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Quantitative Analysis 【Reading 12】Sample

Consider the following estimated regression equation, with calculated t-statistics of the estimates as indicated:
AUTOt = 10.0 + 1.25 PIt + 1.0 TEENt – 2.0 INSt
with a PI calculated t-statstic of 0.45, a TEEN calculated t-statstic of 2.2, and an INS calculated t-statstic of 0.63.

The equation was estimated over 40 companies. Using a 5% level of significance, which of the independent variables significantly different from zero?
A)
PI and INS only.
B)
PI only.
C)
TEEN only.



The critical t-values for 40-3-1 = 36 degrees of freedom and a 5% level of significance are ± 2.028. Therefore, only TEEN is statistically significant.

Consider a study of 100 university endowment funds that was conducted to determine if the funds’ annual risk-adjusted returns could be explained by the size of the fund and the percentage of fund assets that are managed to an indexing strategy. The equation used to model this relationship is:

ARARi = b0 + b1Sizei + b2Indexi + ei
Where:

ARARi
=
the average annual risk-adjusted percent returns for the fund i over the 1998-2002 time period.

Sizei
=
the natural logarithm of the average assets under management for fund i.

Indexi
=
the percentage of assets in fund i that were managed to an indexing strategy.

The table below contains a portion of the regression results from the study.

Partial Results from Regression ARAR on Size and Extent of Indexing


Coefficients

Standard Error

t-Statistic


Intercept

???

0.55

−5.2


Size

0.6

0.18

???


Index

1.1

???

2.1

Which of the following is the most accurate interpretation of the slope coefficient for size? ARAR:
A)
and index will change by 1.1% when the natural logarithm of assets under management changes by 1.0.
B)
will change by 0.6% when the natural logarithm of assets under management changes by 1.0, holding index constant.
C)
will change by 1.0% when the natural logarithm of assets under management changes by 0.6, holding index constant.



A slope coefficient in a multiple linear regression model measures how much the dependent variable changes for a one-unit change in the independent variable, holding all other independent variables constant. In this case, the independent variable size (= ln average assets under management) has a slope coefficient of 0.6, indicating that the dependent variable ARAR will change by 0.6% return for a one-unit change in size, assuming nothing else changes. Pay attention to the units on the dependent variable. <STUDY Session 3, LOS 12.a)

Which of the following is the estimated standard error of the regression coefficient for index?
A)
0.52.
B)
2.31.
C)
1.91.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.
Using the information provided, the estimated coefficient standard error can be computed as bIndex / t = σIndex = 1.1 / 2.1 = 0.5238.
(Study session 3, LOS 12.b)


Which of the following is the t-statistic for size?
A)
0.70.
B)
0.30.
C)
3.33.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.
Using the information provided, the t-statistic for size can be computed as t = bSize / σSize = 0.6 / 0.18 = 3.3333.
(Study session 3, LOS 12.b)


Which of the following is the estimated intercept for the regression?
A)
−9.45.
B)
−0.11.
C)
−2.86.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated parameter, and σi is the parameter’s standard error.
Using the information provided, the estimated intercept can be computed as b0 = t × σ0 = −5.2 × 0.55 = −2.86.
(Study session 3, LOS 12.b)


Which of the following statements is most accurate regarding the significance of the regression parameters at a 5% level of significance?
A)
The parameter estimates for the intercept are significantly different than zero. The slope coefficients for index and size are not significant.
B)
All of the parameter estimates are significantly different than zero at the 5% level of significance.
C)
The parameter estimates for the intercept and the independent variable size are significantly different than zero. The coefficient for index is not significant.



