# Mathematics

Q17. A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size), and education of the head of household (School). House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is in years. The builder randomly selected 50 families and ran the µltiple regression. Microsoft Excel output is provided below:

SUMMARY OUTPUT

Regression Statistics | |

Multiple R | 0.865 |

R Square | 0.748 |

Adjusted R Square | 0.726 |

Standard Error | 5.195 |

Observations | 50 |

ANOVA | df | SS | MS | F | Signif F |

Regression | 3605.7736 | 901.4434 | 0.0001 | ||

Residual | 1214.2264 | 26.9828 | |||

Total | 49 | 4820.0000 |

Coeff | StdError | t Stat | P-value | |

Intercept | -1.6335 | 5.8078 | -0.281 | 0.7798 |

Income | 0.4485 | 0.1137 | 3.9545 | 0.0003 |

Size | 4.2615 | 0.8062 | 5.286 | 0.0001 |

School | -0.6517 | 0.4319 | -1.509 | 0.1383 |

Referring to the tables, one individual in the sample had an annual income of $10,000, a family size of 1, and an education of 8 years. This individual owned a home with an area of 1,000 square feet (House = 10.00). What is the residual (in hundreds of square feet) for this data point?

a. 8.10

b. 5.40

c. -5.40

d. -8.10

Q18. A manager of a product sales group believes the number of sales made by an employee (Y) depends on how many years that employee has been with the company (X_{1}) and how he/she scored on a business aptitude test (X_{2}). A random sample of 8 employees provides the following:

Employee | Y | X_{1} |
X_{2} |

1 | 100 | 10 | 7 |

2 | 90 | 3 | 10 |

3 | 80 | 8 | 9 |

4 | 70 | 5 | 4 |

5 | 60 | 5 | 8 |

6 | 50 | 7 | 5 |

7 | 40 | 1 | 4 |

8 | 30 | 1 | 1 |

Referring to the table, for these data, what is the estimated coefficient for the variable representing years an employee has been with the company, b_{1}?

a. 0.998

b. 3.103

c. 4.698

d. 21.293

Q19. In testing for differences between the means of 2 independent populations, the null hypothesis is:

a. H_{0}: μ1 – μ2 = 2.

b. H_{0}: μ1 – μ2 = 0.

c. H_{0}: μ1 – μ2 > 0.

d. H_{0}: μ1 – μ2 < 2.

Q20. A manager of a product sales group believes the number of sales made by an employee (Y) depends on how many years that employee has been with the company (X_{1}) and how he/she scored on a business aptitude test (X_{2}). A random sample of 8 employees provides the following:

Employee |
Y |
X_{1} |
X_{2} |

1 | 100 | 10 | 7 |

2 | 90 | 3 | 10 |

3 | 80 | 8 | 9 |

4 | 70 | 5 | 4 |

5 | 60 | 5 | 8 |

6 | 50 | 7 | 5 |

7 | 40 | 1 | 4 |

8 | 30 | 1 | 1 |

Referring to Table 14-1, for these data, what is the estimated coefficient for the variable representing scores on the aptitude test, b_{2}?A manager of a product sales group believes the number of sales made by an employee (Y) depends on how many years that employee has been with the company (X_{1}) and how he/she scored on a business aptitude test (X_{2}). A random sample of 8 employees provides the following:

Employee | Y |
X_{1} |
X_{2} |

1 | 100 | 10 | 7 |

2 | 90 | 3 | 10 |

3 | 80 | 8 | 9 |

4 | 70 | 5 | 4 |

5 | 60 | 5 | 8 |

6 | 50 | 7 | 5 |

7 | 40 | 1 | 4 |

8 | 30 | 1 | 1 |

Referring to the table, what is the estimated coefficient for the variable representing scores on the aptitude test, b_{2}?

a. 0.998

b. 3.103

c. 4.698

d. 21.293

Q21. To test the effects of a business school preparation course, 8 students took a general business test before and after the course. The results are given below.

Student | Exam Score Before Course | Exam Score After Course |

1 | 530 | 670 |

2 | 690 | 770 |

3 | 910 | 1,000 |

4 | 700 | 710 |

5 | 450 | 550 |

6 | 820 | 870 |

7 | 820 | 770 |

8 | 630 | 610 |

Referring to the table, at the 0.05 level of significance, the conclusion for this hypothesis test would be:

a. the business school preparation course does improve exam score.

b. the business school preparation course does not improve exam score.

c. the business school preparation course has no impact on exam score.

d. It cannot be drawn from the information given.

Q22. If the p value is less than α in a two-tailed test,

a. the null hypothesis should not be rejected.

b. the null hypothesis should be rejected.

c. a one-tailed test should be used.

d. no conclusion should be reached.

