Exam 16: Simple Linear Regression and Correlation

The graph of a confidence interval for the expected value of y is represented by two parallel lines, one on either side of the regression line.

The width of the confidence interval estimate for the predicted value of y depends on

NARRBEGIN: Oil Quality/Price Oil Quality and Price 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 positive relationship between quality and price per barrel. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial statistical software output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Regression Analysis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 \square.008220 11.59 0.000 S=0.1314R-Sq=92.46\%R-Sq(adj)=91.7\% Analysis of Variance Source DF SS MS F P Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Do the $\rho$ and $\beta$ ₁ tests in the previous two questions provide the same results? Explain.

NARRBEGIN: Comedy Shows Revenues Comedy Shows Revenues A financier whose specialty is investing in comedy shows has observed that, in general, shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid comedians in the show for ten recently staged shows. Show Cost of Two Highest Paid Comedian ( \mil ) 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 0.7 58 NARREND -{Comedy Shows Revenues Narrative} Plot the residuals against the predicted values of y. What does the graph tell you?

The regression line $\tilde { y } = 2 + 3 x$ has been fitted to the data points (4, 11), (2, 7), and (1, 5). The sum of squares for error will be 10.0.

Testing whether the slope of the population regression line could be zero is equivalent to testing whether the:

NARRBEGIN: Wayne Newton Concert Wayne Newton Concert At a recent Wayne Newton concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first 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 An Excel output follows: SUMMARY OUTPUT DESCRIPTIVE STATISTICS Regression Statistics 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 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ $\text { ANOVA }$ df SS MS F Signficance F Regression 1 28.65711 28.65711 32.45653 2.1082-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standerd Emor t Stat Rvalie Lower 95\% Uoper 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 NARREND -{Wayne Newton Concert Narrative} Which interval in the previous two questions is narrower: the confidence interval estimate of the expected value of y or the prediction interval for the same given value of x (10 years) and same confidence level? Why?

In testing the hypotheses: H₀: $\beta$ ₁ = 0 vs. H₀: $\beta$ ₁ $\neq$ 0, the following statistics are available: $n = 10$ , $b _ { 0 } = - 1.8$ , $b _ { 1 } = 2.45$ , $s _ { b _ { 1 } } = 1.20$ , and $\hat { y } = 6$ . The value of the test statistic is:

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of hardware 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. Salesperson Years of Exgperience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 NARREND -{Sales and Experience Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether more experience is related to higher sales, as the manager speculates.

NARRBEGIN: Oil Quality and Price Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S=0.1314R-Sq=92.46\%R-Sq(ad)=91.7\% Analysis of Variance Source DF SS MS F P Regeression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Does it appear that the errors are normally distributed? Explain.

A straight line regression model with only one independent variable is called a(n) ____________________-order linear model.

NARRBEGIN: Marc Anthony Concert Marc Anthony Concert At a recent Marc Anthony concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af 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 An Excel output follows: SUMMARY OUTPUT DESCRIPTIVE STATISTICS Reqression Statistics 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 Samplevariance 2.3447 Count 20 Court 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ df SS MS F Sign\&icance 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 Rvale Lower 95\% Upoer 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 NARREND -{Marc Anthony Concert Narrative} Identify possible outliers.

In order to predict with 90% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

If the coefficient of determination is 0.975, then which of the following is true regarding the slope of the regression line?

In a simple linear regression problem, the following sum of squares are produced: $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 } = 200$ , $\sum \left( y _ { i } - \tilde { y } _ { i } \right) ^ { 2 } = 50$ , and $\sum \left( \tilde { y } _ { i } - \bar { y } \right) ^ { 2 } = 150$ . The percentage of the variation in y that is explained by the variation in x is:

Statisticians have shown that sample y-intercept b₀ and sample slope coefficient b₁ are unbiased estimators of the population regression parameters $\beta$ ₀ and $\beta$ ₁, respectively.

NARRBEGIN: Accidents & Rain Accidents and Rain A statistician investigating the relationship between the amount of rain (in inches) and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year. The results are shown below. Day Rain Number of Accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 NARREND -{Accidents and Rain Narrative} What does the slope of the least squares regression line tell you?

NARRBEGIN: Rock Concert Revenues Rock Concert Revenues A financier whose specialty is investing in rock concerts has observed that, in general, concerts with "big-name" stars seem to generate more revenue than those concerts whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the concert for ten concert tours. Concert Cost of Twa Highest Paid Performers ( \mil ) 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 0.7 58 NARREND -{Rock Concert Revenues Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a positive linear relationship exists between payment to the two highest-paid performers and gross revenue.

NARRBEGIN: Trivia Games & Ed.Trivia Games & Education 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 favorite game show. She records their winnings in dollars and the number of 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 0 16 300 7 13 650 8 14 400 NARREND -{Trivia Games & Education Narrative} Determine the least squares regression line.

NARRBEGIN: Comedy Shows Revenues Comedy Shows Revenues A financier whose specialty is investing in comedy shows has observed that, in general, shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid comedians in the show for ten recently staged shows. Show Cost of Two Highest Paid Comedian ( \mil ) 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 0.7 58 NARREND -{Comedy Shows Revenues Narrative} Use the regression equation $\hat { y } = 4.225 + 8.285 x$ to determine the predicted values of y.