Exam 12: Simple Linear Regression

onfident that the mean amount of time needed to record one additional loan application is somewhere $\text { Regression statistics }$ MultipleR 0.9447 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 Observations 30 $\text { ANOVA }$ df S5 MS F Significance F Regression 1 25.9438 25.9438 232.2200 4.3946-15 Residual 28 3.1282 0.1117 Total 29 29.072 Coefficients Standard Error tStat p-value Lower 95\% Upper 95\% Intercept 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Applications Recorded 0.0126 0.0008 15.2388 4.3946- 15 0.0109 0.0143 $\text { Note: } 4.3946 \mathrm { E } - 15 \text { is } 4.3946 \times 10 ^ { - 15 } \text {. }$ -Referring to Instruction 12.36,there is sufficient evidence that the amount of time needed linearly depends on the number of loan applications at a 5% level of significance.

Instruction 12.34 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Instruction 12.34,the value of the t test statistic and F test statistic should be the same when testing whether the number of customers who make purchases is a good predictor for weekly sales.

Instruction 12.37 A chocolate bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses six country towns and cities and offers the chocolate bar at different prices. Using chocolate bar sales as the dependent variable, the company will conduct a simple linear regression on the data below: $\begin{array} { | l | l | l | } \hline \text { City } & \text { Price(\$) } & \text { Sales } \\\hline \text { Toowoomba } & 1.30 & 100 \\\hline \text { Broken Hill } & 1.60 & 50 \\\hline \text { Bendigo } & 1.80 & 50 \\\hline \text { Kalgoorlie } & 2.00 & 40 \\\hline \text { Launceston } & 2.40 & 38 \\\hline \text { Port Augusta } & 2.90 & 32 \\\hline\end{array}$ -Referring to Instruction 12.37,if the price of the chocolate bar is set at $2,the estimated average sales will be

Instruction 12.34 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Instruction 12.34,construct a 95% confidence interval for the change in average weekly sales when the number of customers who make purchases increases by 1.

Instruction 12.34 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Instruction 12.34,what are the degrees of freedom of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

The slope (b₁)represents

Instruction 12.39 The managers of a brokerage firm are interested in finding out if the number of new customers a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new customers they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Clients Sles 1 27 52 2 11 37 3 42 64 4 33 55 5 15 29 6 15 34 7 25 58 8 36 59 9 28 44 10 30 48 11 17 31 12 22 38 -Referring to Instruction 12.39,suppose the managers of the brokerage firm want to obtain a 99% prediction interval for the sales made by a broker who has brought into the firm 18 new customers.The t critical value they would use is __________.

Instruction 12.2 A chocolate bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses six country towns and cities and offers the chocolate bar at different prices. Using chocolate bar sales as the dependent variable, the company will conduct a simple linear regression on the data below: $\begin{array} { | l | l | l | } \hline \text { City } & \text { Price(\$) } & \text { Syles } \\\hline \text { Toowoomba } & 1.30 & 100 \\\hline \text { Broken Hill } & 1.60 & 50 \\\hline \text { Bendigo } & 1.80 & 50 \\\hline \text { Kalgoorlie } & 2.00 & 40 \\\hline \text { Launceston } & 2.40 & 38 \\\hline \text { Port Augusta } & 2.50 & 32 \\\hline\end{array}$ -Referring to Instruction 12.2,what is ?(X * $\bar { X }$ )² for these data?

Instruction 12.27 The director of cooperative education at a university wants to examine the effect of cooperative education job experience on marketability in the workplace. She takes a random sample of four students. For these four, she finds out how many times each had a cooperative education job and how many job offers they received upon graduation. These data are presented in the table below. Student Coop jobs job Offer 1 1 4 2 2 6 3 1 3 4 0 1 -Referring to Instruction 12.27,the director of cooperative education wanted to test the hypothesis that the true slope was equal to 3.0.For a test with a level of significance of 0.05,the null hypothesis should be rejected if the value of the test statistic is____________

Instruction 12.23 The managing partner of an advertising agency believes that his company's sales are related to the industry sales. He uses Microsoft Excel's Data Analysis tool to analyse the last four years of quarterly data with the following results: $\text { Regression statistics }$ Multiple R 0.802 R Square 0.643 Adjusted R Square 0.618 Standard Error SYX 0.9224 Observations 16 $\text { ANOVA }$ df SS MS F Sig.F Regression 1 21.497 21.497 25.27 0.000 Error 14 11.912 0.851 Total 15 33.409 Predictor Coef StdError t Stat p-value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin-Watson 1.59 Statistic -Referring to Instruction 12.23,if the Durbin-Watson statistic has a value close to 0,which assumption is violated?

