HW06solutions2008 - Homework 6 Stat 408 Winter 2008 1...

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Homework 6 Stat 408 Winter 2008 1. Describe the regression effect. Solution. The regression effect is that an observation on a feature at one point in time---e.g. your test I score---may be a distance—e.g. so many standard deviations from average---is expected to be closer to the subsequent average than at the first measurement. 2. Explain why there is a regression effect. Describe each of the four rationales for this effect. Solution. Again, the regression effect is due to the presence of chance or luck affecting the final observed score. Here are the four rationales for this effect: 1. If you predict a subsequent reading, you would use the actual values. The actual values are, on average, closer to the mean than the measured values. This is due to the fact the variance of the measured values—the average squared distance to the average----is greater than the average squared distance to the average of the actual values. These two facts imply that we expect a subsequent reading to be closer to average that the initial measured value. 2. When the measured values are recorded as thee actual values—no luck involved —your prediction is that a subsequent reading is what the measured value was. On the other hand, when there is no variation in actual values---it’s all luck---then you would use the average as a predictor of a subsequent value. In general, you predict a subsequent reading somewhere between the actual value and the mean. 3. Actual values are more concentrated around the mean than the measured values, and the distribution of measured values is generally more spread out than the distribution of actual values. 4. The fourth rationale is the following formula that gives the magnitude of the regression effet. [E(Y|X)-E(Y)]/SD(Y)=sqrt(R 2 )*[X-E(X)]/SD(X) 3. Are we approaching mediocrity? Explain.
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