MAT 303 Module Three Problem Set Report Template
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Southern New Hampshire University *
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303
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Mathematics
Date
Apr 3, 2024
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Uploaded by mrjwallace
MAT 303 Module Three Problem Set Report
Second Order Models
Joshuah Wallace
joshuah.wallace@snhu.edu
Southern New Hampshire University
Note: Replace the bracketed text on page one (the cover page) with your personal information.
1. Introduction
Discuss the statement of the problem with regard to the statistical analyses that are being performed.
Address the following questions in your analysis:
The data being analyzed for this data assignment is how different variables impact wage growth rate by
comparing it with predictor variables of unemployment rate, inflation rate, recession status, and gdp
growth rate.
This data can be used to make predictions about how the economy in regards to individual consumers
and family level view will be impacted by various factors. This information can be used to determine the
needs of social programs like unemployment and other financial assistance programs.
We will be performing a second order multiple regression analysis of the data in order to find the
answers.
2. Data Preparation
There are some important variables that you have been asked to analyze in this problem set. Identify and
explain these variables. Address the following questions in your analysis:
The important variables are gdp, inflation, and unemployment rate, recession status, and of course the
response variable, wage growth rate.
There are 6 different variables attributing to 6 total columns, and in total 100 rows, or entries, are
available for analysis.
3. Quadratic (Second Order) Model with One Quantitative Variable
Correlation Analysis
Visualize and describe the relationships between the variables in the data set. Address the following
question in your analysis:
The scatterplot between wage growth and unemployment suggests that a second order multiple
regression model is the most appropriate for analysis due to the negative parabolic nature of the plotted
points. A parabolic shape better fits the curve of the plotted points than a line does, suggesting a
nonlinear relationship between wage growth and unemployment that a second order model will
describe better.
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Reporting Results
Report the results of the regression model. Address the following questions in your analysis:
Wage growth = B0 + B1(unemployment) + B2(unemployment^2)
Wage Growth = 12.234206 - 1.743170(X) + 0.067408(X^2)
The R-squared value is 0.9436 and the adjusted R-squared value is 0.9424. This indicates a very strong
model with 94.36% of the relationship between the two variables being explained by this model. A high
adjusted r-squared value like .9424 indicates that additional variables are probably not required since it
is unlikely to become more correlated.
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The beta estimate for unemployment is -1.743170 and the beta estimate for unemployment^2 is
0.067408
Evaluating Model Significance
Evaluate model significance for the regression model. Address the following questions in your analysis:
The model is significant at a 5% level of significance. The null hypothesis is such that no predictor
variables have a significant relationship to the response variable. The alternative hypothesis is such that
at least one predictor variable has a significant relationship with the response variable. The p-value of
2.2e^-16 is significantly lower than the 0.05 value resulting in a rejection of the null hypothesis in favor
of the alternative hypothesis.
Null Hypothesis: (H0): u1=0;
This suggests that the coefficient for unemployment and economy are 0 in the model equation, and thus
are not significant.
Alternative Hypothesis: (H1): u1 =/ 0;
This suggests that the one variable of unemployment does not have a coefficient of 0 thus favoring the
alternative hypothesis over the null hypothesis.
Unemployment has an individual t-test of 2e-16 and a p-value for the model of 2.2e-16 which results in a
rejection of the null hypothesis in favor of the alternative hypothesis that unemployment is a significant
variable in determining wage growth.
All terms are deemed significant to the model since the p-value is near zero and all individual T-tests also
produce values near zero.
Making Predictions Using Model
Make predictions using the regression model. Address the following questions in your analysis:
With an unemployment rate of 2.54, we expect the predicted wage growth rate to be at 8.24.
At a 95% prediction interval, the model expects to see 95% of all points between a value of [6.9071,
9.5758] from the observed value of 2.54 unemployment rate.
At a 95% confidence level, the model expects to see 95% of the mean value between a value of [8.0936,
8.3893] from the observed value of 2.54 unemployment rate.
The fitness value that is expected based on the model is 8.2414 which proves the manual calculation
above correct.
4. Complete Second Order Model with Two Quantitative Variables
Reporting Results
Report the results of the regression model. Address the following questions in your analysis:
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E(Y) = B0 + B1(X1) + B2(X2) + B3(X1:X2) + B4(X1^2) + B5(X2^2)
E(Y) = B0 + B1(gdp) + B2(unemployment) + B3(gdp:unemployment) + B4(gdp^2) + B5(unemployment^2)
Wage Growth = 8.989434 + 0.283691(gdp) - 1.152823(unemployment) - 0.006282(gdp:unemployment) -
0.006599(gdp^2) + 0.037685(unemployment^2)
The r-squared value is .9587 and the adjusted r-squared value is .9565. Compared to the first model
generated, this is a better model at predicting wage growth than looking at unemployment alone. The
multiple r-squared value has increased and the adjusted r-squared value has increased. This means that
95.87% of the model data can explain the growth rate, and the adjusted r-squared value is higher
indicating a higher correlation of this model than the previous with an adjusted r-squared value .9424.
The X1 and X2 coefficients have no meaning in the presence of their second order variables. X1^2, the
second order value for gdp, is negative indicating a negative curvature relationship between gdp and
wage growth. This means as gdp increases, wage growth decreases. Unemployment second order value,
X2^2, has a positive curvature correlation coefficient with wage growth indicating that increased levels
of unemployment have a positive effect on wage growth.