At 5% significance and 97 degrees of freedom (100 − 3), the critical t-value is slightly greater than, but very close to, 1.984.
The t-statistic for the intercept and index are provided as −5.2 and 2.1, respectively, and the t-statistic for size is computed as 0.6 / 0.18 = 3.33.
The absolute value of the all of the regression intercepts is greater than tcritical = 1.984.
Thus, it can be concluded that all of the parameter estimates are significantly different than zero at the 5% level of significance.
(Study session 3, LOS 12.b)


Which of the following is NOT a required assumption for multiple linear regression?
A)
The error term is normally distributed.
B)
The expected value of the error term is zero.
C)
The error term is linearly related to the dependent variable.



The assumptions of multiple linear regression include: linear relationship between dependent and independent variable, independent variables are not random and no exact linear relationship exists between the two or more independent variables, error term is normally distributed with an expected value of zero and constant variance, and the error term is serially uncorrelated. (Study Session 3, LOS 12.d)

TOP

Consider a study of 100 university endowment funds that was conducted to determine if the funds’ annual risk-adjusted returns could be explained by the size of the fund and the percentage of fund assets that are managed to an indexing strategy. The equation used to model this relationship is:

ARARi = b0 + b1Sizei + b2Indexi + ei
Where:

ARARi
=
the average annual risk-adjusted percent returns for the fund i over the 1998-2002 time period.

Sizei
=
the natural logarithm of the average assets under management for fund i.

Indexi
=
the percentage of assets in fund i that were managed to an indexing strategy.

The table below contains a portion of the regression results from the study.

Partial Results from Regression ARAR on Size and Extent of Indexing


Coefficients

Standard Error

t-Statistic


Intercept

???

0.55

−5.2


Size

0.6

0.18

???


Index

1.1

???

2.1

Which of the following is the most accurate interpretation of the slope coefficient for size? ARAR:
A)
and index will change by 1.1% when the natural logarithm of assets under management changes by 1.0.
B)
will change by 0.6% when the natural logarithm of assets under management changes by 1.0, holding index constant.
C)
will change by 1.0% when the natural logarithm of assets under management changes by 0.6, holding index constant.



A slope coefficient in a multiple linear regression model measures how much the dependent variable changes for a one-unit change in the independent variable, holding all other independent variables constant. In this case, the independent variable size (= ln average assets under management) has a slope coefficient of 0.6, indicating that the dependent variable ARAR will change by 0.6% return for a one-unit change in size, assuming nothing else changes. Pay attention to the units on the dependent variable. <STUDY Session 3, LOS 12.a)

Which of the following is the estimated standard error of the regression coefficient for index?
A)
0.52.
B)
2.31.
C)
1.91.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.
Using the information provided, the estimated coefficient standard error can be computed as bIndex / t = σIndex = 1.1 / 2.1 = 0.5238.
(Study session 3, LOS 12.b)


Which of the following is the t-statistic for size?
A)
0.70.
B)
0.30.
C)
3.33.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.
Using the information provided, the t-statistic for size can be computed as t = bSize / σSize = 0.6 / 0.18 = 3.3333.
(Study session 3, LOS 12.b)


Which of the following is the estimated intercept for the regression?
A)
−9.45.
B)
−0.11.
C)
−2.86.



The t-statistic for testing the null hypothesis H0:
βi = 0 is t = (bi − 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated parameter, and σi is the parameter’s standard error.
Using the information provided, the estimated intercept can be computed as b0 = t × σ0 = −5.2 × 0.55 = −2.86.
(Study session 3, LOS 12.b)


Which of the following statements is most accurate regarding the significance of the regression parameters at a 5% level of significance?
A)
The parameter estimates for the intercept are significantly different than zero. The slope coefficients for index and size are not significant.
B)
All of the parameter estimates are significantly different than zero at the 5% level of significance.
C)
The parameter estimates for the intercept and the independent variable size are significantly different than zero. The coefficient for index is not significant.



