 #14, 18