Quantitative Research Assignment: Regression and Correlation Analysis

This assignment solution provides a comprehensive analysis of wage data using quantitative research methods. It begins with descriptive statistics, including measures of central tendency and dispersion, to summarize the data. The solution then explores the relationships between various variables and wage using correlation analysis, identifying the strength and direction of these relationships. Simple and multiple regression models are developed to examine the impact of education and other factors (IQ, hours, experience, tenure, age, siblings) on wage. The analysis includes the interpretation of coefficients, R-squared values, and the significance of independent variables. Hypothesis testing is employed to determine the statistical significance of the relationships, and predictions are made based on the regression models. The results of the regression models are interpreted, highlighting the effects of each independent variable on wage, and the overall fit of the models.

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ASSIGNMENT- QUANTITATIVE RESEARCH
Question 1: Descriptive statistics
The descriptive analysis of the data is provided on the table below. The measures of
central tendency are mean, median and mode. The mean gives the average value for each
variable, while the median is a single number that represents the center of the data set. The
mode indicates the value with the highest frequency for each variable. The measures of
dispersion are range and standard deviation. The range gives the difference of the maximum
and the minimum value for each variable. The standard deviation indicated how dispersed the
data points of each variable are spread out from the mean.
Statistics
WAGE HOURS IQ KWW EDUC EXPER
TENUR
E AGE SIBS MEDUC FED
N Valid 479 479 479 479 479 479 479 479 479 479
Missing 0 0 0 0 0 0 0 0 0 0
Mean 1007.994 44.422 103.816 36.706 13.743 11.628 7.332 33.025 2.754 10.983 10
Median 962.000 40.000 105.000 37.000 13.000 11.000 7.000 33.000 2.000 12.000 11
Mode 1000.0 40.0 109.0 38.0 12.0 9.0 1.0 30.0 1.0 12.0
Std. Deviation 411.5006 7.1181 14.2482 7.4482 2.2358 4.1534 5.0970 3.0852 2.2016 2.6402 3.
Skewness 1.069 1.612 -.213 -.296 .412 .084 .410 .156 1.765 -.378
Range 2571.0 53.0 86.0 43.0 9.0 21.0 22.0 10.0 14.0 17.0
Minimum 200.0 27.0 59.0 13.0 9.0 1.0 .0 28.0 .0 1.0
Maximum 2771.0 80.0 145.0 56.0 18.0 22.0 22.0 38.0 14.0 18.0
There are 479 observations in this data set. The mean wage (monthly earnings) is
$1,007.99, while the median wage is $962.00. Hence, the data appears to be skewed to the
right, since the mean is greater than the median of wage. The skewness statistic is 1.069
which indicates that the wage data is very slightly skewed to the right. The range in the
variable is $2,571.00, with the highest monthly earnings being $2,771.00 and the lowest
monthly earnings being $200.00. The standard deviation for wage is 411.50, which means
that data is highly dispersed from its mean.
The mean hours is 44.42hours. The skewness is at 1.61, which indicates that the hours
data is highly skewed to the right, which means most of the data is close to the maximum
value. The range for the hours variable is 53 hours, with the lowest value being 27 hours and
the highest value being 80 hours.

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For the IQ variable, the mean is 103.82, the median is 105.00, and the mode is 109.00.
the range is 86.0, with the maximum value being 145.0 and the minimum value being 59.0.
Question 2: Correlation
The correlation table helps in finding out what variables affect Wage and how they affect
the wage. First, we find the correlation between all the variables and wage, using the
correlation matrix. The test is to see whether there is a relationship between each variable and
wage. Therefore, the hypothesis test is:
(No, the variable has no relationship with wage)
(Yes, the variable has a relationship with wage).
The decision rule is to reject if the significance is less than 0.05, and fail to reject
, when the significance is greater than 0.05.
If the test result is Yes, then we explain the nature of the relationship between the
variable and wage. First, if its positive or negative, based on the sign on the value of r. Next,
the value of r indicates the magnitude of the relationship. The table below indicate the criteria
used to determine the strength of the relationship between a variable and wage.
Magnitude of relationship Value of r
0.51 to 1.0, -0.51 to -1.0 Strong
0.31 to 0.5, -0.31 to -0.5 Moderate
0.11 to 0.3, -0.11 to -0.3 Weak
-0.1 to 0.1 None, or Very weak
The results for the correlation matrix are shown in the table below.
WAGE Relationship (Strength, Directions and Test results)
WAGE Pearson Correlation 1
Sig. (2-tailed)
HOURS Pearson Correlation -.030
Sig. (2-tailed) .516 >0.05, Fail to reject Ho. No relationship.
IQ Pearson Correlation .298** Moderate, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.
KWW Pearson Correlation .340** Strong, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.