At 5% significance and 97 degrees of freedom (100 − 3), the critical t-value is slightly greater than, but very close to, 1.984.
The t-statistic for the intercept and index are provided as −5.2 and 2.1, respectively, and the t-statistic for size is computed as 0.6 / 0.18 = 3.33.
The absolute value of the all of the regression intercepts is greater than tcritical = 1.984.
Thus, it can be concluded that all of the parameter estimates are significantly different than zero at the 5% level of significance.
(Study session 3, LOS 12.b)


Which of the following is NOT a required assumption for multiple linear regression?
A)
The error term is normally distributed.
B)
The expected value of the error term is zero.
C)
The error term is linearly related to the dependent variable.



The assumptions of multiple linear regression include: linear relationship between dependent and independent variable, independent variables are not random and no exact linear relationship exists between the two or more independent variables, error term is normally distributed with an expected value of zero and constant variance, and the error term is serially uncorrelated. (Study Session 3, LOS 12.d)

TOP

William Brent, CFA, is the chief financial officer for Mega Flowers, one of the largest producers of flowers and bedding plants in the Western United States. Mega Flowers grows its plants in three large nursery facilities located in California. Its products are sold in its company-owned retail nurseries as well as in large, home and garden “super centers”. For its retail stores, Mega Flowers has designed and implemented marketing plans each season that are aimed at its consumers in order to generate additional sales for certain high-margin products. To fully implement the marketing plan, additional contract salespeople are seasonally employed.
For the past several years, these marketing plans seemed to be successful, providing a significant boost in sales to those specific products highlighted by the marketing efforts. However, for the past year, revenues have been flat, even though marketing expenditures increased slightly. Brent is concerned that the expensive seasonal marketing campaigns are simply no longer generating the desired returns, and should either be significantly modified or eliminated altogether. He proposes that the company hire additional, permanent salespeople to focus on selling Mega Flowers’ high-margin products all year long. The chief operating officer, David Johnson, disagrees with Brent. He believes that although last year’s results were disappointing, the marketing campaign has demonstrated impressive results for the past five years, and should be continued. His belief is that the prior years’ performance can be used as a gauge for future results, and that a simple increase in the sales force will not bring about the desired results.
Brent gathers information regarding quarterly sales revenue and marketing expenditures for the past five years. Based upon historical data, Brent derives the following regression equation for Mega Flowers (stated in millions of dollars):

Expected Sales = 12.6 + 1.6 (Marketing Expenditures) + 1.2 (# of Salespeople)

Brent shows the equation to Johnson and tells him, “This equation shows that a $1 million increase in marketing expenditures will increase the independent variable by $1.6 million, all other factors being equal.” Johnson replies, “It also appears that sales will equal $12.6 million if all independent variables are equal to zero.”In regard to their conversation about the regression equation:
A)
Brent’s statement is correct; Johnson’s statement is correct.
B)
Brent’s statement is incorrect; Johnson’s statement is correct.
C)
Brent’s statement is correct; Johnson’s statement is incorrect.



Expected sales is the dependent variable in the equation, while expenditures for marketing and salespeople are the independent variables. Therefore, a $1 million increase in marketing expenditures will increase the dependent variable (expected sales) by $1.6 million. Brent’s statement is incorrect.Johnson’s statement is correct. 12.6 is the intercept in the equation, which means that if all independent variables are equal to zero, expected sales will be $12.6 million. (Study Session 3, LOS 12.a)


Using data from the past 20 quarters, Brent calculates the t-statistic for marketing expenditures to be 3.68 and the t-statistic for salespeople at 2.19. At a 5% significance level, the two-tailed critical values are tc = +/- 2.127. This most likely indicates that:
A)
the t-statistic has 18 degrees of freedom.
B)
both independent variables are statistically significant.
C)
the null hypothesis should not be rejected.



Using a 5% significance level with degrees of freedom (df) of 17 (20 – 2 – 1), both independent variables are significant and contribute to the level of expected sales. (Study Session 3, LOS 12.a)

Brent calculated that the sum of squared errors (SSE) for the variables is 267. The mean squared error (MSE) would be:
A)
14.831.
B)
15.706.
C)
14.055.