Evaluating Model Significance
Evaluate model significance for the regression model. Address the following questions in your analysis:
The model is significant at a 5% level of significance with a P-value of 2.2e-16. The null hypothesis is such
that no predictor variables have a significant relationship with the response variable, growth rate. The
alternative hypothesis is such that at least one variable is significantly correlated with the response
variable. With a P-value less than 0.05, we reject the null hypothesis in favor of the alternative
hypothesis.
Null Hypothesis: (H0): u1=u2=0;
This suggests that the coefficient for unemployment and gdp are 0 in the model equation, and thus are
not significant.
Alternative Hypothesis: (H1): u1 =/ 0 or u2 =/ 0:
This suggests that any one variable of unemployment or gdp does not have a coefficient of 0 making any
one of the two variables significant thus favoring the alternative hypothesis over the null hypothesis.
Gdp has an individual t-test of .04682 and unemployment has an individual t-test of 8.26e-06 and the
p-value of the overall model is 2.2e-16 resulting in a rejection of the null hypothesis in favor of the
alternative hypothesis.
Those significant variables in the model, based on individual t-tests of each variable, are all except for
the second order variable of gdp with a t-test of 0.12815, and the interaction term of gdp and
unemployment with a t-test of .76678. All other terms are less than 0.05 representing significance with a
5% level of significance.
Making Predictions Using Model
Make predictions using the regression model. Address the following questions in your analysis:
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The expected value of wage growth with an unemployment rate of 2.50 and a gdp growth rate of 6.50 is
7.806 wage growth rate.
At a 95% prediction interval, we expect to see 95% of all data points gathered to be between the
intervals of [6.6315, 8.9805].
At a 95% confidence interval, we expect to see the mean value fall between [7.583, 8.0289] 95% of the
time.
5. Complete Second Order Model with One Quantitative and One Qualitative Variable
Reporting Results
Report the results of the regression model. Address the following questions in your analysis:
E(Y) = B0 + B1(X1) + B2(X2) + B3(X1)(X2) + B4(X1^2) + B5(X1^2)(X2)
E(Y) = B0 + B1(unemployment) + B2(economy) + B3(unemployment)(economy) + B4(unemployment^2) +
B5(unemployment^2)(economy)
Growth Rate = 12.36072 - 1.8083(unemployment) - 2.70404(economyrecession) +
0.69359(unemployment)(economyrecession) + 0.07574(unemployment^2) -
0.04358(unemployment)(economyrecession)
The r-squared value is .9475 and the adjusted r-squared value is .9446. This means that this model
explains 94.75% of the variation observed in wage growth. The both values are slightly lower than the
previous model examining a complete second order regression using unemployment and gdp leaving the
second model to be the more accurate model.
Evaluating Model Significance
Evaluate model significance for the regression model. Address the following questions in your analysis:
Null Hypothesis: (H0): u1=u2=0;
This suggests that the coefficient for unemployment and economy are 0 in the model equation, and thus
are not significant.
Alternative Hypothesis: (H1): u1 =/ 0 or u2 =/ 0:
This suggests that any one variable of unemployment or economy does not have a coefficient of 0
making any one of the two variables significant thus favoring the alternative hypothesis over the null
hypothesis.
Both variables have a t-test that of 2e-16 which is well under the .05 level of significance required
making both variables significant and allowing us to reject the null hypothesis. The P-value is also 2.2e-16
resulting in a rejection of the null hypothesis altogether.
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All terms except for the interaction term between the second order variable of unemployment is
significant to the model due to having an individual t-test of greater than the significance level of .05.
0.512 > .05 meaning this value is not significant to the model.
Making Predictions Using Model
Make predictions using the regression model. Address the following questions in your analysis:
The predicted value the model outputs for an unemployment rate of 2.50 and without a recession is
8.3132 growth rate.
The 95% prediction interval indicates that 95% of values are expected to be between [7.003, 9.6235].
The 95% confidence interval indicates that 95% of the observed mean value is expected to be between
[8.1573, 8.4692].
A prediction interval is the total range of values a point is expected to be found within the data while a
confidence interval specifically looks at where the mean of the data is found. Mean is a much tighter
range of values and is the result of the prediction interval so the prediction interval will always be wider
than the confidence interval.
6. Conclusion
Describe the results of the statistical analyses and address the following questions:
I would say that the 100 samples is plenty significant in assessing the accuracy of the model. All models
generated had an r-squared value greater than .9 which indicates that all models are very accurate, and
that there is little to no noise in the data sufficiently blocking the effect of outliers.
The results indicate that the predictor variables chosen to describe the response variable of wage
growth, are significant to a satisfactory level of significance to explain almost all variation with wage
growth. No additional predictor variables will need to be considered since it has been discovered across
three models, that unemployment, gdp, and recessionary status alone can explain over 94% of the
results seen in wage growth.
Using wage growth alone as a response variable seems to have little use at large except for, in turn, being
used as a predictor variable for another response. As the response variable, understanding wage growth
models can help determine what will happen to the purchasing power of individuals. This is likely a very
well correlated feedback loop as I can see how increased or decreased wage growth/decay can result in a
change in unemployment, impending recession, gdp, and very closely with inflation. The models indicate
that as inflation increases, so does wage growth and this makes sense. It also makes sense that as wage
growth increases, inflation increases where the opposite may not be directly related for many other
variables.
7. Citations
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You are not required to use external resources for this report. If none were used, remove this entire
section. However, if you used any resources to help you with your interpretation, you must cite them. Use
proper APA format for citations.
Insert references here in the following format:
Author's Last Name, First Initial. Middle Initial. (Year of Publication). Title of book: Subtitle of book,
edition. Place of Publication: Publisher.
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