Exam 18: Simple Linear Regression and Correlation

Would the test of the significance of the overall equation have the same conclusion as the test of significance of the slope in a simple linear regression model of weekly sales in a fast food restaurant on number of vouchers printed in the local newspaper? Explain.

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Predict with 95% confidence the winnings of a contestant who has 15 years of education.

In a simple linear regression, which of the following is equivalent to testing the significance of the population slope? A Testing the signi ficance of the population intercept. B Testing the significance of the mean. C Testing the significance of the coefficient of correlation. D All of these choices are correct.

A financier whose specialty is investing in movie productions has observed that, in general, movies with 'big-name' stars seem to generate more revenue than those movies whose stars are less well known. To examine his belief, he records the gross revenue and the payment (in $ million) given to the two highest-paid performers in the movie for 10 recently released movies. Movie Cost of two highest- paid performers (\ ) Gross revenue (\ ) 1 5.3 48 2 7.2 65 3 1.3 18 4 1.8 20 5 3.5 31 6 2.6 26 7 8.0 73 8 2.4 23 9 4.5 39 10 6.7 58 Compute the standardised residuals.

A medical statistician wanted to examine the relationship between the amount of sunshine (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100 000 of population and the average daily sunshine in eight country towns around NSW. These data are shown below. Average daily sunshine (hours) 5 7 6 7 8 6 4 3 Skin cancer per 100000 7 11 9 12 15 10 7 5 Find the least squares regression line.

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between TV game show contestants' years of education and their winnings.

Given that cov(x,y) = 10, $s _ { y } ^ { 2 }$ = 15, $S _ { x } ^ { 2 }$ = 8 and n = 12, the value of the standard error of estimate, $S _ { \varepsilon }$ , is 2.75.

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. Book Number of pages Selling price (\ ) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 Determine the least squares regression line.

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Assume that the conditions for the tests conducted in the previous two questions are not met. Do the data allow us to infer at the 5% significance level that years of education and income are linearly related?

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Predict with 95% confidence the average income of all individuals with 10 years of education.

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. Book Number of pages Selling price (\ ) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 Estimate with 90% confidence the selling price of a book with 900 pages.

The coefficient of determination is the coefficient of correlation squared.

In the first-order linear regression model, the population parameters of the y-intercept and the slope are estimated by: A. and B. and C. and D. and

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be: A. 1.0 B. -1.0 C. 0.0 D. 2.0

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: $\sum ( x - \bar { x } ) ( y - \bar { y } )$ = 2250, $S _ { x }$ = 10, $\sum x$ = 50, $\sum y$ = 75 The least squares estimates of the slope and y-intercept are respectively: A. 225 and -1117.5 B. 2.5 and -5 . C. 25 and -117.5 D. 25 and 117.5

The manager of a fast food restaurant wants to determine how sales in a given week are related to the number of discount vouchers (#) printed in the local newspaper during the week. The number of vouchers and sales ($000s) from 10 randomly selected weeks is given below with Excel regression output. Number of vouchers Sales 4 12.8 7 15.4 5 13.9 3 11.2 19 18.7 10 17.9 8 16.8 6 15.9 3 11.5 5 13.9 Use the regression equation to determine the predicted sales for number of vouchers in the sample, given, Estimated Sales = 11.5676 + 0.4618.Vouchers

A medical statistician wanted to examine the relationship between the amount of sunshine (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100 000 of population and the average daily sunshine in eight country towns around NSW. These data are shown below. Average daily sunshine (hours) 5 7 6 7 8 6 4 3 Skin cancer per 100000 7 11 9 12 15 10 7 5 Draw a scatter diagram of the data and plot the least squares regression line on it.

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between years of education and income.

Which of the following best describes the residuals in regression analysis? \begin{array}{|l|l|}\hline\text { A. } & \text {The residuals are the difference between \mathrm{x} data values observed }\\& \text {and \( \mathrm{x} \) predicted model values. } \\\hline \text { B. } & \text {The residuals are the difference between the actual slope and the predicted model slope. } \\\hline \text { C. } &\text {The residuals are the difference between the \( y \) data values observed }\\& \text {and the predicted \( y \) model values. } \\\hline \text { D. } &\text {The residuals are the addition of the \( \mathrm{y} \) data values observed and predicted model values. }\\\hline\end{array}