Ch14Lesson - Chapter 14 Inference for Regression Section 1...

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Chapter 14 Inference for Regression Section 1 Inference about the Model Example 14.1: Crying and IQ Infants who cry easily may be more easily stimulated than others and this may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants four to ten days old and their later IQ test scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test.
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Least regression line equation: x bx a y 493 . 1 91 ˆ + = + =
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The ’s or predicted values for the IQ, attempts to estimate the actual values of the population’s IQ measured several years later. y ˆ The predictions usually won’t agree exactly with the actual values.
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According to the r 2 , only about 21% of the variation in IQ scores is explained by the crying intensity. The prediction will not be very accurate. We need to do inference procedures on our statistics to assess the confidence level of our prediction and to test how statistically significant our findings are based on a significance level. The regression Model Statistics : slope b and a intercept calculated from the sample , both unbiased estimators. What are our statistics, slope, b and y- intercept, a trying to estimate?
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Parameters : slope β and α intercept calculated from the population . In other words : The sample regression line estimates the population regression line. We could take more samples, calculate more regression lines and would end up with many ’s, one per sample. y ˆ Then, take the “average” of all the ’s and compute the sampling distribution of all the ’s. y ˆ y ˆ This sampling distribution’s mean would attempt to estimate the y μ of the population regression line whose slope and y intercept are and respectively.
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The values of y that we do observe vary about their means according to a normal distribution. If we hold x fixed and take many observations on y , the normal pattern will eventually appear. Let’s say for x = 20 1 ˆ y =91.27+1.49(20)=121.07 2 ˆ y =83.91+1.49(20)=115.51 3 ˆ y =114.72+1.49(20)=120.92 4 ˆ y =94.78+1.49(20)=121.18 5 ˆ y =86.82+1.49(20)=125.82 6 ˆ y =120.72+1.49(20)=125.52 7 ˆ y =108.19+1.49(20)=117.99 8 ˆ y =89.43+1.49(20)=118.43 9 ˆ y =91.41+1.49(20)=123.21 10 ˆ y =115.52+1.49(20)=119.12
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This note was uploaded on 12/09/2011 for the course STAT 101 taught by Professor O during the Fall '08 term at Lake Land.

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Ch14Lesson - Chapter 14 Inference for Regression Section 1...

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