MATH1278 Statistics

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MATH1278 Statistics

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MATH1278 Statistics

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Course Code: MATH1278
University: Royal Melbourne Institute Of Technology

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Country: Australia

Question:
Conducting a multiple regression and report on the test results in APA format
Answer:
The existence of a linear relationship between the Age and Errors variables. This is evident from the alignment of the data points which appear to be clustering around a straight line. The clustering of data points around a straight line is an indication of the existence of association between the variables in the scatterplot (Theus & Urbanek, 2008). With the Errors variable increasing with increase in the Age variable, the linear relationship can be described as positive.
The data points however, can only be described as reasonably close to one another around the straight line. Thus, the relationship is reasonably strong in general. The Age and Errors variables are not too closely associated, they are reasonably associated.
When the value of the Pearson Correlation (r) is close to one, we infer that there is a reasonably strong correlation between the variables (Barbara & Susan, 2014). In our case, we can conclude that there is a reasonably strong correlation between the Age and Errors variables with the relationship being positive.
Correlation Matrix above also show that the correlation is significant at the 0.01 level (2 – tailed). To test the hypothesis for correlation at 0.05 level of significance, we have:
H0: The correlation between the Age and Errors Variable = 0
H1: The correlation between the Age and Errors Variable ≠ 0
If the correlation is significant at 0.01 level as indicated in Table 1: Correlation Matrix, then p < 0.01. Given p < 0.01, hence p < 0.05 as well. We therefore reject the null hypothesis that the correlation between the Age and Errors Variable = 0, and conclude that the correlation between Age and Errors is statistically significant at 0.05 level.  Simple Regression  The plot in Figure 2: Scatterplot of Age against Intercourse indicates there is very little existence of an association between the Age and Intercourse variables. When data points on a scatter plots occur to be in a sparse pattern and not clustered up around a straight line, then there is poor goodness of fit (Hastie, Tibshir, & Jerome, 2009). Therefore we can conclude that the goodness of fit for the relationship between the Age and Intercourse variables is poor. The data point however have a direction that indicates a decrease in the Intercourse variable with increase in the Age variable, hence a negative association. There also appears to be an outlier in the dataset.   The results in Table 3: p values for Linear Model of Age and Intercourse show that the p value for the Age variable = 0.000. This means that taking 5% significance level, p < 0.05 implies the variable is not statistically significant (O'Neil & Schutt, 2013). Thus the Age variable is statistically significant in predicting the Intercourse variable at 5% level of significance.  The results in Table 4: Adjusted R Square for Linear Model of Age and Intercourse indicate that the Adjusted R Square = 0.357. This implies that the model explains 35.7% of the relationship between Age and Intercourse. The adjusted R Square gives information on the model fitness (Howitt & Cramer, 2010), hence we can conclude that the model is not good fit.  The results in the plot in Figure 3: Scatterplot of Age against Residuals above indicate that there is an increase in the variability of the residuals as the Age variable increases. An increase in residual variability with increase in the independent variable suggests the assumptions of homogeneity of variance is not met (Freedman, 2009). This model thus violates this assumption. The plot also indicates nonlinearity, since the points are not clustered around a straight line, thus violating the assumption of linearity. The normality, however cannot be described clearly from the plot.   The results in Table 5: Coefficients of Regression Model of Age and Intercourse show that the p value for the Age variable = 0.000. This means that taking 5% significance level, p < 0.05 implies the variable is not statistically significant (O'Neil & Schutt, 2013). Thus the Age variable is statistically significant in predicting the Intercourse variable at 5% level of significance.  The results in Table 4: Adjusted R Square for Linear Model of Age and Intercourse indicate that the Adjusted R Square = 0.357. This implies that the model explains 35.7% of the relationship between Age and Intercourse. Hence we can conclude that the model is not a good fit.   The results in Table 7: Coefficients of Multiple Regression Model of Age and Intercourse show that the p value for the Age variable = 0.000. This means that taking 5% significance level, p < 0.05 implies the variable is not statistically significant (O'Neil & Schutt, 2013). Thus the Age variable is statistically significant in predicting the Intercourse variable at 5% level of significance. Also from Table 7: Coefficients of Multiple Regression Model of Age and Intercourse the p value for the Pain (Back Pain) variable = 0.790. This means that taking 5% significance level, p > 0.05 implies the variable is not statistically significant (O’Neil & Schutt, 2013). Thus the Pain variable is not statistically significant in predicting the Intercourse variable at 5% level of significance. 
Adjusted R Square for Linear Model of Age and Intercourse indicate that the Adjusted R Square = 0.354. This implies that the model explains 35.4% of the relationship between Age, Pain and Intercourse. Hence we can conclude that the model is not a good fit.
In comparison, the single predictor model is a better fit for predicting the average frequency of Intercourse as compared to the two-predictor model. Firstly, the value of the R Square is higher (although slightly) for the single predictor model as compared to the two-predictor model (0.357 > 0.354). Secondly, the added variable in the two-predictor model has been proven to have no statistical significance in predicting the average frequency of Intercourse.   
References
Barbara, I., & Susan, D. (2014). Introductory Statistics (1st ed.). New York: OpenStax CNX.
Freedman, D. A. (2009). Statistical Models: Theory and Practice. London: Cambridge University Press.
Hastie, T., Tibshir, R., & Jerome, H. (2009). The Elements of Statistical Learning (2nd ed.). New York: Springer.
Howitt, D., & Cramer, D. (2010). Introduction to Descriptive Statistics in Psycology, 5th Edition. New York: Prentice Hall.
O’Neil, C., & Schutt, R. (2013). Doing Data Science (3rd ed.). London: O’Reily.
Theus, M., & Urbanek, S. (2008). Inteactive Graphics For Data Analysis (1st ed.). Boca Raton: CRC Press.

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