**8.9.1**Consider the data in the following table. ... ... Plot yield (

*Y*) against weight (

*W*). Suppose we think that the line

*Y*=

*W*+ 0

*.*6 describes these data. Plot the line on your graph of yield against weight. Find and plot the residuals.

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**8.9.2**Consider the data in the following table. ... ... Plot height (

*H*) against weight (

*W*). Suppose we think that the line

*H*= 8

*W*+ 5 describes the data. Plot the line on your graph of height against weight. Find and plot the residuals.

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**8.9.3**Consider the data in the following table. ... ... Find SSE for the model used in Exercise 1

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**8.9.4**Consider the data in the following table. ... ... Find SSE for the model used in Exercise 2

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**8.9.5**Consider the data in the following table. ... ... Find the null model that best fits

*Y*as a function of

*W*and find SST.

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**8.9.6**Consider the data in the following table. ... ... Find the null model that best fits

*H*as a function of

*W*and find SST.

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**8.9.7**Consider the data in the following table. ... ... Find

*...*for the model in Exercise 1.

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**8.9.8**Consider the data in the following table. ... ... Find

*...*for the model in Exercise 2.

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**8.9.9**Consider the data in the following table. ... ... Compute the best fitting line for yield as a function of weight. Graph the line.

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**8.9.10**Consider the data in the following table. ... ... Compute the best fitting line for height as a function of weight. Graph the line.

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**8.9.11**Consider the data in the following table. ... ... Find SSE for the line in Exercise 9, find

*...*, and compare with the model in Exercise 1.

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**8.9.12**Consider the data in the following table. ... ... Find SSE for the line in Exercise 10, find

*...*, and compare with the model in Exercise 2.

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**8.9.13**Consider the data in the following table. ... ... Find the correlation between weight and yield. Check that its square is equal to the value of

*...*found in Exercise 11.

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**8.9.14**Consider the data in the following table. ... ... Find the correlation between weight and height. Check that its square is equal to the value of

*...*found in Exercise 12.

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**8.9.15**Linear regression has important connections with other techniques in statistics, such as testing whether two populations differ. In the following data set, the independent variable takes on only two values. Find the best fitting line and

*...*, and then test whether the two sets of points differ in their mean distribution using the techniques in Section 8.6. Assume that the data are normally distributed with known variance of 25. In each case, graph the regression line and the data. ... Find the best fitting line and

*...*for replicate 1 and then test whether the diet has a significant effect.

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**8.9.16**Linear regression has important connections with other techniques in statistics, such as testing whether two populations differ. In the following data set, the independent variable takes on only two values. Find the best fitting line and

*...*, and then test whether the two sets of points differ in their mean distribution using the techniques in Section 8.6. Assume that the data are normally distributed with known variance of 25. In each case, graph the regression line and the data. ... Find the best fitting line and

*...*for replicate 2 and then test whether the diet has a significant effect. Compare with the previous problem.

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**8.9.18**Best fit regression lines have many nice properties. Show that the best linear fit from Theorem 8.6 passes through the center of the data in the sense that the sum of the residuals is 0.

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**8.9.18**Best fit regression lines have many nice properties. Show that the best linear fit from Theorem 8.6 passes through the center of the data in the sense that the sum of the residuals is 0.

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**8.9.19**Best fit regression lines have many nice properties. Consider models of the form

*Y*=

*b*. Show that the sum of the squares of the residuals is minimized when

*...*the sample mean of the

*...*

*.*

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**8.9.20**Best fit regression lines have many nice properties. Consider models of the form

*Y*=

*aX*. Find the slope that minimizes the sum of the squares of the residuals.

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**8.9.21**Consider the following measurements. ... Find the best linear fit. Plot the line and find ...How good is the model?

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**8.9.22**Consider the following measurements. ... Use the principle of least squares to write the expression you would use to fit a curve of the form

*...*. One easy way to solve this is to think of a new measurement

*...*and find the linear regression of

*Y*on

*Z*. Plot the linear regression of

*Y*against

*Z*and the curved regression of

*Y*against

*...*. Which model does better?

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**8.9.23**Consider the following measurements. ... Find the dimensions of the slope ...

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**8.9.24**Consider the following measurements. ... Find the dimensions of the intercept ...and check that all the parts of the equation match.

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**8.9.25**Consider the following data describing change in a bacterial population. ... Find

*...*. Graph the data and the line.

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**8.9.26**Consider the following data describing change in a bacterial population. ... Find the best fitting line, and compare with a mathematically idealized model. Which makes more sense ?

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**8.9.27**Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for population 1 as a function of time and

*...*.

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**8.9.28**Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for population 2 as a function of time and compute

*...*

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**8.9.29**Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for the logarithm of population 1 as a function of time and compute

*...*. Is this a better fit?

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**8.9.30**Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for the logarithm of population 2 as a function of time and compute

*...*. Is this a better fit?

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**8.9.31**Consider the following data which include one outlying point. Find the best fitting line with and without that point. How much difference does that point make? The idea of removing one point and testing how much the fit changes is an important tool in regression, and is sometimes called the leverage of that point. ... Find the best fitting line and

*...*for replicate 1 with and without the fourth point. Graph the two regression lines and the data.

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**8.9.32**Consider the following data which include one outlying point. Find the best fitting line with and without that point. How much difference does that point make? The idea of removing one point and testing how much the fit changes is an important tool in regression, and is sometimes called the leverage of that point. ... Find the best fitting line and

*...*for replicate 2 with and without the last point. Graph the two regression lines and the data. Why do you think the outlier affects this regression line more?

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**8.9.33**The following table gives the winning Olympic times for men and women in the 400 m race. ...

a. Use your computer to find the best linear regression for men and for women.

b. Plot the residuals for each. Does the linear model fit well?

c. Predict the times in the 2000 Olympics for women and men. How well did it actually work?

d. Predict when women will outrun men. Do you believe this?

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