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Exam 15: Simple Linear Regression and Correlation

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Question 41

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Teacher A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course. $\begin{array} { | c | c | c | } \hline \text { Student } & \text { Hours Studyine } & \text { Final Grade } \\\hline 1 & 42 & 92 \\\hline 2 & 58 & 95 \\\hline 3 & 32 & 81 \\\hline 4 & 39 & 78 \\\hline 5 & 37 & 75 \\\hline 6 & 51 & 88 \\\hline 7 & 49 & 85 \\\hline 8 & 45 & 85 \\\end{array}$ -In testing the hypotheses $H _ { 0 } : \beta _ { 1 } = 0$ vs. $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0$ ,what is the value of the test statistic?

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Question 42

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In publishing the results of some research work,the following values of the coefficient of determination were listed.Which one would appear to be incorrect?

A) 0.91
B) 0.06
C) 0.47
D) -0.64
E) 0.00

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Question 43

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______________________________ measures the strength of the relationship between the dependent and independent variables.

The ____________________ requires that the sum of the squared deviations between y values in the scatter diagram and y values predicted by the equation be minimized.

One way to examine whether two variables might be linearly related is to construct a ______________________________.

For n = 16 data points, $r ^ { 2 } = 0.81$ At the 0.05 level of significance,can we conclude that the true coefficient of correlation could be zero?

The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Sallepersan } & \text { Yeare af Experience } & \text { Salez } \\\hline 1 & 0 & 7 \\\hline 2 & 2 & 9 \\\hline 3 & 10 & 20 \\\hline 4 & 3 & 15 \\\hline 5 & 8 & 18 \\\hline 6 & 5 & 14 \\\hline 7 & 12 & 20 \\\hline σ & 7 & 17 \\\hline 9 & 20 & 30 \\\hline 10 & 15 &25\\\hline\end{array}$ Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate.

What happens to the width of a prediction interval for y as the x value on which the interval estimate is based gets farther away from the mean of x? Why?

If the sum of squares due to regression (SSR) is 60,which of the following must be true?

The coefficient of correlation assumes values between ____________________ and ____________________,inclusive.

The simple linear regression model assumes that regardless of the value for x,the standard deviation of the distribution of y values about the regression line is the same.

If all the points in a scatter diagram lie on the least squares regression line,then the coefficient of correlation:

Household The following data was collected by a particular company to determine if a relationship exists between the number of people in a household and weekly food expenditures. $\begin{array} { | c | c | c } \hline \text { Househald } & \text { Number in Hausehald } & \text { Weakly Fadd Expenses } \\\hline 1 & 2 & \$ 95 \\\hline 2 & 3 & \$ 137 \\\hline 3 & 5 & \$ 165 \\\hline 4 & 2 & \$ 105 \\\hline 5 & 4 & \$ 227 \\\hline 6 & 4 & \$ 240 \\\hline 7 & 5 & \$ 185 \\\hline\end{array}$ -Determine the standard error of estimate.

Consider the following data values of x and y. $\begin{array} { | c | c c c c c c | } \hline \boldsymbol { x } & 97 & 113 & 81 & 68 & 90 & 79 \\\hline y & 103 & 103 & 105 & 115 & 127 & 104 \\\hline\end{array}$ Use Excel or Minitab to perform a regression analysis on the data.

Testing the null hypothesis that the slope of the true regression line equals zero is equivalent to testing whether the true coefficient of correlation could be zero.Why?

Salesperson The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Ealesperson } & \text { Years of Experience } & \text { Eales } \\\hline 1 & 0 & 7 \\\hline 2 & 2 & 9 \\\hline 3 & 10 & 20 \\\hline 4 & 3 & 15 \\\hline 5 & 8 & 18 \\\hline 0 & 5 & 14 \\\hline 7 & 12& 20 \\\hline 8 &7 & 17 \\\hline 9 &20 & 30 \\\hline 10 & 15& 25 \\\hline\end{array}$ -Determine the least squares regression line. $\hat{ Y }$ = _____________________________

A scatter diagram includes the following data points: $\begin{array} { | l | l | l | l | l | l | } \hline \mathbf { x } & \mathbf { 3 } & 2 & 5 & 4 & 5 \\\hline \mathbf { y } & \mathbf { 8 } & 6 & 12 & 10 & 14 \\\hline\end{array}$ Two regression models are proposed: Model 1: $\hat { y } =$ 1.2 + 2.5x Model 2: $\hat { y } =$ 5.5 + 4.0x Using the least squares method,which of these regression models provide the better fit to the data? Why?

What does the least-squares criterion have to do with obtaining a regression line for a given set of data?

Teacher A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course. $\begin{array} { | c | c | c | } \hline \text { Student } & \text { Hours Studyine } & \text { Final Grade } \\\hline 1 & 42 & 92 \\\hline 2 & 58 & 95 \\\hline 3 & 32 & 81 \\\hline 4 & 39 & 78 \\\hline 5 & 37 & 75 \\\hline 6 & 51 & 88 \\\hline 7 & 49 & 85 \\\hline 8 & 45 & 85 \\\end{array}$ -In testing the hypotheses $H _ { 0 } : \beta _ { 1 } = 0$ vs. $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0$ ,what is the conclusion at the 0.05 significance level?

Number of years Data was collected to describe the relationship between salary and number of years of working experience at a particular organization and is shown below in the following table. $\begin{array} { | c | c | c | } \hline \text { Employee } & \text { Salary } & \text { Years af Experience } \\\hline 1 & \$ 35,000 & 4 \\\hline 2 & \$ 7,200 & 13 \\\hline 3 & \$ 45,900 & 6 \\\hline 4 & \$ 42,200 & 7 \\\hline 5 & \$ 65,600 & 10 \\\hline 6 & \$ 51,500 & 9 \\\hline 7 & \$ 57,200 & 5 \\\hline 8 & \$ 71,400 & 7 \\\hline\end{array}$ -Using the least-squares regression line,predict the salary of an employee with 11 years of experience.