The MSE is calculated as SSE / (n – k – 1). Recall that there are twenty observations and two independent variables. Therefore, the MSE in this instance [267 / (20 – 2 – 1)] = 15.706. (Study Session 3, LOS 11.i)

Brent is trying to explain the concept of the standard error of estimate (SEE) to Johnson. In his explanation, Brent makes three points about the SEE:
  • Point 1: The SEE is the standard deviation of the differences between the estimated values for the independent variables and the actual observations for the independent variable.
  • Point 2: Any violation of the basic assumptions of a multiple regression model is going to affect the SEE.
  • Point 3: If there is a strong relationship between the variables and the SSE is small, the individual estimation errors will also be small.

How many of Brent’s points are most accurate?
A)
2 of Brent’s points are correct.
B)
1 of Brent’s points are correct.
C)
All 3 of Brent’s points are correct.



The statements that if there is a strong relationship between the variables and the SSE is small, the individual estimation errors will also be small, and also that any violation of the basic assumptions of a multiple regression model is going to affect the SEE are both correct.
The SEE is the standard deviation of the differences between the estimated values for the dependent variables (not independent) and the actual observations for the dependent variable. Brent’s Point 1 is incorrect.
Therefore, 2 of Brent’s points are correct. (Study Session 3, LOS 11.f)


Assuming that next year’s marketing expenditures are $3,500,000 and there are five salespeople, predicted sales for Mega Flowers will be:
A)
$11,600,000.
B)
$24,200,000.
C)
$2,400,000.



Using the regression equation from above, expected sales equals 12.6 + (1.6 x 3.5) + (1.2 x 5) = $24.2 million. Remember to check the details – i.e. this equation is denominated in millions of dollars. (Study Session 3, LOS 12.c)

Brent would like to further investigate whether at least one of the independent variables can explain a significant portion of the variation of the dependent variable. Which of the following methods would be best for Brent to use?
A)
The multiple coefficient of determination.
B)
The F-statistic.
C)
An ANOVA table.



To determine whether at least one of the coefficients is statistically significant, the calculated F-statistic is compared with the critical F-value at the appropriate level of significance. (Study Session 3, LOS 12.e)

TOP

Consider the following estimated regression equation, with the standard errors of the slope coefficients as noted:
Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi – 2.0 COMPi + 8.0 CAPi
where the standard error for the estimated coefficient on R&D is 0.45, the standard error for the estimated coefficient on ADV is 2.2 , the standard error for the estimated coefficient on COMP is 0.63, and the standard error for the estimated coefficient on CAP is 2.5.

The equation was estimated over 40 companies. Using a 5% level of significance, which of the estimated coefficients are significantly different from zero?
A)
R&D, COMP, and CAP only.
B)
R&D, ADV, COMP, and CAP.
C)
ADV and CAP only.


The critical t-values for 40-4-1 = 35 degrees of freedom and a 5% level of significance are ± 2.03.
The calculated t-values are:
t for R&D = 1.25 / 0.45 = 2.777
t for ADV = 1.0/ 2.2 = 0.455
t for COMP = -2.0 / 0.63 = -3.175
t for CAP = 8.0 / 2.5 = 3.2

TOP

Consider the following regression equation:
Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi – 2.0 COMPi + 8.0 CAPi
where Sales is dollar sales in millions, R&D is research and development expenditures in millions, ADV is dollar amount spent on advertising in millions, COMP is the number of competitors in the industry, and CAP is the capital expenditures for the period in millions of dollars.  

Which of the following is NOT a correct interpretation of this regression information
A)
If a company spends $1 million more on capital expenditures (holding everything else constant), Sales are expected to increase by $8.0 million.
B)
One more competitor will mean $2 million less in Sales (holding everything else constant).
C)
If R&D and advertising expenditures are $1 million each, there are 5 competitors, and capital expenditures are $2 million, expected Sales are $8.25 million.



Predicted sales = $10 + 1.25 + 1 – 10 + 16 = $18.25 million.