Q23. If the Type I error (α) for a given test is to be decreased, then for a fixed sample size n

a. the Type II error (β) will also decrease.

b. the Type II error (β) will increase.

c. the power of the test will increase.

d. a one-tailed test must be utilized.

Q24. Psychologists have found that people are generally reluctant to transmit bad news to their peers.

This phenomenon has been termed the “MM effect.” To investigate the cause of the MM effect, 40 undergraduates

at Duke University participated in an experiment. Each subject was asked to administer an IQ test to another student

and then provide the test taker with his or her percentile score. Unknown to the subject, the test taker was a bogus

student who was working with the researchers. The experimenters manipulated two factors: subject visibility and

success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of

the test taker was either visible or not visible to the test taker. Success of the test taker was either

top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions, then

the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test

taker was measured. (This variable is called the latency to feedback.) The data were subjected to appropriate analyses

with the following results.

Source | df | SS | MS | F | PR > F |

Subject visibility | 1 | 1380.24 | 1380.24 | 4.26 | 0.043 |

Test taker success | 1 | 1325.16 | 1325.16 | 4.09 | 0.050 |

Interaction | 1 | 3385.80 | 3385.80 | 10.45 | 0.002 |

Error | 36 | 11,664.00 | 324.00 | ||

Total | 39 | 17,755.20 |

Referring to the table, in the context of this study, interpret the statement: “Subject visibility and test taker success interact.”

a. The difference between the mean feedback time for visible and nonvisible subjects depends on the success of the test taker.

b. The difference between the mean feedback time for test takers scoring in the top 20% and bottom 20% depends on the visibility of the subject.

c. The relationship between feedback time and subject visibility depends on the success of the test taker.

d. All of the above are correct interpretations.

Q25. A candy 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 6 small cities and offers the candy bar at different prices. Using candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below:

City | Price ($) | Sales |

River Falls | 1.30 | 100 |

Hudson | 1.60 | 90 |

Ellsworth | 1.80 | 90 |

Prescott | 2.00 | 40 |

Rock Elm | 2.40 | 38 |

Stillwater | 2.90 | 32 |

Referring to the table, what is the coefficient of correlation for these data?

a. -0.8854

b. -0.7839

c. 0.7839

d. 0.8854

Q26. A ____________ is a numerical quantity computed from the data of a sample and is used in reaching a decision on whether or not to reject the null hypothesis.

a. significance level

b. critical value

c. test statistic

d. parameter

Q27. An investment specialist claims that if one holds a portfolio that moves in opposite direction to the market index like the S&P 500, 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 S&P 500 index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the S&P 500 index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio (Y) on the returns of S&P 500 index (X) to prove that the private prisons stock portfolio is negatively related to the S&P 500 index at a 5% level of significance. The results are given in the following EXCEL output.

Coefficients | Standard Error | T Stat | P-value | |

Intercept | 4.866004258 | 0.35743609 | 13.61363441 | 8.7932E-13 |

S&P | -0.502513506 | 0.071597152 | -7.01862425 | 2.94942E-07 |

Referring to the table, which of the following will be a correct conclusion?

a. We cannot reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prison stock portfolio and S&P 500 index are negatively related.

b. We can reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prison stock portfolio and S&P 500 index are negatively related.

c. We cannot reject the null hypothesis and, therefore, conclude that there is not sufficient evidence to show that the prison stock portfolio and S&P 500 index are negatively related.

d. We can reject the null hypothesis and conclude that there is not sufficient evidence to show that the prison stock portfolio and S&P 500 index are negatively related.

Q28. One criterion used to evaluate employees in the assembly section of a large factory is the number of defective pieces per 1,000 parts produced. The quality control department wants to find out whether there is a relationship between years of experience and defect rate. Since the job is repetitious, after the initial training period any improvement due to a learning effect might be offset by a loss of motivation. A defect rate is calculated for each worker in a yearly evaluation. The results for 100 workers are given in the table below.

