Exam 12: Simple Linear Regression

The coefficient of determination represents the ratio of SSR to SST.

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 the percentage of the total variation in chocolate bar sales explained by the regression model?

Instruction 12.26 The manager of the purchasing department of a large savings and loan organization would like to develop a model to predict the amount of time (measured in hours) it takes to record a loan application. Data are collected from a sample of 30 days, and the number of applications recorded and completion time in hours is recorded. Below is the regression output: $\text { Regression statistics }$ 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 Note: 4.3946E-15 is 4.3946 × 10^-15. -Referring to Instruction 12.26,there is no evidence of positive autocorrelation if the Durbin-Watson test statistic is found to be 1.78.

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,the coefficient of determination is____________.

Instruction 12.38 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 Oifer 1 1 4 2 2 6 3 1 3 4 0 1 -Referring to Instruction 12.38,suppose the director of cooperative education wants to obtain a 95% prediction interval estimate for the number of job offers received by students who have had exactly one cooperative education job.The prediction interval is from __________ to __________.

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: ρ = 0 against the two-sided alternative H₁: ρ ≠ 0.To test H₀: ρ = 0 against the one-sided alternative H₁: ρ > 0 at a significance level of 0.2,the p-value is

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 is the p-value of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

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 is the value of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

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,which is the correct null hypothesis for testing whether the number of customers who make purchases affects weekly sales?

Instruction 12.31 An investment specialist claims that if one holds a portfolio that moves in opposite direction to the market index like the All Ordinaries Index, then it is possible to reduce the variability of the portfolio's return. In other words, one can create a portfolio with positive returns but less exposure to risk. A sample of 26 years of the All Ordinaries index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the All Ordinaries index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio (Y) on the returns of All Ordinaries index (X) to prove that the prison stocks portfolio is negatively related to the All Ordinaries index at a 5% level of significance. The results are given in the following Microsoft Excel output. Coefflelents Standard Error tStat p -vahse Intercept 4.866004258 0.35743609 13.61363441 8.7932-13 S\&P -0.502513506 0.071597152 -7.01862425 2.94942-07 -Referring to Instruction 12.31,which of the following will be a correct conclusion?

When using a regression model to make predictions,the term 'relevant range' refers to____________.

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,to test whether a change in price will have any impact on average sales,what would be the critical values? Use ? = 0.05.

In performing a regression analysis involving two numerical variables,you are assuming

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,what is the predicted average value of average grade when university entrance exam score = 20?

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 0.The value of the test statistic is ____________.

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,the least squares estimate of the slope is_____________.

Instruction 12.13 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 Sales 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 12.13,the standard error of the estimated slope coefficient is ____________.

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,to test the claim that the mean amount of time depends positively on the number of loan applications recorded against the null hypothesis that the mean amount of time does not depend linearly on the number of invoices processed,the p-value of the test statistic is

In a simple linear regression problem,r and b₁

Instruction 12.28 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 5les 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 12.28,the managers of the brokerage firm wanted to test the hypothesis that the true slope was equal to 0.At a level of significance of 0.01,the null hypothesis should be ____________(rejected or not rejected).