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| The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year. |
| Car | Age (years) | Selling Price ($000) | Car | Age (years) | Selling Price ($000) |
| 1 | 9 | 8.1 | 7 | 8 | 7.6 |
| 2 | 7 | 6.0 | 8 | 11 | 8.0 |
| 3 | 11 | 3.6 | 9 | 10 | 8.0 |
| 4 | 12 | 4.0 | 10 | 12 | 6.0 |
| 5 | 8 | 5.0 | 11 | 6 | 8.6 |
| 6 | 7 | 10.0 | 12 | 6 | 8.0 |
Click here for the Excel Data File
| a. | If we want to estimate selling price on the basis of the age of the car, which variable is the dependent variable and which is the independent variable? |
| is the independent variable and is the dependent variable. |
| b-1. | Determine the correlation coefficient. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.) |
| X | Y | ( |
( |
( |
||
| 9.0 | 8.1 | 1.192 | 0.007 | 1.420 | 0.099 | |
| 7.0 | 6.0 | -0.908 | 3.674 | 0.825 | 1.741 | |
| 11.0 | 3.6 | 2.083 | 4.340 | 10.945 | -6.892 | |
| 12.0 | 4.0 | 3.083 | 9.507 | 8.458 | -8.967 | |
| 8.0 | 5.0 | -0.917 | -1.908 | 3.642 | 1.749 | |
| 7.0 | 10.0 | -1.917 | 3.092 | 9.558 | -5.926 | |
| 8.0 | 7.6 | -0.917 | 0.692 | 0.840 | -0.634 | |
| 11.0 | 8.0 | 2.083 | 1.092 | 4.340 | 2.274 | |
| 10.0 | 8.0 | 1.083 | 1.092 | 1.174 | 1.192 | |
| 12.0 | 6.0 | 3.083 | -0.908 | 9.507 | 0.825 | |
| 6.0 | 8.6 | -2.917 | 1.692 | 8.507 | 2.862 | -4.934 |
| 6.0 | 8.0 | -2.917 | 1.092 | 8.507 | 1.192 | -3.184 |
| 107.000 | 82.900 | |||||
| = | = | sx | = | sy | = |
| r | = |
| b-2. | Determine the coefficient of determination. (Round your answer to 3 decimal places.) |
| c. | Interpret the correlation coefficient. Does it surprise you that the correlation coefficient is negative? (Round your answer to nearest whole number.) |
| correlation between age of car and selling price. So, % of the variation in the selling price is explained by the variation in the age of the car. |
| The Student Government Association at Middle Carolina University wanted to demonstrate the relationship between the number of beers a student drinks and his or her blood alcohol content (BAC). A random sample of 18 students participated in a study in which each participating student was randomly assigned a number of 12-ounce cans of beer to drink. Thirty minutes after they consumed their assigned number of beers, a member of the local sheriff’s office measured their blood alcohol content. The sample information is reported below. |
| Student | Beers | BAC | Student | Beers | BAC |
| 1 | 6 | 0.10 | 10 | 3 | 0.07 |
| 2 | 7 | 0.09 | 11 | 3 | 0.05 |
| 3 | 7 | 0.09 | 12 | 7 | 0.08 |
| 4 | 4 | 0.10 | 13 | 1 | 0.04 |
| 5 | 5 | 0.10 | 14 | 4 | 0.07 |
| 6 | 3 | 0.07 | 15 | 2 | 0.06 |
| 7 | 3 | 0.10 | 16 | 7 | 0.12 |
| 8 | 6 | 0.12 | 17 | 2 | 0.05 |
| 9 | 6 | 0.09 | 18 | 1 | 0.02 |
| Use a statistical software package to answer the following questions. |
1.
value: 10.00 points
Required information
| a-1. | Choose a scatter diagram that best fits the data. |
2.
value: 10.00 points
Required information
| b. | Fill in the blanks below. (Round your answers to 3 decimal places.) |
| |
|
| |
|
| sx | |
| sy | |
| c. | Determine the coefficient of correlation and coefficient of determination. (Round your answers to 3 decimal places.) |
| Coefficient of correlation | |
| Coefficient of determination | |
References
eBook & Resources
Worksheet Difficulty: 2 Intermediate Learning Objective: 13-02 Calculate a correlation coefficient to test and interpret the relationship between two variables.
3.
value: 10.00 points
Required information
| c-1. | State the decision rule for .01 significance level: H0: ρ ≤ 0; H1: ρ > 0. (Round your answer to 3 decimal places.) |
| Reject H0 if t > |
| c-2. | Compute the value of the test statistic. (Round your answer to 2 decimal places.) |
| Value of the test statistic |
| c-3. | What is the p-value? (Hint: use Megastat) (Round p-value to 4 decimal places.) |
| p-value |
| c-4. | At the .01 significance level, is it reasonable to conclude that there is a positive relationship in the population between the number of beers consumed and the BAC? |
| H0 . There is between beers consumed and BAC. |
1.
value: 10.00 points
| The following sample observations were randomly selected. |
| X: | 5 | 3 | 6 | 3 | 4 | 4 | 6 | 8 |
| Y: | 13 | 15 | 7 | 12 | 13 | 11 | 9 | 5 |
| a. | Determine the regression equation. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.) |
| X | Y | ( |
( |
( |
||
| 5 | 13 | 2.375 | 5.641 | |||
| 3 | 15 | −1.875 | 3.516 | −8.203 | ||
| 6 | 7 | 13.141 | −4.078 | |||
| 3 | 12 | −1.875 | 1.375 | |||
| 4 | 13 | −0.875 | 0.766 | −2.078 | ||
| 4 | 11 | 0.375 | 0.141 | |||
| 6 | 9 | 1.125 | −1.625 | |||
| 8 | 5 | 31.641 | −17.578 | |||
| |
= | |
= | sx | = |
| sy | = | r | = |
| b | = | a | = |
| Y’ = + X |
| b. | Determine the value of |
| The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year. |
| Car | Age (years) | Selling Price ($000) | Car | Age (years) | Selling Price ($000) |
| 1 | 9 | 8.1 | 7 | 8 | 7.6 |
| 2 | 7 | 6.0 | 8 | 11 | 8.0 |
| 3 | 11 | 3.6 | 9 | 10 | 8.0 |
| 4 | 12 | 4.0 | 10 | 12 | 6.0 |
| 5 | 8 | 5.0 | 11 | 6 | 8.6 |
| 6 | 7 | 10.0 | 12 | 6 | 8.0 |
| The regression equation is |
| H0 and conclude the slope is zero. |
| The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year. |
| Car | Age (years) | Selling Price ($000) | |||
| 1 | 9 | 8.1 | |||
| 2 | 7 | 6.0 | |||
| 3 | 11 | 3.6 | |||
| 4 | 12 | 4.0 | |||
| 5 | 8 | 5.0 | |||
| 6 | 7 | 10.0 | |||
| 7 | 8 | 7.6 | |||
| 8 | 11 | 8.0 | |||
| 9 | 10 | 8.0 | |||
| 10 | 12 | 6.0 | |||
| 11 | 6 | 8.6 | |||
| 12 | 6 | 8.0 | |||
Click here for the Excel Data File
| a. | Determine the standard error of estimate. (Round your answer to 3 decimal places.) |
| Standard error of estimate |
| b. | Determine the coefficient of determination. (Round your answer to 3 decimal places.) |
| c. | Interpret the coefficient of determination. (Round your answer to the nearest whole number.) |
| percent of the variation in the selling price is explained by the variation in the age of the car. |
1.
