11+Regression

multiple linear regression most practical

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Unformatted text preview: $6119] with probability P = (1 − α) = 0.95. b. The 95% prediction interval for an individual new value of y at the point x = xp = 6 is 4.1 ± 3.182 · 0.605 · 1+ 1 (6 − 3)2 + = 4.1 ± 2.79. 5 10 Therefore, we predict with 95% confidence that the sales revenue next month will fall in the interval from $1310 to $6890. Confidence strip for the mean value of y . The confidence interval for the mean value of y gives the confidence strip for the graph y = y + β1 (x − x) ± tα/2 · s · ¯ˆ ¯ 8 ¯ 1 (x − x)2 + . n SSxx A. Zhensykbaev Regression For our case y = 2 + 0.7(x − 3) ± 3.182 · 0.605 · 1 (x − 3)2 + . 5 10 • 6 " " " " " " " " " " " " " " " " " " " " " " " •""" • " " " " " " " " " " " " " " " " " " " " • """• " " " " " " " " " " " " - Pic. 2 Residual analysis. The goal is to know, whether the data indicate significant departures from the assumptions: εi (i = 1 : n) are independent and Estimates for errors are ˆ ˆ εi = yi − (β0 + β1 xi ). ˆ Residuals are εi (i = 1 : n). ˆ To verify our assumptions we have to use statistical tests: for normality and independence (say, chi-square and Spearman’s rank test, runs, etc.). Multiple linear regression. Most practical applications use regression models which are more complex than the simple linear regression. A realistic probabilistic model for monthly sales revenue includes not only the advertising expenditure but also such factors as season, prices, etc. One may model such situations by the multiple linear regression. 9 A. Zhensykbaev Regression The model is k y = β 0 + β 1 x1 + · · · + β k xk + ε = βi xi + ε, i=0 where y is the dependent random variable, βi are parameters (unkn...
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