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Chapter 11: Q11E (page 753)

Question:For the conditions of Exercise \({\bf{7}}\), and the data in Table \({\bf{11}}{\bf{.1}}\), determine the value of \({{\bf{R}}^{\bf{2}}}\), as defined by Eq.\({\bf{(11}}{\bf{.5}}{\bf{.26)}}\).

Short Answer

Expert verified

The value of \({R^2}\)is \(0.6440\).

Step by step solution

01

Define linear statistical models

A linear model describes the correlation between the dependent and independent variables as a straight line.

\(y = {a_0} + {a_1}{x_1} + {a_2}{x_2} + \tilde A,\hat A1/4 + {a_n}{x_n}\)

Models using only one predictor are simple linear regression models. Multiple predictors are used in multiple linear regression models. For many response variables, multiple regression analysis models are used.

02

Find the value of \({{\bf{R}}^{\bf{2}}}\)

For the model of one of the previous exercises and the given data, we need to determine the value of multiple \({R^2}\).

It can be calculated as

For this we have fitted the regression function,

\(y = - 0.745 + 0.619x + 0.013{x^2}\)

Now we can find the mean.

Now we can calculate that

which means

\(\begin{array}{c}{R^2} = 1 - \frac{{9.366}}{{26.309}}\\ = 0.6440{\rm{\;or\;}}64.40{\rm{\% }}\end{array}\)

Hence the value of \({R^2}\)is \(0.6440\).

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Most popular questions from this chapter

Suppose that each of two different varieties of corn is treated with two different types of fertilizer in order to compare the yields, and that \(K\)independent replications are obtained for each of the four combinations. Let \({X_{ijk}}\)denote the yield on the \(K\)The replication of the combination of variety \(i\) with fertilizer\(j(i = 1,2;j = 1,2\);\(k = 1, \ldots ,K\)). Assume that all the observations are independent and normally distributed, each distribution has the same unknown variance, and \({\bf{E}}\left( {{{\bf{X}}_{{\bf{ijk}}}}} \right){\bf{ = }}{{\bf{\mu }}_{{\bf{ij}}}}\)for \(k = 1, \ldots ,K.\) Explain in words what the following hypotheses mean, and describe how to carry out a test of them:

\({{\bf{H}}_{\bf{0}}}{\bf{:}}\;\;\;{{\bf{\mu }}_{{\bf{11}}}}{\bf{ - }}{{\bf{\mu }}_{{\bf{12}}}}{\bf{ = }}{{\bf{\mu }}_{{\bf{21}}}}{\bf{ - }}{{\bf{\mu }}_{{\bf{22}}}}{\bf{, }}\)

\({H_1}\): The hypothesis \({H_0}\) is not true.

Question:For the conditions of Exercise \({\bf{12}}\), and the data in Table \({\bf{11}}{\bf{.2}}\), carry out a test of the following hypotheses.

\(\begin{array}{*{20}{c}}{{{\bf{H}}_{\bf{0}}}{\bf{:}}{{\bf{\beta }}_{\bf{2}}}{\bf{ = - 1}}}\\{{{\bf{H}}_{\bf{1}}}{\bf{:}}{{\bf{\beta }}_{\bf{2}}} \ne {\bf{ - 1}}}\end{array}\)

Prove that \({{\bf{\sigma }}^{{\bf{'2}}}}\), defined in Eq.\({\bf{(11}}{\bf{.5}}{\bf{.8)}}\)is an unbiased estimator of \({{\bf{\sigma }}^{\bf{2}}}\). You may assume that \({{\bf{S}}^{\bf{2}}}\)has a \({{\bf{X}}^{\bf{2}}}\) distribution with \({\bf{n - p}}\) degrees of freedom.

Suppose it is desired to estimate the proportion of persons in a large population with a certain characteristic. A random sample of 100 persons is selected from the population without replacement, and the proportion \(\overline X \)of persons in the sample who have the characteristic is observed. Show that, no matter how large the population, the standard deviation \(\overline X \)is at most 0.05.

Consider again the conditions of Exercise 12, and suppose that it is desired to estimate the value of \(\theta = 5 - 4{\beta _0} + {\beta _1}\). Find an unbiased estimator \(\hat \theta \) of

\(\theta \). Determine the value of \(\hat \theta \)and the M.S.E. of\(\hat \theta \).
See all solutions

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