TOP

Damon Washburn, CFA, is currently enrolled as a part-time graduate student at State University. One of his recent assignments for his course on Quantitative Analysis is to perform a regression analysis utilizing the concepts covered during the semester. He must interpret the results of the regression as well as the test statistics. Washburn is confident in his ability to calculate the statistics because the class is allowed to use statistical software. However, he realizes that the interpretation of the statistics will be the true test of his knowledge of regression analysis. His professor has given to the students a list of questions that must be answered by the results of the analysis.
Washburn has estimated a regression equation in which 160 quarterly returns on the S&P 500 are explained by three macroeconomic variables: employment growth (EMP) as measured by nonfarm payrolls, gross domestic product (GDP) growth, and private investment (INV). The results of the regression analysis are as follows:
Coefficient Estimates
Parameter Coefficient Standard Error
of Coefficient
Intercept 9.50 3.40
EMP -4.50 1.25
GDP 4.20 0.76
INV -0.30 0.16

Other Data:
  • Regression sum of squares (RSS) = 126.00
  • Sum of squared errors (SSE) = 267.00
  • Durbin-Watson statistic (DW) = 1.34

Abbreviated Table of the Student’s t-distribution (One-Tailed Probabilities)

df p = 0.10 p = 0.05 p = 0.025 p = 0.01 p = 0.005
3 1.638 2.353 3.182 4.541 5.841
10 1.372 1.812 2.228 2.764 3.169
50 1.299 1.676 2.009 2.4032.678
100 1.290 1.660 1.984 2.364 2.626
120 1.289 1.658 1.980 2.358 2.617
200 1.286 1.653 1.972 2.345 2.601

Critical Values for Durbin-Watson Statistic (α = 0.05)

K=1 K=2K=3K=4 K=5
ndldudldudldu dl du dl du
201.201.411.101.541.001.68 0.90 1.83 0.79 1.99
501.501.591.461.631.421.67 1.38 1.72 1.34 1.77
>1001.651.691.631.721.611.74 1.59 1.76 1.57 1.78
How many of the three independent variables (not including the intercept term) are statistically significant in explaining quarterly stock returns at the 5.0% level?
A)
One of the three is statistically significant.
B)
Two of the three are statistically significant.
C)
All three are statistically significant.



To determine whether the independent variables are statistically significant, we use the student’s t-statistic, where t equals the coefficient estimate divided by the standard error of the coefficient. This is a two-tailed test. The critical value for a 5.0% significance level and 156 degrees of freedom (160-3-1) is about 1.980, according to the table.
The t-statistic for employment growth = -4.50/1.25 = -3.60.
The t-statistic for GDP growth = 4.20/0.76 = 5.53.
The t-statistic for investment growth = -0.30/0.16 = -1.88.
Therefore, employment growth and GDP growth are statistically significant because the absolute values of their t-statistics are larger than the critical value, which means two of the three independent variables are statistically significantly different from zero. (Study Session 3, LOS 12.a)


Can the null hypothesis that the GDP growth coefficient is equal to 3.50 be rejected at the 1.0% confidence level versus the alternative that it is not equal to 3.50? The null hypothesis is:
A)
rejected because the t-statistic is less than 2.617.
B)
not rejected because the t-statistic is equal to 0.92.
C)
accepted because the t-statistic is less than 2.617.



The hypothesis is:

H0: bGDP = 3.50
Ha: bGDP ≠ 3.50

This is a two-tailed test. The critical value for the 1.0% significance level and 156 degrees of freedom (160 − 3 − 1) is about 2.617. The t-statistic is (4.20 − 3.50)/0.76 = 0.92. Because the t-statistic is less than the critical value, we cannot reject the null hypothesis. Notice we cannot say that the null hypothesis is accepted; only that it is not rejected. (Study Session 3, LOS 12.b)


The percentage of the total variation in quarterly stock returns explained by the independent variables is closest to:
A)
32%.
B)
47%.
C)
42%.