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EDUC Pearson Correlation .331** Strong, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.
EXPER Pearson Correlation -.029
Sig. (2-tailed) .531 >0.05, Fail to reject Ho. No relationship.
TENURE Pearson Correlation .111* Weak, positive relationship
Sig. (2-tailed) .015 <0.05, Reject Ho; Yes relationship.
AGE Pearson Correlation .161** Weak, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.
SIBS Pearson Correlation -.120** Weak, negative relationship
Sig. (2-tailed) .009 <0.05, Reject Ho; Yes relationship.
MEDUC Pearson Correlation .224** Weak, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.
FEDUC Pearson Correlation .264** Weak, positive relationship
Sig. (2-tailed) .000 <0.05, Reject Ho; Yes relationship.
From the correlation matrix above, we find that the variables with a significant
relationship with wage at 0.05 significance level are IQ, KWW, educ, tenure, age, sibs,
meduc, feduc. Of these, only KWW and educ have a strong relationship with wage.
Question 3: Regression
We start with a simple regression, where we regress wage on education.
The results are shown in the table below. The regression model follows the format:
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1 EDUCb . Enter
a. Dependent Variable: WAGE
b. All requested variables entered.

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. The R-square for this model is 0.110*100%=11.0%. Thus, 11% of the
variation in monthly earning (wage) is explained by years of education (educ).
The F-Test is not explained for a simple model where we have one independent
variable.The table below gives the coefficients for the simple regression model above.
The significance level for the independent variable is 0.00 < 0.05, hence there is a
significant relationship between educ and wage, with a coefficient of . The
constant coefficient is . Therefore, the regression model becomes:
Hence, one more year of education increases the wage by $60.93, while a person with
zero years of education is expected to earn $170.60 monthly.
Effect:
If the years of education increase by two, the wage will increase by $121.86 (that is,
$60.93*2 years). If the years of education increase by five, the wage will increase by $304.65
(that is, $60.93*5 years).
Prediction:
The predicated wage for a person with 18 years of education, is found as:
Model Summary
Model R R Square
Adjusted R
Square
Std. Error of
the Estimate
1 .331a .110 .108 388.7030
a. Predictors: (Constant), EDUC
Coefficientsa
Model
Unstandardized
Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) 170.596 110.719 1.541 .124
EDUC 60.932 7.952 .331 7.663 .000
a. Dependent Variable: WAGE

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Question 4: Simple Regression
We have a simple regression model, where we regress wage on IQ score.
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1 IQb . Enter
a. Dependent Variable: WAGE
b. All requested variables entered.
The simple regression model follows the format: . From the results
tables below, we find that the R-square for this model is 0.089*100%=8.9%. Thus, 8.9% of
the variation in monthly earnings (wage) is explained by IQ score (IQ).
F-Test is not explained for a simple model where we have one independent variable. The
table below gives the coefficients for the simple regression model above.
Coefficientsa
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) 114.061 132.265 .862 .389
IQ 8.611 1.262 .298 6.822 .000
a. Dependent Variable: WAGE
The test the relationship between IQ score and wage, we set the hypothesis:
(No, IQ score has no relationship with wage)
(Yes, IQ score has a relationship with wage).
Model Summary
Model R R Square
Adjusted R
Square
Std. Error of the
Estimate
1 .298a .089 .087 393.1971
a. Predictors: (Constant), IQ

