4-25-2011 Prediction Intervals

4-25-2011 Prediction Intervals - 4/25/11 Lecture 24: April...

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•4/25/11 •1 Lecture 24: April 25, 2011 • Chapter 12: – Bottom page 515 - top page 518 • Prediction interval for an individual value of the dependent variable – Middle page 518 - page 522 • Exploring the behavior of the error term in a regression equation • Connect Quiz 4: Chapter 12 – Coverage: page 504-top page 518 – These are topics from last class. – 15 questions; two Minitab runs; calculations based on Minitab results; a lot regarding understanding regression output. – Opens Tuesday, April 26 at 9:00 a.m. – Closes Friday, April 29 at 9:00 a.m. – You have 2 two-hour sessions to complete the quiz. • Next class: Chapter 13, Multiple Regression Returning to the regression model and the estimated regression equation • The population model is: • The estimated regression equation is: First item today: – We will construct an interval around a fitted value or prediction . Second item today: – We will explore the behavior of the estimated error term . i0 1 ii Y= X+  i ˆ Y=b bX iii ˆ eYY  i ˆ Y Characteristics of a single point prediction • A point prediction has limitations: • It can be too high or too low. • It’s much more informative to construct an interval around this single point estimate: – The interval estimate provides a range of values representing possibilities that the item being predicted, i.e., Y i , can be expected to have. i ˆ Y i ˆ Y
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•4/25/11 •2 Constructing a prediction interval for an individual value of Y . • The prediction itself – – is a random variable , and we will construct an interval around it. • There are some rather complicated formulas relating to prediction intervals. We will not go this route. • Instead, we will use an approximation to the true formula. • This is Formula (12.31) on page 517. It is called the Quick Prediction Interval for Individual Y . i ˆ Y Here’s how the prediction interval looks • Quick Prediction Interval for Individual Y: –Formula (12.31) on page 517: –where is the point prediction •t α /2 is the t-value for a (1- α ) prediction interval • s is the standard error of the estimate i/ 2 ˆ Yt s  i ˆ Y An example: Let’s construct a 95% prediction interval for a family’s food expenditures with income $40 (thousand) The givens: Quick Prediction Interval for Individual Y: – Formula (12.31) on page 517: – Making substitutions: 14.12 ± 3.182 (2.82) 14.12 – 3.182 (2.82) to 14.12 + 3.182 (2.82) 14.12 – 8.97 to 14.12 + 8.97 5.15 to 23.09 Interpretation: A family with an annual income of $40,000 can be expected to spend somewhere between $5,150 and $23,090 annually on food with 95% accuracy.
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4-25-2011 Prediction Intervals - 4/25/11 Lecture 24: April...

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