11+Regression

# The 1 prediction interval for an individual new value

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Unformatted text preview: th, which represents the proportion of the total variability around y that is ¯ explained by the regression line. The coeﬃcient of determination is the proportion of total sample variability explained by the linear relationship SSyy − SSE SSE =1− SSyy SSyy and it is equal to the square of the simple linear coeﬃcient of correlation r: r2 = 1 − SSE =1− SSyy (yi − yi )2 ˆ (yi − yi )2 ¯ Remark. 1. A high correlation does not necessarily imply that a causal relationship exists between x and y - only that a linear trend may exist; 2. A low correlation does not necessarily imply that x and y are unrelated - only that x and y are not strongly linearly related. Prediction. Let we need to predict the future possible mean value of y for a speciﬁc value of x, say xp . The (1 − α)% conﬁdence interval for mean value of y at the point x = xp is deﬁned as y ± tα/2 · s · ˆ 1 (xp − x)2 ¯ + , n SSxx ˆ where y = β0 + β1 xp , tα/2 is based on (n − 2) degree of freedom. ˆˆ The (1 − α)% prediction interval for an individual new value of y at the point x = xp is y ± tα/2 · s · ˆ 1+ ¯ 1 (xp − x)2 + , n SSxx ˆ where y = β0 + β1 x, tα/2 is based on (n − 2) degree of freedom. ˆˆ 7 A. Zhensykbaev Regression Remark. The term &quot;prediction interval&quot; is used when the interval formed is intended to enclose the value of a random variable. The term &quot;conﬁdence interval&quot; is reserved for estimation of population parameters. Example 4. Refer to example 1. a. Find 95% conﬁdence interval for mean monthly sales when the appliance store spends \$600 on advertaising. b. Using 95% prediction interval predict the monthly sales for next month, if \$600 is spent on advertaising. Solution. For \$600 advertaising x = xp = 6 and y = 0.7x − 0.1. Then y = 0.7 · 6 − 0.1 = 4.1. ˆ a. The 95% conﬁdence interval for the mean value of y at the point x = xp = 6 is (α = 0.05) 4.1 ± 3.182 · 0.605 · 1 (6 − 3)2 + = 4.1 ± 2.019. 5 10 Thus, the expected mean value of sales revenue will belong to the interval [\$2081,...
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## This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.

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