MAT 303 Module Three Problem Set Report Template

MAT 303 Module Three Problem Set Report Second Order Models Brian Tynan Brian.Tynan@snhu.edu Southern New Hampshire University

1. Introduction I am working for the government as an analyst and my employer requested that I use a CVS file that holds historical economic data to complete a statistical analysis that will analyze the wage growth for the labor force. The data set can be used to determine and analyze how wage growth affects inflation, the economy, and how it affects pay. It can also be used to ensure that when the costs increase to make and sell products that the wages for the laborers are accurately increased. This data can be extremely beneficial as it can be used to help prevent the economy from crashing and it help prevent a recession or a depression. I will be developing scatterplots so that a visible analysis can be performed for specific variables in this data set. A quadratic (second order) model with one quantitative variable will be created. I will be using wage growth as the response variable and GDP (growth rate) as the predictor variable for this model. As a part of my analysis, I will be interpreting the beta estimates and prediction intervals. I will also be creating a model with two quantitative variables. I will analyze and interpret the results. The last model that will be created will be done so that I can perform a complete second order regression analysis for a model with a quantitative and qualitative variable. 2. Data Preparation This data set consists of some very important variables. The test response variable is the wage growth. The predictor variable consists of the GDP growth rate, inflation, rate the unemployment rate, and the recession for the economy. This data set consists of six rows and six columns. The columns of the data set consist of the prediction variables. 3. Quadratic (Second Order) Model with One Quantitative Variable Correlation Analysis 2

R 2 = 0.8338 and the Adjusted R 2 = 0.8303 As you can see the R-squared and the adjusted R squared are almost the same and this is due to there is only one input variable. The R-squared value shows me that approximately 90% of the variation in wage growth can be expressed by using a model that uses unemployment as the predictor variable. I believe that this data fits my model. The beta estimates that the term for unemployment is +1.1144 and the beta estimates for the term unemployment 2 is shown as -0.0326. Due to the one number is a positive number and the one is a negative the numbers indicate a curved relationship with a downward concavity. Evaluating Model Significance H 0 : β 1 = β 2 =0 H a : at least one β 1 ≠ 0 for I = 1, 2 The p-value of the overall F-test can is 2.2e-16 which is lower than the .05 significance level. With this information we need to reject the null hypothesis. Upon reviewing this data, I can conclude that there is a statistically significant relationship that exists between wage growth and unemployment. I will use an additional t-test to determine which terms are significant within the model. The following shows the null hypothesis and alternated for this test: H 0 : β i =0 for some i=1, 2,,,,n H a : β i ≠ 0 The p-value for unemployment is 2e-16 and the p-value for unemployment 2 is 2.64e-13 and both of these values are less than the significance level of .05 or 5%. I have determined that both variables are statistically significant with this model so with that information I will not reject the null hypothesis. Making Predictions Using Model The predicted wage growth of unemployment for this data set is 2.54 is 8.2414. The interval is between 6.9071 and 9.5758. This leads me to believe that when taking the regression error into account, I can be 90% certain that the individual data point for wage growth will fall between 6.9071 and 9.5758. The confidence interval for this data set is 8.2414 for wage growth when the unemployment is 2.54. The interval with 90% certainty will fall between 8.0936 and 8.3893 for the wage growth. 4. Complete Second Order Model with Two Quantitative Variables Reporting Results Below is the general form equation for the complete second order regression model with the response variable of wage growth and unemployment and gpd growth as the predictor variables. E(y)=β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x 2 1 + β 5 x 2 2 The prediction equation for this model is: ŷ =β 0 ^ + β 1 ^ x 1 + β 2 ^ x 2 + β 3 ^ x 1 x 2 + β 4 ^ x 2 1 + β 5 ^ x 2 2 Here is the prediction model equation with the outputs from the model the model I created. Y=8.989434-1.152823x 1 +0.283691x 2 + 0.037685x 1 x 2 -+0.006599x 2 1 -0.006282x 2 2 The R 2 =0.9587 and the adjusted R 2 -0.9565. The R-squared value shows that approximately 96% of the variance for the wage growth for this data set can be explained by using a model that consists of 4

unemployment and GDP as the predictor variables. The beta estimate for this data set for unemployment 2 =0.037685 and gdp 2 =-0.006599. Since the unemployment 2 is positive it relates to an upward and concavity and since the gdp 2 is negative it relates to a downward concavity. Evaluating Model Significance The overall F-test will be performed by identifying the null and alternate hypothesis and by documenting them below: H 0 : β 1 =β 2 = ⋯ =β n =0 H a : at least one β i ≠0 for i=1, 2, .... , n The p-value for this data set is 2.2e-16, which again is less than the significant level of 5% or .05. With this information I will reject the null hypothesis and I have determined that at least one of the predictor variables and response variables are statistically related. I will perform a t-test at a 5% level of significance to determine which ones. Here are the null hypothesis and the alternate hypothesis: H 0 : β i =0 for some i=1, 2, …, n H a : β i ≠0 The p-value for unemployment is shown as 8.26e-06 and this is less than the significant value of 5% or .05. With this data I will reject the null hypothesis and determine that there is a significant statistical relationship between wage growth and unemployment. The p-value for GDP is shown as 0.00489 which is also lower than the p-value of .05 making it statistically significant in this model and gdp 2 is showing with a p-value of 0.12815 and this is higher than the significance level of .05 so I am able to determine that it is not significant with this model. The same can be said for the unemployment: gdp due to it has a p-value that is shown as 0.76678 which is also higher than the .05 significance level. Making Predictions Using Model When the unemployment is 2.50 and GDP growth grown is 6.50 the predicted wage growth will show as 7.806. The wage growth prediction interval with a 95% prediction interval is between 6.6315 and 8.9805. This lets me know that I can be 95% certain that an individual data point for wage growth for this model will gall between 6.6315 and 8.9805. The confidence interval for this data set for this specific model provides me with a 95% confidence that a series of the data points will fall between 7.583 and 8.0289. 5. Complete Second Order Model with One Quantitative and One Qualitative Variable Reporting Results The general form of the equation: E(y)=β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x 2 1 + β 5 x 2 1 x 2 The prediction equation for this model: ŷ =β 0 ^ + β 1 ^ x 1 + β 2 ^ x 2 + β 3 ^ x 1 x 2 + β 4 ^ x 2 1 + β 5 ^ x 2 1 x 2 ŷ =12.36072 -1.80834 x 1 -2.70404 x 2 + 0.07574 x 1 x 2 + 0.69359 x 2 1 -0.04358 x 2 1 x 2 R 2 -0.9475 Adjusted R 2 =0.9446 R-squared value shows that approximately 95% of the variance for wage growth can be expressed by using a model that uses the unemployment data and economy data as the predictor variables. Evaluating Model Significance H 0 : β 1 =β 2 = ⋯ =β n =0 5