nolanheinzen3e

Chapter 16

16.1 Simple linear regression is a statistical tool that lets statisticians predict the score on a scale dependent variable from the score on a scale independent variable.
16.2 The regression line allows us to make predictions about one variable based on what we know about another variable. It gives us a visual representation of what we believe is the underlying relation between the variables, based on the data we have available to us.
16.3
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16.4
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16.5

a. The y intercept, 2.586, is the GPA we might expect if someone played no minutes. The slope of 0.016 is the increase in GPA that we would expect for each one-minute increase in playing time. Because the correlation is positive, it makes sense that the slope is also positive.

b. The standardized regression coefficient is equal to the correlation coefficient for simple linear regression, 0.344. We can also check that this is correct by computing β:

X	(X − M_X)	(X − M_X)²	Y	(Y − M_Y)	(Y − M_Y)²
29.70	13.159	173.159	3.20	0.343	0.118
32.14	15.599	243.329	2.88	0.023	0.001
32.72	16.179	261.760	2.78	−0.077	0.006
21.76	5.219	27.238	3.18	0.323	0.104
18.56	2.019	4.076	3.46	0.603	0.364
16.23	−0.311	0.097	2.12	−0.737	0.543
11.80	−4.741	22.477	2.36	−0.497	0.247
6.88	−9.661	93.335	2.89	0.033	0.001
6.38	−10.161	103.246	2.24	−0.617	0.381
15.83	−0.711	0.506	3.35	0.493	0.243
2.50	−14.041	197.150	3.00	0.143	0.020
4.17	−12.371	153.042	2.18	−0.677	0.458
16.36	−0.181	0.033	3.50	0.643	0.413
Σ(X − M_X)² = 1279.448			Σ(Y − M_Y)² = 2.899

There is some difference due to rounding decisions, but in both cases, these numbers would be expressed as 0.34.

c. Strong correlations indicate strong linear relations between variables. Because of this, when we have a strong correlation, we have a more useful regression line, resulting in more accurate predictions.
d. According to the hypothesis test for the correlation, this r value, 0.34, fails to reach statistical significance. The critical values for r with 11 degrees of freedom at a p level of 0.05 are − 0.553 and 0.553. In this chapter, we learned that the outcome of hypothesis testing is the same for simple linear regression as it is for correlation, so we know we also do not have a statistically significant regression.

16.6 The standard error of the estimate is a statistic that indicates the typical distance between a regression line and the actual data points. When we do not have enough information to compute a regression equation, we often use the mean as the “best guess.” The error of prediction when the mean is used is typically greater than the standard error of the estimate.
16.7 Strong correlations mean highly accurate predictions with regression. This translates into a large proportionate reduction in error.

16.8

X	Y	(M_Y) MEAN FOR Y	(Y − M_Y) ERROR	(Y − M_Y)² SQUARED ERROR
5	6	5.333	0.667	0.445
6	5	5.333	−0.333	0.111
4	6	5.333	0.667	0.445
5	6	5.333	0.667	0.445
7	4	5.333	−1.333	1.777
8	5	5.333	−0.333	0.111
			SS_total = Σ(Y − M_Y)² = 3.334

b.
c. We have reduced error from 3.334 to 1.322, which is a reduction of 2.012. Now we calculate this reduction as a proportion of the total error:

This can also be written as:

We have reduced 0.603, or 60.3%, of error using the regression equation as an improvement over the use of the mean as our predictor.
d. r²=(−0.77)(−0.77)=0.593, which closely matches our calculation of r² above, 0.603. These numbers are slightly different due to rounding decisions.

16.9 Tell Coach Parcells that prediction suffers from the same limitations as correlation. First, just because two variables are associated doesn’t mean one causes the other. This is not a true experiment, and if we didn’t randomly assign athletes to appear on a Sports Illustrated cover or not, then we cannot determine whether a cover appearance causes sporting failure. Moreover, we have a limited range; by definition, those lauded on the cover are the best in sports. Would the association be different among those with a wider range of athletic ability? Finally, and most important, there is the very strong possibility of regression to the mean. Those chosen for a cover appearance are at the very top of their game. There is nowhere to go but down, so it is not surprising that those who merit a cover appearance would soon thereafter experience a decline. There’s likely no need to avoid that cover, Coach.
16.10 Multiple regression is a statistical tool that predicts a dependent variable by using two or more independent variables as predictors. It is an improvement over simple linear regression, which only allows one independent variable to inform predictions.
16.11
16.12
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16.13
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- c. The negative sign in the slope (−0.072) tells us that those with higher levels of admiration for Pamela Anderson tend to have lower GPAs, and those with lower levels of admiration for Pamela Anderson tend to have higher GPAs.