Exam 15: Simple Linear Regression and Correlation

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 Can we conclude at the 1% significance level that there is a linear relationship between sunshine and skin cancer?

Which of the following best describes the residuals in regression analysis?

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.

In simple linear regression, the divisor of the standard error of estimate, $S _ { \varepsilon }$ , is n - 1.

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.

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 Use the regression equation to determine the predicted values of y.

Which of the following is not a required condition for the error variable $\varepsilon$ in the simple linear regression model?

In regression analysis, if the coefficient of determination is 1.0, then:

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

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 Determine the coefficient of determination, and discuss what its value tells you about the two variables.

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.

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 Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between payment to the two highest-paid performers and gross revenue.

In a simple linear regression problem, the least squares line is $\ddot {p}$ = -3.75 + 1.25 $x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

The value of the sum of squares for regression, SSR, can never be equal to the value of total sum of squares, SST.

A regression analysis between weight $y$ (in kilograms) and height $x$ (in centimetres) yielded the least squares line $\hat { y }$ = 135 + 6 $x$ . We estimate that for each 1 cm additional height, weight will decrease by 6 kilograms, on average.

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 Can we infer at the 5% significance level that the editor is correct?

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. Oil degrees API Price per barrel (in \ ) 27.0 12.02 28.5 12.04 30.8 12.32 31.3 12.27 31.9 12.49 34.5 12.70 34.0 12.80 34.7 13.00 37.0 13.00 41.0 13.17 41.0 13.19 38.8 13.22 39.3 13.27 A partial computer output follows. Descriptive Statistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Frice 13 12.730 0.457 0.127 Covariances Degrees Price Degrees 21.281667 Price 2.026750 0.208833 Regression Analysis Fredictor Coef StDev Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7% Analysis of Variance Source DF SS MS Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between the quality of oil and price per barrel.

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. Age 62 57 40 49 67 54 43 65 54 41 Number of concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 a. Use the regression equation $\hat { y }$ = -3.0115 + 0.1257x to determine the predicted values of y. b. Use the predicted values and the actual values of y to calculate the residuals. c. Plot the residuals against the predicted values $\hat { y }$ . d. Does it appear that heteroscedasticity is a problem? Explain. e. Draw a histogram of the residuals. f. Does it appear that the errors are normally distributed? Explain. g. Use the residuals to compute the standardised residuals. h. Identify possible outliers.

In a simple linear regression model, if r2 is 0.75, then 75% of the variation in the dependent variable y can be explained by the regression line, on the independent variable x.

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 Interpret the value of the slope of the regression line.