The R2 is the percentage of variation in the dependent variable explained by the independent variables. The R2 is equal to the SSRegession/SSTotal, where the SSTotal is equal to SSRegression + SSError. R2 = 126.00/(126.00+267.00) = 32%. (Study Session 3, LOS 12.f)

According to the Durbin-Watson statistic, there is:
A)
no significant positive serial correlation in the residuals.
B)
significant positive serial correlation in the residuals.
C)
significant heteroskedasticity in the residuals.



The Durbin-Watson statistic tests for serial correlation in the residuals. According to the table, dl = 1.61 and du = 1.74 for three independent variables and 160 degrees of freedom. Because the DW (1.34) is less than the lower value (1.61), the null hypothesis of no significant positive serial correlation can be rejected. This means there is a problem with serial correlation in the regression, which affects the interpretation of the results. (Study Session 3, LOS 12.i)



What is the predicted quarterly stock return, given the following forecasts?
  • Employment growth = 2.0%
  • GDP growth = 1.0%
  • Private investment growth = -1.0%
A)
5.0%.
B)
4.4%.
C)
23.0%.



Predicted quarterly stock return is 9.50% + (-4.50)(2.0%) + (4.20)(1.0%) + (-0.30)(-1.0%) = 5.0%. (Study Session 3, LOS 12.c)

What is the standard error of the estimate?
A)
1.71.
B)
0.81.
C)
1.31.



The standard error of the estimate is equal to [SSE/(n − k − 1)]1/2 = [267.00/156]1/2 = approximately 1.31. (Study Session 3, LOS 11.i)

TOP

Consider the following regression equation:
Salesi = 20.5 + 1.5 R&Di + 2.5 ADVi – 3.0 COMPi
where Sales is dollar sales in millions, R&D is research and development expenditures in millions, ADV is dollar amount spent on advertising in millions, and COMP is the number of competitors in the industry.

Which of the following is NOT a correct interpretation of this regression information?
A)
One more competitor will mean $3 million less in sales (holding everything else constant).
B)
If R&D and advertising expenditures are $1 million each and there are 5 competitors, expected sales are $9.5 million.
C)
If a company spends $1 more on R&D (holding everything else constant), sales are expected to increase by $1.5 million.



If a company spends $1 million more on R&D (holding everything else constant), sales are expected to increase by $1.5 million. Always be aware of the units of measure for the different variables.

TOP

Henry Hilton, CFA, is undertaking an analysis of the bicycle industry. He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV). All data are measured in millions of units. Hilton gathers data for the last 20 years. Which of the follow regression equations correctly represents Hilton’s hypothesis?
A)
SALES = α x β1 POP x β2 INCOME x β3 ADV x ε.
B)
SALES = α + β1 POP + β2 INCOME + β3 ADV + ε.
C)
INCOME = α + β1 POP + β2 SALES + β3 ADV + ε.



SALES is the dependent variable. POP, INCOME, and ADV should be the independent variables (on the right hand side) of the equation (in any order). Regression equations are additive.

TOP

Henry Hilton, CFA, is undertaking an analysis of the bicycle industry.
He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV).
All data are measured in millions of units.
Hilton gathers data for the last 20 years and estimates the following equation (standard errors in parentheses):  

SALES = α + 0.004 POP + 1.031 INCOME + 2.002 ADV

(0.005)

(0.337)

(2.312)

  

The critical t-statistic for a 95% confidence level is 2.120.
Which of the independent variables is statistically different from zero at the 95% confidence level?

A)
ADV only.
B)
INCOME only.
C)
INCOME and ADV.



The calculated test statistic is coefficient/standard error. Hence, the t-stats are 0.8 for POP, 3.059 for INCOME, and 0.866 for ADV. Since the t-stat for INCOME is the only one greater than the critical t-value of 2.120, only INCOME is significantly different from zero.

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