Exam 16: Simple Linear Regression and Correlation

For a regression analysis to be valid, the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of Designer 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. Sales person 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 -{Sales and Experience Narrative} Estimate with 95% confidence the average monthly sales of all salespersons with 10 years of experience.

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} Calculate the Pearson correlation coefficient and interpret.

In performing a regression analysis which of the following must be true about the distribution of the error variable?

Suppose the slope of a simple linear regression line between hours studying and exam score is 5. That means as ____________________ increases by one, ____________________ increases by 5.

If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero.

The standardized residual is defined as:

Error terms that are correlated over time are said to be ____________________.

You use a(n) ____________________ interval whenever you want to estimate a one-time occurrence for a particular value of y when x is a given value.

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} Plot the least squares regression line on the scatter diagram.

In a regression problem, if the coefficient of determination is 0.95, this means that:

If the standard error of estimate s _{$\varepsilon$} = 20 and n = 10, then the sum of squares for error, SSE, is:

NARRBEGIN: Game Winnings & Ed.Game Winnings & 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 -{Game Winnings & Education Narrative} Estimate with 95% confidence the average winnings of all contestants who have 10 years of education.

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} Use the predicted and actual values of y to calculate the residuals.

Consider the following data values of variables x and y. 2 4 6 g 10 13 7 11 17 21 27 36 a.Determine the least squares regression line. b.Find the predicted value of y for x = 9. c.What does the value of the slope of the regression line tell you?

NARRBEGIN: Theatre Revenues Theatre Revenues A financier whose specialty is investing in stage productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those plays 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 play for ten recently staged plays. Play Cost of Two 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 -{Theatre Revenues Narrative} Are the two highest paid performers worth all the money paid for them? Comment using the statistical analyses you have done.

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: $\hat { y } = 77 + 8 x$ . This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $125,000.

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} Calculate the Pearson correlation coefficient. Interpret this result.

There is more error in estimating a mean value of y as opposed to predicting an individual value of y.

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.