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Chapter 13: Problem 31
What are the degrees of freedom for a simple linear regression model?
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A population data set produced the following information. $$ \begin{aligned} &N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \\ &\Sigma x^{2}=485,870, \text { and } \Sigma y^{2}=135,675 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).
A car rental company charges $$\$ 50$$ a day and 20 cents per mile for renting a car. Let \(y\) be the total rental charges (in dollars) for a car for one day and \(x\) be the miles driven. The equation for the relationship between \(x\) and \(y\) is $$ y=50+.20 x $$ a. How much will a person pay who rents a car for one day and drives it 100 miles? b. Suppose each of 20 persons rents a car from this agency for one day and drives it 100 miles. Will each of them pay the same amount for renting a car for a day or do you expect each person to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?
The following information is obtained for a sample of 16 observations taken from a population. $$ \mathrm{SS}_{x x}=340.700, \quad s_{e}=1.951, \quad \text { and } \quad \hat{y}=12.45+6.32 x $$ a. Make a \(99 \%\) confidence interval for \(B\). b. Using a significance level of .025, can you conclude that \(B\) is positive? c. Using a significance level of .01, can you conclude that \(B\) is different from zero? d. Using a significance level of .02, test whether \(B\) is different from 4.50. (Hint: The null hypothesis here will be \(H_{0}: B=4.50\), and the alternative hypothesis will be \(H_{1}: B \neq 4.50\). Notice that the value of \(B=4.50\) will be used to calculate the value of the test statistic \(t\).)
Explain each of the following concepts. You may use graphs to illustrate each concept. a. Perfect positive linear correlation b. Perfect negative linear correlation c. Strong positive linear correlation d. Strong negative linear correlation e. Weak positive linear correlation f. Weak negative linear correlation g. No linear correlation
The following data give the experience (in years) and monthly salaries (in hundreds of dollars) of nine randomly selected secretaries. $$ \begin{array}{l|rrrrrrrrr} \hline \text { Experience } & 14 & 3 & 5 & 6 & 4 & 9 & 18 & 5 & 16 \\ \hline \text { Monthly salary } & 62 & 29 & 37 & 43 & 35 & 60 & 67 & 32 & 60 \\\ \hline \end{array} $$ a. Find the least squares regression line with experience as an independent variable and monthly salary as a dependent variable. b. Construct a \(98 \%\) confidence interval for \(B\). c. Test at the \(2.5 \%\) significance level whether \(B\) is greater than zero.
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