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The decision rule is to reject if the significance is less than 0.05, and fail to reject
, when the significance is greater than 0.05. The significance level for the independent
variable, IQ, is 0.000 < 0.05, hence we reject . There is a significant relationship between
IQ and wage, with a coefficient of . The constant coefficient is .
Therefore, the regression model becomes:
The resulting regression equation means that, one more point of IQ score increases the
wage by $8.61, while a person with zero points in IQ score is expected to earn a wage of
$114.06 monthly.
Effect:
If a person’s IQ score increases by 10 points, their wage is expected to increase by
$86.10 (that is, 8.61*10 points).
Prediction:
Using the minimum value of IQ score form our data set, 59, the predicted wage is:
Using the maximum value of IQ score form our data set, 145, the predicted wage is:
Question 5: Multiple Regression
Here we conduct a multiple regression analysis, where we regress wage on hours, IQ,
educ, exper, tenure, age, and sibs.
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1 SIBS, AGE,
HOURS, EDUC,
TENURE, IQ,
EXPERb
. Enter
a. Dependent Variable: WAGE
b. All requested variables entered.
The multiple regression model follows the format:

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.
From the results tables below, we find that the R-square for this model is
0.164*100%=16.4%. Thus, 16.4% of the variation in monthly earnings (wage) is explained
by average weekly hours of work (hours), IQ score (IQ), years of education (educ), years of
work experience (exper), years with current employer (tenure), age in years (age), and
number of siblings (sibs).
Model Summary
Model R R Square Adjusted R Square
Std. Error of the
Estimate
1 .405a .164 .151 379.0821
a. Predictors: (Constant), SIBS, AGE, HOURS, EDUC, TENURE, IQ, EXPER
The F-Test is used to evaluate the relationship between the seven independent variable
and the dependent variable, wage. We set the hypothesis as, at 0.05 significance level:
(The model is not a good fit)
(the mode is a god fit)
The decision rule is to reject when the p-value ≤ 0.05, and fail to reject , when the
p-value ≥ 0.05. From the ANOVA table below, F = 13.179, p-value < 0.0001. We conclude
that since p-value < 0.0001 ≤ 0.05, we reject . Therefore, there exists enough statistical
evidence to conclude that at least one of the predictors is useful for predicting wage; hence
the model is a good fit.
ANOVAa
Model Sum of Squares df Mean Square F Sig.
1 Regression 13256834.718 7 1893833.531 13.179 .000b
Residual 67684232.263 471 143703.253
Total 80941066.981 478
a. Dependent Variable: WAGE
b. Predictors: (Constant), SIBS, AGE, HOURS, EDUC, TENURE, IQ, EXPER
The table below shows the coefficients for the multiple regression model.Thus, the
model becomes:
Coefficientsa

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Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) -669.035 264.943 -2.525 .012
HOURS -2.085 2.453 -.036 -.850 .396
IQ 4.794 1.492 .166 3.214 .001
EDUC 47.838 10.811 .260 4.425 .000
EXPER 5.699 5.847 .058 .975 .330
TENURE 4.558 3.606 .056 1.264 .207
AGE 16.008 7.021 .120 2.280 .023
SIBS -5.034 8.159 -.027 -.617 .538
a. Dependent Variable: WAGE
We perform a T-Test for each of the predictors to evaluate if each independent variable
has a significant relationship with wage. The hypothesis for the T-Test is:
(No, the predictor has no relationship with wage)
(Yes, the predictor has a relationship with wage).
The decision rule is to reject if the significance is less than 0.05, and fail to reject
, when the significance is greater than 0.05. We conclude that for the predictors, IQ, educ,
and age, we reject since Sig < 0.05. Hence, there is a relationship between IQ score, years
of education, age in years, and wage. For the predictors; hours, exper, tenurw and sibs, the
significance level > 0.05, thus we fail to reject . Therefore, there is no significant
relationship between wage and each of the variables; hours of work, years of work
experience, years with current employer, and number of siblings.
Effect:
From the model, one more year of education increases the wage by $47.84. Hence, if
education increases by 10 years, the wage will increase by $478.40 (that is, $47.84*10 years).
One more year in current employer increases the wage by $4.56. Thus, if tenure increases
by 10 years, then the wage will increase by $45.60 (that is, #4.56*10 years).
One more IQ score point, increases the wage by $4.79. Hence, if the person’s IQ score
increased by 20 points, the wage will increase by $95.80 (that is, $4.79*20 points).
If education increases by 10 years and tenure increases by by 5 years, then the wage will
increase by $501.20 (that is, {($47.84*10 years)+($4.56*5 years)}.