value: 10.00 points
| Thompson Photo Works purchased several new, highly sophisticated processing machines. The production department needed some guidance with respect to qualifications needed by an operator. Is age a factor? Is the length of service as an operator (in years) important? In order to explore further the factors needed to estimate performance on the new processing machines, four variables were listed: |
| X1 = Length of time an employee was in the industry |
| X2 = Mechanical aptitude test score |
| X3 = Prior on-the-job rating |
| X4 = Age |
| Performance on the new machine is designated y. |
| Thirty employees were selected at random. Data were collected for each, and their performances on the new machines were recorded. A few results are: |
| Name | Performance on New Machine, Y | Length of Time in Industry, X1 | Mechanical Aptitude Score, X2 | Prior On-the-Job Performance, X3 | Age, X4 | |||||
| Mike Miraglia | 112 | 12 | 312 | 121 | 52 | |||||
| Sue Trythall | 113 | 2 | 380 | 123 | 27 | |||||
| The equation is: |
| a. | What is this equation called? | ||||||
|
| b. | How many dependent and independent variables are there? |
| dependent, independent |
| c. | What is the number 0.286 called? | ||||||||
|
| d. | As age increases by one year, how much does estimated performance on the new machine increase? (Round your answer to 3 decimal places.) |
| e. | Carl Knox applied for a job at Photo Works. He has been in the business for 6 years and scored 280 on the mechanical aptitude test. Carl’s prior on-the-job performance rating is 97, and he is 35 years old. Estimate Carl’s performance on the new machine. (Round your answer to 3 decimal places.) |
1.
value: 10.00 points
| Consider the ANOVA table that follows. |
| Analysis of Variance | |||||||||
| Source | DF | SS | MS | F | |||||
| Regression | 5 | 3710.00 | 742.00 | 12.89 | |||||
| Residual Error | 46 | 2647.38 | 57.55 | ||||||
| Total | 51 | 6357.38 | |||||||
| a-1. | Determine the standard error of estimate. (Round your answer to 2 decimal places.) |
| Standard error of estimate |
| a-2. | About 95% of the residuals will be between what two values? (Round your answers to 2 decimal places.) |
| 95% of the residuals will be between and . |
| b-1. | Determine the coefficient of multiple determination. (Round your answer to 3 decimal places.) |
| Coefficient of multiple determination value is . |
| b-2. | Determine the percentage variation for the independent variables. (Round your answer to 1 decimal place. Omit the “%” sign in your response.) |
| The independent variables explain % of the variation. |
| c. | Determine the coefficient of multiple determination, adjusted for the degrees of freedom. (Round your answer to 3 decimal places.) |
| Coefficient of multiple determination |
1.
value: 10.00 points
| The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars. |
| Predictor | Coeff | SE Coeff | t | p-value | ||||
| Constant | 7.987 | 2.967 | 2.690 | 0.010 | ||||
| X1 | 0.122 | 0.031 | 3.920 | 0.000 | ||||
| X2 | –1.120 | 0.053 | –2.270 | 0.028 | ||||
| X3 | –0.063 | 0.039 | –1.610 | 0.114 | ||||
| X4 | 0.523 | 0.142 | 3.690 | 0.001 | ||||
| X5 | –0.065 | 0.040 | –1.620 | 0.112 | ||||
| Analysis of Variance | ||||||||||
| Source | DF | SS | MS | F | p-value | |||||
| Regression | 5 | 3710.00 | 742.00 | 12.89 | 0.000 | |||||
| Residual Error | 46 | 2647.38 | 57.55 | |||||||
| Total | 51 | 6357.38 | ||||||||
| X1 is the number of architects employed by the company. |
| X2 is the number of engineers employed by the company. |
| X3 is the number of years involved with health care projects. |
| X4 is the number of states in which the firm operates. |
| X5 is the percent of the firm’s work that is health care–related. |
| a. | Write out the regression equation. (Round your answers to 3 decimal places. Negative answers should be indicated by a minus sign.) |
| Ŷ = + X1 + X2 + X3 + X4 + X5. |
| b. | How large is the sample? How many independent variables are there? |
| Sample n | |
| Independent variables k | |
|
|
| c-1. | State the decision rule for .05 significance level: H0: β1 = β2 = β3 =β4 =β5 =0; H1: Not all β’s are 0. (Round your answer to 2 decimal places.) |
