Exam 18: Simple Linear Regression and Correlation

In regression analysis, the coefficient of determination, $R ^ { 2 }$ , measures the amount of variation in y that is: A. caused by the variation in x . B. explained by the variation in x . C. unexplained by the variation in x . D. caused by the variation in x or explained by the variation in x .

In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing $H _ { 0 } : \beta _ { 1 } = 0$ in a simple linear regression follows the Student t-distribution with n - 1 degrees of freedom.

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 Predict with 95% confidence the incidence of skin cancers per 100 000 in a town with a daily average of 6.5 hours of sunshine.

In regression analysis, if the coefficient of correlation is -1.0, then: A the sum of squares for error is -1.0 . B the sum of squares for regression is 1.0 . C the sum of squares for error and sum of squares for regressi on are equal. D the sum of squares for regression and total variation in y are equal.

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value .

Predict weekly sales in the fast food restaurant if 10 vouchers are printed in the local newspaper, given Estimated Sales = 11.5676 + 0.4618.Vouchers Is this a good estimate?

In developing a 95% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 10, the appropriate table value would be 2.306.

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 Identify possible outliers.

The value of the sum of squares for regression, SSR, can never be smaller than 0.0.

Which of the following statements best describes correlation analysis in a simple linear regression? A. Correlation analysis measures the strength of a relationship between two numerical variables. B. Correlation analysis measures the strength and direction of a linear elationship between two numerical variables. C. Correlation analysis measures the direction of a relationship between two numerical variables. D. Correlation analysis measures the strength of a relationship between two categorical variables.

A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error of estimate was: A. 2.1788. B. 1.5648. C. 1.5009. D. 2.2716.

When the sample size n is greater than 30, the Spearman rank correlation coefficient $r _ { s }$ is approximately normally distributed with: \begin{array}{|l|l|}\hline\text { A. } & \text { mean 0 and standard deviation 1 .}\\\hline \text { B. } & \text { mean 1 and standard deviation \sqrt{n-1} .} \\\hline \text { C. } &\text {mean 1 and standard deviation \( 1 / \sqrt{n-1} \). }\\\hline \text { D. } &\text {mean 0 and standard deviation \( 1 / \sqrt{n-1} \). }\\\hline\end{array}

If a simple linear regression model has no y-intercept, then: A. when x=0 , so does y . B. y is al ways zero. C. when y=0 , so does x . D. y is al ways equal to x .

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 Predict with 95% confidence the gross revenue of a movie whose top two stars earn $5.0 million.

Given that ss_x = 2500, ss_y = 3750, ss_xy = 500 and n = 6, the standard error of estimate is: A 12.247 B 24.933 C 30.2076 D 11.180

Which of the following statements best describes the slope in the simple linear regression model? \begin{array}{|l|l|}\hline\text { A. } & \text {The estimated average change in \mathrm{x} per one unit increase in \( \mathrm{y} \). }\\\hline \text { B. } & \text {The estimated average change in \( \mathrm{y} \) when \( \mathrm{x}=1 \). } \\\hline \text { C. } &\text {The estimated average value of \( \mathrm{y} \) when \( \mathrm{x}=0 \). }\\\hline \text { D. } &\text {The estimated average change in y per one unit increase in \( \mathrm{x} \). }\\\hline\end{array}

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 Minitab 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 F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 a. Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the quality of oil and price per barrel. b. Determine the least squares regression line. c. Redraw the scatter diagram and plot the least squares regression line on it. d. Interpret the value of the slope of the regression line.

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 SUMMARY OUTPUT DESCRIPTIVE STATISTICS Reqression Statiatics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 Age Concerts Mean 53 Mean 3.65 Standard Error 2.1849 Standard Error 0.3424 Standard Deviation 9.7711 Standard Deviation 1.5313 Sample Variance 95.4737 Sample Variance 2.3447 Count 20 Count 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ ANOVA df SS MS F Significance F Regression 1 28.65711 28.65711 32.45653 2.1082-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between age and number of concerts attended.

In developing a 90% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 15, the appropriate table value would be 1.761.