Years Since Training Period | ||||

< 1 Year | 1 – 4 Years | 5 – 9 Years | ||

Defect Rate | High | 6 | 9 | 9 |

Average | 9 | 19 | 23 | |

Low | 7 | 8 | 10 |

Referring to the table, find the rejection region necessary for testing at the 0.05 level of significance whether there is a relationship between defect rate and years of experience.

a. Reject H_{0} if chi-square > 16.919

b. Reject H_{0} if chi-square > 15.507

c. Reject H_{0} if chi-square > 11.143

d. Reject H_{0} if chi-square > 9.488

Q29. A campus researcher wanted to investigate the factors that affect visitor travel time in a complex, multilevel building on campus. Specifically, he wanted to determine whether different building signs (building maps versus wall signage) affect the total amount of time visitors require to reach their destination and whether that time depends on whether the starting location is inside or outside the building. Three subjects were assigned to each of the combinations of signs and starting locations, and travel time in seconds from beginning to destination was recorded. An Excel output of the appropriate analysis is given below:

ANOVA | ||||||

Source of Variation | SS | df | MS | F | P-value | F crit |

Signs | 14008.33 | 14008.33 | 0.11267 | 5.317645 | ||

Starting Location | 12288 | 2.784395 | 0.13374 | 5.317645 | ||

Interaction | 48 | 48 | 0.919506 | 5.317645 | ||

Within | 35305.33 | 4413.167 | ||||

Total | 61649.67 | 11 |

Referring to the table, the within (error) degrees of freedom is

a. 1

b. 4

c. 8

d. 11

Q30. If we wish to determine whether there is evidence that the proportion of successes is higher in group 1 than in group 2, the appropriate test to use is

a. the Z test.

b. the chi-squared test.

c. the W test.

d. the X test.

Q31. A researcher randomly sampled 30 graduates of an MBA program and recorded data concerning their starting salaries. Of primary interest to the researcher was the effect of gender on starting salaries. Analysis of the mean salaries of the females and males in the sample is given below.

Size | Mean | Std Dev | |

Females | 18 | 48,266.7 | 13,577.63 |

Males | 12 | 55,000 | 11,741.29 |

Std Error = 4,764.82 | |||

Means Diff = -6,733.3 | |||

Z = -1.4528 2-tailed p value = 0.1463 | |||

T = -1.4221 2-tailed p value = 0.1574 |

Referring to the table, the researcher was attempting to show statistically that the female MBA graduates have a significantly lower mean starting salary than the male MBA graduates. From the analysis in the table, the correct test statistic is:

a. 4,634.72

b. -1.4221

c. -1.4528

d. -6,733.33

Q32. If we are performing a two-tailed test of whether μ = 100, the probability of detecting a shift of the mean to 105 will be ________ the probability of detecting a shift of the mean to 110.

a. less than

b. greater than

c. equal to

d. not comparable to

Q33. Parents complain that children read too few storybooks and watch too µch television nowadays. A survey of 1,000 children reveals the following information on average time spent watching TV and average time spent reading storybooks

Average time spent reading storybooks | |||

Average time spent watching TV |
Less than 1 hour |
Between 1 and 2 hours |
More than 2 hours |

Less than 2 hours | 90 | 85 | 130 |

More than 2 hours | 655 | 32 | 8 |

Referring to the table, to test whether there is any relationship between average time spent watching TV and average time spent reading storybooks, the value of the measured test statistic is:

a. -12.59

b. 1.61

c. 481.49

d. 1,368.06

Q34. A large national bank charges local companies for using their services. A bank official reported the results of a regression analysis designed to predict the bank’s charges (Y) — measured in dollars per month — for services rendered to local companies. One independent variable used to predict service charge to a company is the company’s sales revenue (X) — measured in millions of dollars. Data for 21 companies who use the bank’s services were used to fit the model:

E(Y) = ß_{0} + ß_{1}X

The results of the simple linear regression are provided below.

Y = -2,700+20X, s_{yx} = 65, two-tailed p value = 0.034 (for testing ß_{1})

Referring to Table 13-1, interpret the _{p} value for testing whether ß_{1} exceeds 0.

a. There is sufficient evidence (at the α = 0.05) to conclude that sales revenue (X) is a useful linear predictor of service charge (Y).

b. There is insufficient evidence (at the α = 0.10) to conclude that sales revenue (X) is a useful linear predictor of service charge (Y).

c. Sales revenue (X) is a poor predictor of service charge (Y).

d. For every $1 million increase in sales revenue, we expect a service charge to increase $0.034.

Q35. A study published in the American Journal of Public Health was conducted to determine whether the use of seat belts in motor vehicles depends on ethnic status in San Diego County. A sample of 792 children treated for injuries sustained from motor vehicle accidents was obtained, and each child was classified according to (1) ethnic status (Hispanic or non-Hispanic) and (2) seat belt usage (worn or not worn) during the accident. The number of children in each category is given in the table below.

Hispanic | Non-Hispanic | |

Seat belts worn | 31 | 148 |

Seat belts not worn | 283 | 330 |

Referring to the table, which test would be used to properly analyze the data in this experiment?

a. chi-square test for independence in a two-way contingency table.

b. chi-square test for equal proportions in a one-way table.

c. ANOVA F test for interaction in a 2 x 2 factorial design.

d. chi-square test for a 2 x 2 factorial design.