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Prediction:
The predicted wage using the average values for all variables is found as:
The predicted wage using the minimum values for all variables is found as:
Question 6: Multiple Regression
Here we conduct a multiple regression analysis, where we regress wage on all the
variables in our data set.
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1 FEDUC,
TENURE,
HOURS, SIBS,
AGE, IQ,
EXPER,
MEDUC, KWW,
EDUCb
. Enter
a. Dependent Variable: WAGE
b. All requested variables entered.
The multiple regression model follows the format:
From the results tables below, we find that the R-square for this model is
0.199*100%=19.9%. Thus, 19.9% of the variation in monthly earnings (wage) is explained
by all the 10 variables in our data set.

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Model Summary
Model R R Square
Adjusted R
Square
Std. Error of the
Estimate
1 .446a .199 .182 372.2849
a. Predictors: (Constant), FEDUC, TENURE, HOURS, SIBS, AGE, IQ,
EXPER, MEDUC, KWW, EDUC
F-Test: We have that at 0.05 significance level, the null hypothesis is that there is no
linear relationship between the variables in our model, while the alternative hypothesis is that
there is a linear relationship between the variables. The ANOVA table below indicate that F =
11.601, p-value < 0.001 ≤ 0.05, therefore we reject the null hypothesis. There is sufficient
statistical evidence to conclude that there is a significant relationship between the variables,
thus the model is a good fit.
ANOVAa
Model Sum of Squares df Mean Square F Sig.
1 Regression 16078133.245 10 1607813.325 11.601 .000b
Residual 64862933.736 468 138596.012
Total 80941066.981 478
a. Dependent Variable: WAGE
b. Predictors: (Constant), FEDUC, TENURE, HOURS, SIBS, AGE, IQ, EXPER, MEDUC, KWW,
EDUC
The table below shows the coefficients of the multiple regression for all variables.
Coefficientsa
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) -565.236 271.943 -2.079 .038
HOURS -2.743 2.421 -.047 -1.133 .258
IQ 3.090 1.519 .107 2.034 .043
KWW 8.973 2.990 .162 3.001 .003
EDUC 32.209 11.169 .175 2.884 .004
EXPER 6.770 5.750 .068 1.177 .240
TENURE 3.695 3.547 .046 1.042 .298
AGE 7.997 7.557 .060 1.058 .290
SIBS 2.123 8.204 .011 .259 .796

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MEDUC 5.945 8.126 .038 .732 .465
FEDUC 15.592 6.857 .122 2.274 .023
a. Dependent Variable: WAGE
The regression model becomes:
The significant predictors in the model are:
IQ score: Significance = 0.043 < 0.05, reject ,Yes relationship; Coefficient = 3.09. One
more point in IQ score increases wage by $3.09.
Knowledge of World Score: Significance = 0.003 < 0.05, reject ,Yes relationship;
Coefficient = 8.97. One more point in knowledge of world score increases wage by $8.97.
Years of Education: Significance = 0.004 < 0.05, reject ,Yes relationship; Coefficient =
32.21. One more year of education increases wage by $32.21.
Father’s Education: Significance = 0.023 < 0.05, reject ,Yes relationship; Coefficient =
15.59. One more year in a persons father’s education increases wage by $15.59.
Prediction:
The predicted wage using the median values for all the values was calculated as: