MAT 303 Module Three Problem Set Report Template
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Southern New Hampshire University *
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Apr 3, 2024
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MAT 303 Module Three Problem Set Report
Second Order Models
Diego Beteta
diego.beteta@snhu.edu
Southern New Hampshire University
Note: Replace the bracketed text on page one (the cover page) with your personal information.
1. Introduction
The economic dataset includes variables such as wage growth, inflation, unemployment, economic conditions (recession or no recession), levels of education, and GDP. This data is likely a historical record intended to study how different economic factors are associated with wage growth in the labor force. Our results could be vital for policymakers and economists to understand the dynamics of wage growth, helping to inform decisions and create strategies for economic development. The analyses will likely involve statistical methods to determine correlations, trends, and potentially predictive modeling to estimate future wage growth under different economic scenarios. We might employ regression analysis to understand the relationships between wage growth and other factors and time-series analysis if the data is chronological to look at the trends over time.
2. Data Preparation The important variables in this dataset that we're focusing on include:
Wage Growth
: This measures the percentage increase in labor force wages. Understanding this helps to evaluate workers' standard of living and economic prosperity.
Inflation
: Inflation represents the rate at which the general level of prices for goods and services
rises and, subsequently, how purchasing power is falling. Analyzing inflation alongside wage growth can indicate whether wage increases keep pace with the cost of living.
GDP (Gross Domestic Product) Growth
: GDP growth is the increase in the production and consumption of goods and services in an economy. It's a broad measure of overall economic activity and health.
These variables are crucial as they interplay to define a country's or region's economic condition, influencing policy decisions.
Regarding the structure of the dataset, it consists of 99 rows and 6 columns. Each row represents an entry (potentially a year or other time frame). In contrast, the columns represent the variables mentioned, including wage growth, inflation, GDP growth, and other related economic factors.
3. Quadratic (Second Order) Model with One Quantitative Variable
Correlation Analysis
2
Our scatterplot shows how wage growth compares with the unemployment rate. The relationship between the two isn't a straight line, indicating that the connection isn't just a simple increase or decrease. The pattern suggests that as unemployment changes, the effect on wage growth might increase initially and then decrease, or the opposite, forming more of a curve than a straight line.
The scatterplot shows that the relationship between wage growth and unemployment is not perfectly linear. The data points do not align in a straight line, which suggests that a first order (linear) model might not be the best fit. Instead, the data points show a pattern that could be a curve, hinting that unemployment's impact on wage growth isn't constant as unemployment changes.
Given this observation, a second order (quadratic) model might be more appropriate, which would allow
for a curve that can bend upwards or downwards. This model can account for a more complex relationship where the effect of unemployment on wage growth could increase or decrease at different unemployment rates rather than changing at a constant rate, as a linear model would suggest.
Reporting Results Report the results of the regression model. Address the following questions in your analysis:
General Form: y
=
β
0
+
β
1
x
+
β
2
x
2
Prediction Equation: ^
y
=
^
β
0
+
^
β
1
unemployment
+
^
β
2
unemployment
2
Second-order regression model for wage growth using unemployment as the independent variable:
^
wage
growth
=
12.2342
−
1.7432
unemployment
+
0.0674
unemployment
2
3
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R-squared value = 0.9436
This value tells us that the model explains about 94.4% of the variance in wage growth. It's a measure of how well the observed outcomes are replicated by the model, based on the proportion of total variation
of outcomes explained by the model.
Adjusted R-squared value = 0.9424
This value adjusts the R-squared for the number of predictors in the model and the number of observations. It's approximately 94.2%, close to the R-squared value. This similarity suggests that the number of predictors is appropriate for the number of observations in the model.
Both statistics indicate that our second-order model does an excellent job of explaining how wage growth changes with unemployment. The high values mean that the model fits the data well, and the slight difference between R-squared and Adjusted R-squared implies that we are not penalized much for
any extra complexity in the model; in other words, our model is appropriately complex given the data.
Interpret the beta estimates for the terms unemployment
and unemployment
2
(
unemployment
squared).
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Evaluating Model Significance Evaluate model significance for the regression model. Address the following questions in your analysis:
Is the model significant at a 5% level of significance? Carry out the overall F-test by identifying the null hypothesis, the alternative hypothesis, the P-value, and the conclusion of the test.
Which terms are significant in the model based on individual T-tests? Use a 5% level of significance.
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Making Predictions Using Model Make predictions using the regression model. Address the following questions in your analysis:
What is the predicted wage growth if unemployment is 2.54?
What is the 95% prediction interval for the wage growth? Interpret the interval.
What is the 95% confidence interval for the wage growth? Interpret the interval.
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. 4. Complete Second Order Model with Two Quantitative Variables
4
Reporting Results Report the results of the regression model. Address the following questions in your analysis:
Write the general form and the prediction equation of the complete second order regression model for wage growth as the response variable, and unemployment and GDP growth as predictor variables.
Create this second order regression model for wage growth as the response variable, and unemployment and GDP growth as predictor variables. Write the prediction model equation using outputs obtained from your R script.
What are the values of (R-squared) and (Adjusted R-squared) for the model? Provide your interpretation of these statistics.
Interpret the beta estimates for GDP
2
(GDP squared) and unemployment 2
(unemployment squared).
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Evaluating Model Significance Evaluate model significance for the regression model. Address the following questions in your analysis:
Is the model significant at a 5% level of significance? Carry out the overall F-test by identifying the null hypothesis, the alternative hypothesis, the P-value, and the conclusion of the test.
Which terms are significant in the model based on individual T-tests? Use a 5% level of significance.
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Making Predictions Using Model Make predictions using the regression model. Address the following questions in your analysis:
What is the predicted wage growth if unemployment is 2.50 and GDP growth is 6.50?
What is the 95% prediction interval for the wage growth? Interpret the interval.
What is the 95% confidence interval for the wage growth? Interpret the interval.
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. 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:
5
Write the general form and the prediction equation of the complete second order regression model for wage growth using unemployment and economy as predictor variables.
Create this second order regression model for wage growth using unemployment and economy as predictors.
Write the prediction model equation using outputs obtained from your R script.
What are the values of (R-squared) and (Adjusted R-squared) for the model? Provide your interpretation of these statistics. Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Evaluating Model Significance Evaluate model significance for the regression model. Address the following questions in your analysis:
Is the model significant at a 5% level of significance? Carry out the overall F-test by identifying the null hypothesis, the alternative hypothesis, the P-value, and the conclusion of the test.
Which terms are significant in the model based on individual T-tests? Use a 5% level of significance.
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. Making Predictions Using Model Make predictions using the regression model. Address the following questions in your analysis:
What is the predicted wage growth if unemployment is 2.50 and the economy is not in recession? (be sure to use single quotes when setting the value for economy)
What is the 95% prediction interval for the wage growth? Interpret the interval.
What is the 95% confidence interval for the wage growth? Interpret the interval.
Why is the prediction interval wider than the confidence interval?
Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. 6. Conclusion Describe the results of the statistical analyses and address the following questions:
Based on the analysis that you have performed and assuming that the sample size is sufficiently large, would you recommend using this model? Why or why not?
Fully describe what these results mean in your scenario using proper statistical terms and concepts.
What is the practical importance of the analyses that were performed?
6
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Answer the questions in a paragraph response. Remove all questions and this note before submitting! Do not include R code in your report. 7. Citations 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. 7