If you collected data over time,avoid using the Durbin-Watson test to check the independence assumption.

Instruction 12.33 It is believed that the average numbers of hours spent studying per day (HOURS) during undergraduate education should have a positive linear relationship with the starting salary (SALARY, measured in thousands of dollars per month) after graduation. Given below is the Microsoft Excel output for predicting starting salary (Y) using number of hours spent studying per day (X) for a sample of 51 students. NOTE: Only partial output is shown. $\text { Regression statistics }$ Multiple R 0.8857 R Square 0.7845 Adjusted R Square 0.7801 Standard Error 1.3704 Observations 51 $\text { ANOVA }$ df SS MS F Significance F Regression 1 335.0472 335.0473 178.3859 Residual 1.8782 Total 50 427.0798 Coefficients Standard Error t Stat p-value Lower 95\% Upper 95\% Intercept -1.8940 0.4018 -4.7134 2.051-05 -2.7015 -1.0865 Hours 0.9795 0.0733 13.3561 5.944-18 0.8321 1.1269 Note: 2.051E-05 = 2.051 * 10^-0.5 and 5.944E-18 = 5.944 * 10^-18. -Referring to Instruction 12.33,the 90% confidence interval for the average change in SALARY (in thousands of dollars)as a result of spending an extra hour per day studying is

Instruction 12.15 The following Microsoft Excel tables are obtained when 'Score received on an exam (measured in percentage points)' (Y) is regressed on 'percentage attendance' (X) for 22 students in a Statistics for Business and Economics course. $\text { Regression Statistics }$ MultipleR 0.142620229 R Square 0.02034053 Adjusted R Square -0.028642444 Standard Error 20.25979924 Observations 22 Coefficients Standard Error t Stat p -value Intercept 39.39027309 37.24347659 1.057642216 0.302826622 Attendance 0.340583573 0.52852452 0.644404489 0.526635689 -Referring to Instruction 12.15,which of the following statements is true?

If the plot of the residuals is fan shaped,which assumption is violated?

Instruction 12.25 A computer software developer would like to use the number of downloads (in thousands) for the trial version of his new shareware to predict the amount of revenue (in thousands of dollars) he can make on the full version of the new shareware. Following is the output from a simple linear regression along with the residual plot and normal probability plot obtained from a data set of 30 different sharewares that he has developed: $\text { Regression statistics }$ MultipleR 0.8691 R Square 0.7554 Adjusted R Square 0.7467 Standard Error 44.4765 Observations 30.0000 $\text { ANOVA }$ df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55388.4309 1978.1582 Total 29 226451.3503 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Intercept -95.0614 26.9183 -3.5315 0.0015 -150.2009 -39.9218 Download 3.7297 0.4011 9.2992 0.0000 2.9082 4.5513 -Referring to Instruction 12.25,there appears to be autocorrelation in the residuals.

Instruction 12.9 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Instruction 12.9,generate the scatter plot.

Instruction 12.3 The director of cooperative education at a university wants to examine the effect of cooperative education job experience on marketability in the workplace. She takes a random sample of four students. For these four, she finds out how many times each had a cooperative education job and how many job offers they received upon graduation. These data are presented in the table below. Student Coop jobs job Offer 1 1 4 2 2 6 3 1 3 4 0 1 -Referring to Instruction 12.3,set up a scatter diagram.

Instruction 12.16 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Instruction 12.16,what is the value of the coefficient of determination?

Instruction 12.7 It is believed that average grade (based on a four-point scale) should have a positive linear relationship with university entrance exam scores. Given below is the Microsoft Excel output from regressing average grade on university entrance exam scores using a data set of eight randomly chosen students from a large university. -Referring to Instruction 12.7,the interpretation of the coefficient of determination in this regression is

Instruction 12.14 The managing partner of an advertising agency believes that his company's sales are related to the industry sales. He uses Microsoft Excel's Data Analysis tool to analyse the last four years of quarterly data with the following results: $\text { Regression statistics }$ Multiple R 0.802 R Square 0.643 Adjusted R Square 0.618 Standard Error SYX 0.9224 Observations 16 $\text { ANOVA }$ df SS MS F Sig.F Regression 1 21.497 21.497 25.27 0.000 Error 14 11.912 0.851 Total 15 33.409 Predictor Coef StdError t Stat p-value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin-Watson 1.59 Statistic -Referring to Instruction 12.14,the coefficient of determination is ____________.