# Data analysisIntroduction Simple linear regression analysis is a linear model which involves one response variable and only one explanatory variable (Bolin J. H. (2014). Hayes Andrew 2013). This study tries to develop a simple linear regression model using Drug Arrests as the independent variable and Prison Incarcerations as the response variable. The main aim of this paper is to establish a linear regression model using excel software. The research questions are: What is the R-square value for the regression model? What does the R-square value tell you about the regression model? What is the significance F value of the model? What does the significance F- value tell you about the statistical significance of the model? What is the coefficient (t stat value) for the X-variable drug arrests? What does the coefficient (t stat value) mean? In other words how do you interpret the coefficient. What is the p-value for the X variable drug arrests? What does the p-value for drug arrests tell you about its ability to predict prison incarcerations at a level of statistical significance? Is the residual plot a random pattern nonrandom U pattern or an inverse U pattern? What does the type of pattern tell you about the “fit” of the regression model to the data? Hypotheses H_0:the Drug Arrest has no ability to predict Prison Incarcerations H_1:the Drug Arrest has ability to predict Prison Incarcerations Appropriate alpha significance level α=0.05 The table below contains the dataset for both explanatory and dependent variables. SUBJECT # DRUG ARRESTS (X) # PRISON INCARCERATIONS (Y) 1 7 3 2 10 4 3 3 1 4 5 4 5 5 4 6 6 2 7 9 6 8 8 5 9 4 1 Data Analysis and Results Interpretation The provided data in the table above is typed into the excel file before analyzing it using excel data tool Pak. The excel output was copied into the word document as shown in the tables below. The tables below show excel regression output for the two variables under study. Table 1: Regression Statistics Simple R 0.75557423 R Square 0.57089241 Adjusted R Square 0.50959133 Standard Error 1.21854359 Observations 9 Table 2: ANOVA df SS MS F Significance F Regression 1 13.82828283 13.82828 9.312925 0.018539 Residual 7 10.39393939 1.484848 Total 8 24.22222222 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -0.1060606 1.232312883 -0.08607 0.933824 -3.02002 2.807896 # DRUG ARRESTS (X) 0.56060606 0.183702358 3.051709 0.018539 0.126219 0.994993 The R-square value for the regression model is 0.7556. This value tells us that 75.76% of the variation between the two variables is explained by the model while 24.24% is explained by other factors. The significance F value = 0.018539. Since this significance F value is less the calculated F -test = 9.312925 it means that the model is statistically significant. The coefficient (t stat value) for the X -variable drug arrests is 3.0517709. The t coefficient value is used to determine the p-value which will be used to test whether the explanatory variable in the model is significant in the estimated regression model or not. The p-value for the X variable drug arrests is 0.018539. This p-value for drug arrests is used to check whether the explanatory variable is statistically significant or not. From the regression output results in the tables above the p=0.018539 is less than the significance level for this study (α=0.05). This means that the null hypothesis is rejected and the alternative hypothesis is adopted. It also indicates that the there is sufficient evidence to conclude that the drug arrest (predictor) variable has the ability to predict the response variable (Prison Incarcerations). Estimated regression model Prison Incarcerations = -0.1060606 + 0.56060606(Drug Arrests). Residual plot The residual plots are used to check whether the estimated linear regression model fits the given data for both predictor and response variable (OConnor Mahar Laughlin & Jackson 2011). Table 3: RESIDUAL OUTPUT Observation Predicted # PRISON INCARCERATIONS (Y) Residuals 1 3.81818182 0.181818182 2 5.5 -1.5 3 1.57575758 -0.57575758 4 2.6969697 1.303030303 5 2.6969697 1.303030303 6 3.25757576 -1.25757576 7 4.93939394 1.060606061 8 4.37878788 0.621212121 9 2.13636364 -1.13636364 The residual plot for Drug Arrests variable exhibited a fairly random pattern. This residual pattern is positive. This random pattern indicates that the estimated linear model provides a decent fit to the data. Conclusion The major aim of this paper was to conduct a simple linear regression model using Drug arrests as the predictor variable and Prison Incarcerations as the response variable and use the regression results to test the significance of the predictor variable as well as the model significance as whole. The value of significance F value and p-value corresponding to independent variable (Drug Arrests) shown that the explanatory variable is statistically significant in the estimated regression model. The hypothesis test indicated that the independent variable in the developed model is also statistically significant and able to predict the Prison Incarcerations. The residual plot confirmed that indeed the estimated simple linear regression equation for the fits the data for the two variables. Reference Bolin J. H. (2014). Hayes Andrew F. (2013). Introduction to Mediation Moderation and Conditional Process Analysis: A Regression‐Based Approach. New York NY: The Guilford Press. Journal of Educational Measurement 51(3) 335-337. OConnor D. P. Mahar M. T. Laughlin M. S. & Jackson A. S. (2011). The Bland-Altman method should not be used in regression cross-validation studies. Research quarterly for exercise and sport 82(4) 610-616.

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