| Reject H0 if F > |
| c-2. | Compute the value of the F statistic. (Round your answer to 2 decimal places.) |
| Computed value of F is |
| c-3. | Can we conclude that the set of regression coefficients could be different from 0? Use the .05 significance level. |
| H0. of the regression coefficients are zero. |
| For X1 | For X2 | For X3 | For X4 | For X5 |
| H0: β1 = 0 | H0: β2 = 0 | H0: β3 = 0 | H0: β4 = 0 | H0: β5 = 0 |
| H1: β1 ≠ 0 | H1: β2 ≠ 0 | H1: β3 ≠ 0 | H1: β4 ≠ 0 | H1: β5 ≠ 0 |
| d-1. | State the decision rule for .05 significance level. (Round your answers to 3 decimal places.) |
| Reject H0 if t < or t > . |
| d-2. | Compute the value of the test statistic. (Round your answers to 2 decimal places. Negative answers should be indicated by a minus sign.) |
| t − value | |
| X1 | |
| X2 | |
| X3 | |
| X4 | |
| X5 | |
|
|
| d-3. | Which variable would you consider eliminating? |
| Consider eliminating variables . |
| We are studying mutual bond funds for the purpose of investing in several funds. For this particular study, we want to focus on the assets of a fund and its five-year performance. The question is: Can the five-year rate of return be estimated based on the assets of the fund? Nine mutual funds were selected at random, and their assets and rates of return are shown below. |
| Assets | Return | Assets | Return | ||
| Fund | ($ millions) | (%) | Fund | ($ millions) | (%) |
| AARP High Quality Bond | $622.2 | 10.8 | MFS Bond A | $494.5 | 11.6 |
| Babson Bond L | 160.4 | 11.3 | Nichols Income | 158.3 | 9.5 |
| Compass Capital Fixed Income | 275.7 | 11.4 | T. Rowe Price Short-term | 681.0 | 8.2 |
| Galaxy Bond Retail | 433.2 | 9.1 | Thompson Income B | 241.3 | 6.8 |
| Keystone Custodian B-1 | 437.9 | 9.2 | |||
Click here for the Excel Data File
| b-1. | Compute the coefficient of correlation. (Round your answer to 3 decimal places. Negative amount should be indicated by a minus sign.) |
| r = |
| b-2. | Compute the coefficient of determination. (Round your answer to 3 decimal places.) |
| r2 = |
| c. | Give a description of the degree of association between the variables. |
| There is association between the variables. |
| d. | Determine the regression equation. Use assets as the independent variable. (Round your answers to 4 decimal places. Negative amounts should be indicated by a minus sign.) |
| b = | |
| a = |
| e. | For a fund with $400.0 million in sales, determine the five-year rate of return (in percent). (Round your answer to 4 decimal places.) |
| Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWhinney gathered the following sample information. The X column indicates the number of client contacts last month, and the Y column shows the value of sales ($ thousands) last month for each client sampled. |
| Number of Contacts, X | Sales ($ thousands), Y | Number of Contacts, X | Sales ($ thousands), Y | ||||
| 14 | 24 | 23 | 30 | ||||
| 12 | 14 | 48 | 90 | ||||
| 20 | 28 | 50 | 85 | ||||
| 16 | 30 | 55 | 120 | ||||
| 46 | 80 | 50 | 110 | ||||
| a. | Determine the regression equation. (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round final answers to 2 decimal places.) |
| X | Y | ( |
( |
( |
||
| 14 | 376.36 | 1376.41 | 719.74 | |||
| 12 | 14 | −21.4 | −47.1 | |||
| 20 | −13.4 | 179.56 | 443.54 | |||
| 16 | 30 | −31.1 | 967.21 | |||
| 46 | 12.6 | 357.21 | ||||
| 23 | −10.4 | 967.21 | ||||
| 48 | 90 | 28.9 | 213.16 | 421.94 | ||
| 50 | 85 | 23.9 | 275.56 | 396.74 | ||
| 55 | 466.56 | 3469.21 | 1,272.24 | |||
| 50 | 110.0 | 16.6 | 48.9 | |||
| |
= | |
= | sx | = |
| sy | = | r | = |
| b = |
| a = | Y’ = + X |
| b. | Determine the estimated sales if 40 contacts are made. (Do not round intermediate calculations. Round final answers to 2 decimal places.) |
| On the first statistics exam, the coefficient of determination between the hours studied and the grade earned was 80%. The standard error of estimate was 10. There were 20 students in the class. Develop an ANOVA table for the regression analysis of hours studied as a predictor of the grade earned on the first statistics exam. |
| Source | DF | SS | MS |
| Regression | |||
| Error | |||
| Total |
0
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