Exam 16: Simple Linear Regression and Correlation

We standardize residuals by subtracting their mean and dividing by their variance.

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} Determine the standard error of estimate and describe what this statistic tells you about the regression line.

You cannot interpret the ____________________ of the simple linear regression line unless the value of x = 0 lies within the range of where data was collected.

In regression analysis, the coefficient of determination R² measures the amount of variation in y that is:

The sample correlation coefficient between x and y is 0.375. It has been found out that the p-value is 0.256 when testing H₀: $\rho$ = 0 against the two-sided alternative H₁: $\rho$ $\neq$ 0. To test H₀: $\rho$ = 0 against the one-sided alternative H₁: $\rho$ > 0 at a significant level of 0.193, the p-value will be equal to

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} Estimate with 95% confidence the average number of concerts attended by all 45 year-old individuals.

In the first-order linear regression model, the population parameters of the y-intercept and the slope are, respectively,

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 $\text { Covariances }$ Degrees Price Degrees 21.281667 Price 2.026750 0208837 Regression Analysis 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(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 0.017 Total 12 2.5060 NARREND -{Oil Quality and Price 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?

If there is no linear relationship between two variables x and y, the coefficient of determination must be -1.0.

NARRBEGIN: Income and Education Income and Education A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample 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 NARREND -{Income and Education Narrative} Interpret the value of the slope of the regression line.

NARRBEGIN: Income and Education Income and Education A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample 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 NARREND -{Income and Education Narrative} Determine the least squares regression line.

If a single point has a large impact on the equation of the regression line, it is called a(n) ____________________ point.

The first-order linear model is sometimes called the simple linear regression model.

The method statisticians use to produce a straight line that minimizes the SSE is called the ____________________ method.

If the regression line is horizontal, the slope is ____________________ and x and y are not related.

For a regression analysis to be valid, the error variable must have a(n) ____________________ distribution.

A scatter diagram includes the following data points: x 3 2 5 4 5 y 8 6 12 10 14 Two regression models are proposed: (1) $\hat { y } = 1.2 + 2.5 x$ , and (2) $\hat { y } = 4.0 x$ . Using the least squares method, which of these regression models provides the better fit to the data? Why?

NARRBEGIN: Allman Brothers Concert Allman Brothers Concert At a recent Allman Brothers 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 Reqression Statistics Mutiple 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 Sianificance 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 tStat Pvalue 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 -{Allman Brothers Concert Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate to describe the relationship between the age and number of concerts attended by the respondents.

NARRBEGIN: Telemarketing Sales and E Telemarketing Sales and Experience The general manager of a telemarketing company believes that experience is the most important factor in determining the level of success of a telemarketer. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected telemarketers. These data are listed below. Telemarketer Years of Experience 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 -{Telemarketer Sales and Experience Narrative} Identify possible outliers.

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