Exam 16: Simple Linear Regression and Correlation

One method of diagnosing heteroscedasticity is to plot the residuals against the predicted values of y, then look for a change in the spread of the plotted values.

The vertical spread of the data points about the regression line is measured by the y-intercept.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value as opposed to predicting an individual value.

The confidence interval estimate of the expected value of y for a given value x, compared to the prediction interval of y for the same given value of x and confidence level, will be:

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of estimate s _{$\varepsilon$} will be 0.0.

In simple linear regression, the coefficient of correlation r and the least squares estimate b₁ of the population slope $\beta$ ₁:

Given the least squares regression line $\hat { y } = 2.48 - 1.63 x$ , and a coefficient of determination of 0.81, the coefficient of correlation is:

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} Does it appear that the errors are normally distributed? Explain.

In the least squares regression line $\tilde { y } = 3 - 2 x$ , the predicted value of y equals:

A regression analysis between sales (in $1,000) and advertising (in $1,000) resulted in the following least squares line: $\hat { y } = 80 + 5 x$ . This implies that:

The plot of residuals vs. predicted values should show no patterns if the conditions of a regression analysis are met.

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 other variables might be associated with accidents, besides or along with rain?

The probability distribution of the error variable $\varepsilon$ is normal, with mean E( $\varepsilon$ ) = 0, and standard deviation $\sigma$ _{$\varepsilon$} =1.

In simple linear regression, most often we perform a two-tail test of the population slope $\beta$ ₁ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as:

An outlier is an observation that is unusually small or unusually large.

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} Draw a histogram of the residuals.

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 heteroscedasticity is a problem? Explain.

NARRBEGIN: Grateful Dead Concert Grateful Dead Concert At a recent Grateful Dead 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. It is suspected that older concert goers tend to go to more of his concerts in one year than younger concert goers. The data and analysis are shown below. 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 Regression Statistics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 $\text { DESCRIPTIVE STATISTICS }$ 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 Court 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ ANOVA df SS MS F Signficance F Regression 1 28.65711 28.65711 32.45653 2.1082E-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error tStat 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 NARREND -{Grateful Dead Concert Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables.

NARRBEGIN: Truck Speed & Gas Mileage Truck Speed and Gas Mileage An economist wanted to analyze the relationship between the speed of a truck (x) and its gas mileage (y). As an experiment a truck is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 NARREND -{Truck Speed and Gas Mileage Narrative} Does this data provide sufficient evidence at the 5% significance level to infer that a linear relationship exists between speed and gas mileage?