# MATHEMATICS

**12.** TABLE 13-1

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 the 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*) = *β*_{0} + *β*_{1}*X*

The results of the simple linear regression are provided below.

= -2,700 + 20*X*, S* _{YX}* = 65, two-tailed

*p*value = 0.034 (for testing

*β*

_{1})

Referring to Table 13-1, interpret the estimate of σ, the standard deviation of the random error term (standard error of the estimate) in the model.

[removed]A) For every $1 million increase in sales revenue, we expect a service charge to increase $65.

[removed]B) About 95% of the observed service charges equal their corresponding predicted values.

[removed]C) About 95% of the observed service charges fall within $130 of the least squares line.

[removed]D) About 95% of the observed service charges fall within $65 of the least squares line.

**13.** In a simple linear regression problem, *r* and *b*_{1}

[removed]A) must have opposite signs.

[removed]B) may have opposite signs.

[removed]C) must have the same sign.

[removed]D) are equal.

**14.** The sample correlation coefficient between *X* and *Y* is 0.375. It has been found that the *p*-value is 0.256 when testing *H*_{0} : *ρ* = 0 against the two-sided alternative *H*_{1} : *ρ* ≠ 0. To test *H*_{0} : *ρ* = 0 against the one-sided alternative *H*_{1} : *ρ* < 0 at a significance level of 0.2, the *p*-value is:

[removed]A) 0.256/2

[removed]B) 1 – 0.256/2

[removed]C) (0.256)2

[removed]D) 1 – 0.256

**15.** Assuming a linear relationship between *X* and *Y*, if the coefficient of correlation (*r*) equals – 0.30,

[removed]A) the slope (*b*_{1}) is negative.

[removed]B) variable *X* is larger than variable *Y*.

[removed]C) there is no correlation.

[removed]D) the variance of *X* is negative.

**16.** In a multiple regression model, the adjusted *r*^{2}

[removed]A) cannot be negative.

[removed]B) has to fall between 0 and +1.

[removed]C) can sometimes be greater than +1.

[removed]D) can sometimes be negative.

**17.** TABLE 14-5

A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression.

SUMMARY OUTPUT

ANOVA

Referring to Table 14-5, what fraction of the variability in sales is explained by spending on capital and wages?

[removed]A) 68.9%

[removed]B) 50.9%

[removed]C) 83.0%

[removed]D) 27.0%

**18.** TABLE 14-1

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:

Referring to Table 14-1, if an employee who had been with the company 5 years scored a 9 on the aptitude test, what would his estimated expected sales be?

[removed]A) 60.88

[removed]B) 79.09

[removed]C) 17.98

[removed]D) 55.62

**19.** To explain personal consumption (CONS) measured in dollars, data is collected for

A regression analysis was performed with CONS as the dependent variable and ln(CRDTLIM), ln(APR), ln(ADVT), and SEX as the independent variables. The estimated model was

= 2.28 – 0.29 ln(CRDTLIM) + 5.77 ln(APR) + 2.35 ln(ADVT) + 0.39 SEX

What is the correct interpretation for the estimated coefficient for SEX?

[removed]A) Holding everything else fixed, personal consumption for females is estimated to be 0.39% higher than males on the average.

[removed]B) Holding everything else fixed, personal consumption for males is estimated to be 0.39% higher than females on the average.

[removed]C) Holding everything else fixed, personal consumption for females is estimated to be $0.39 higher than males on the average.

[removed]D) Holding everything else fixed, personal consumption for males is estimated to be $0.39 higher than females on the average.

**20.** The logarithm transformation can be used

[removed]A) to overcome violations of the homoscedasticity assumption.

[removed]B) to test for possible violations of the homoscedasticity assumption.

[removed]C) to overcome violations of the autocorrelation assumption.

[removed]D) to test for possible violations of the autocorrelation assumption.

**21.** Which of the following is NOT used to determine observations that have an influential effect on the fitted model?

[removed]A) the hat matrix elements *h _{i}*

[removed]B) the

*C*statistic

_{p}[removed]C) the Studentized deleted residuals

*t*

_{i}[removed]D) Cook’s distance statistic

**22.** TABLE 15-2

A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they will gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model:

Y = *β*_{0} + *β*_{1}*X* + *β*_{2}*X*^{2} + ε

where Y = demand (in thousands) and X = retail price per carat.

This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below:

SUMMARY OUTPUT

ANOVA

Referring to Table 15-2, what is the value of the test statistic for testing whether there is an upward curvature in the response curve relating the demand (*Y*) and the price (*X*)?

[removed]A) -5.14

[removed]B) 373

[removed]C) 0.95

[removed]D) none of the above

**23.** After estimating a trend model for annual time-series data, you obtain the following residual plot against time:

The problem with your model is that

[removed]A) the irregular component has not been accounted for.

[removed]B) the trend component has not been accounted for.

[removed]C) the cyclical component has not been accounted for.

[removed]D) the seasonal component has not been accounted for,

**24.**

TABLE 16-4

Given below are Excel outputs for various estimated autoregressive models for Coca-Cola’s real operating revenues (in billions of dollars) from 1975 to 1998. From the data, we also know that the real operating revenues for 1996, 1997, and 1998 are 11.7909, 11.7757 and 11.5537, respectively.

AR(1) Model:

AR(2)Model:

AR(3) Model:

Referring to Table 16-4, if one decides to use AR(3), what will the predicted real operating revenue for Coca-Cola be in 2001?

[removed]A) $11.68 billion

[removed]B) $11.59 billion

[removed]C) $12.47 billion

[removed]D) $11.84 billion

**25.**TABLE 16-3

The following table contains the number of complaints received in a department store for the first 6 months of last year.

Referring to Table 16-3, suppose the last two smoothed values are 81 and 96. (Note: they are not.) What would you forecast as the value of the time series for September?

[removed]A) 96

[removed]B) 81

[removed]C) 91

[removed]D) 86

**26.** A tabular presentation that shows the outcome for each decision alternative under the various states of nature is called:

[removed]A) a payback period matrix.

[removed]B) a decision tree.

[removed]C) a payoff table.

[removed]D) a decision matrix.

**27.** Blossom’s Flowers purchases roses for sale for Valentine’s Day. The roses are purchased for $10 a dozen and are sold for $20 a dozen. Any roses not sold on Valentine’s Day can be sold for $5 per dozen. The owner will purchase 1 of 3 amounts of roses for Valentine’s Day: 100, 200, or 400 dozen roses. If the probability of selling 100 dozen roses is 0.2 and 200 dozen roses is 0.5, then the probability of selling 400 dozen roses is

[removed]A) 0.7

[removed]B) 0.5

[removed]C) 0.2

[removed]D) 0.3

**28.** TABLE 17-2

The following payoff matrix is given in dollars.

Suppose the probability of Event 1 is 0.5 and Event 2 is 0.5.

Referring to Table 17-2, the *EVPI* is

[removed]A) 600

[removed]B) 400

[removed]C) 0

[removed]D) 300

**29.** TABLE 18-1

A local newspaper has 10 delivery boys who each deliver the morning paper to 50 customers every day. The owner decides to record the percentage of papers delivered on time for a 10-day period and construct a *p* chart to see whether the percentage is too erratic.

Referring to Table 18-1, which expression best characterizes the *p* chart?

[removed]A) increasing trend

[removed]B) cycles

[removed]C) in-control

[removed]D) individual outliers