Question One:
1.Open the file PRICEINCOME
This file contains data on the price of a good and the income of its consumers
a.Regress PRICE on INC and report your results (2 marks) Report the sample regression function and R squared (write them out, not insert the output)
Price = .827 + .014 x Income
R2= .000
The significance value of 0.895 (greater than 0.05) indicates that the regression model does not significantly predict the price. This is also indicated by the R squared value of .000 which indicates that about 0% of the variation in price is explained by income.
b.Does the model in a) exhibit heteroscedasticity? Explain your answer
(4 marks) Quote appropriate regression output, carry out test and give result of test with explanation
Based on the below scatterplot, the model does exhibit heteroscedasticity. This was interpreted as such because the points in the above graph are not scattered equally, as would be seen in homoscedasticity.
c. Are the variables PRICE and INC normally distributed? Justify your answer (4 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
Price - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is not normally distributed. This is because p = 0.001, and thus p < 0.05, indicating that p is significant, and thus that the data is not normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is not distributed around a central value; the data has a right skew. Furthermore, in the Q-Q plot, the data points stray from the line of normality, indicating again that the data is not normal.
Alpha value = 0.05.
Income - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is normally distributed. This is because p = 0.061, and thus p > 0.05, indicating that p is insignificant, and thus that the data is normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is distributed around a central value. Furthermore, in the Q-Q plot, the data points remain relatively in line with the line of normality, indicating again that the data is normal.
Alpha value = 0.05.
d. Are the residuals in your model normally distributed? Justify your answer (3 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
Unstandardized Residual - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is not normally distributed. This is because p = 0.002, and thus p < 0.05, indicating that p is significant, and thus that the data is not normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is not distributed around a central value; the data has a right skew. Furthermore, in the Q-Q plot, the data points stray from the line of normality, indicating again that the data is not normal.
Alpha value = 0.05.
e.Is the model in a) mis-specified? Explain your answer (3 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
The regression model is mis-specified. When the variables income and price are plotted, they do not appear to have a linear relationship (please see below in a plot made on spss). This can also be seen in the plot of residuals and predicted values, also pasted below. As the variables do not follow a linear relationship, this violates a key assumption of linear regression, and thus our model is mis-specified.
Total Marks: 16 marks
Question 2:
2. Open the file PRICEINCOME
a. Regress LOGPRICE on LOGINC and report your results (2 marks) Report the sample regression function and R squared (write them out, not insert the output)
LogPrice = (-0.06)LogIncome + 0.192
R2= .001
The significance value of 0.795 (greater than 0.05) indicates that the regression model does not significantly predict the price. This is also indicated by the R squared value of .001 which indicates that about 0.1% of the variation in price is explained by income.
b. Does the model in a) exhibit heteroscedasticity? Explain your answer (4 marks) Quote appropriate regression output, carry out test and give result of test with explanation
Based on the above scatterplot, the model does exhibit heteroscedasticity. This was interpreted as such because the points in the above graph are not scattered equally, as would be seen in homoscedasticity.
c. Are the variables LOGPRICE and LOGINC normally distributed? Justify your answer (4 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
Logprice - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is normally distributed. This is because p = 0.200, and thus p > 0.05, indicating that p is insignificant, and thus that the data is normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is distributed around a central value. Furthermore, in the Q-Q plot, the data points remain consistent with the line of normality, indicating again that the data is normal.
Alpha value = 0.05.
Logincome - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is normally distributed. This is because p = 0.200, and thus p > 0.05, indicating that p is insignificant, and thus that the data is normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is distributed around a central value. Furthermore, in the Q-Q plot, the data points remain consistent with the line of normality, indicating again that the data is normal.
Alpha value = 0.05.
d. Are the residuals in your model normally distributed? Justify your answer (3 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
Unstandardized Residual - Looking at the Tests of Normality, specifically the Kolmonogorov-Smirnov test (as our sample size is greater than 50), the p value indicates that the data is normally distributed. This is because p = 0.200, and thus p > 0.05, indicating that p is insignificant, and thus that the data is normally distributed. This may also be indicated by looking at the histogram and Q-Q plot. Looking at the distribution of the data in the histogram, it is evident that the data is distributed around a central value. Furthermore, in the Q-Q plot, the data points remain in line with the line of normality, indicating again that the data is normal.
Alpha value = 0.05.
e. the model in a) mis-specified? Explain your answer (3 marks) Provide appropriate graphs/diagrams (can cut/paste) and explain your answer
The regression model is mis-specified. When the variables Logincome and logprice are plotted, they do not appear to have a linear relationship (please see below in a plot made on spss). This can also be seen in the plot of residuals and predicted values, also pasted below. As the variables do not follow a linear relationship, this violates a key assumption of linear regression, and thus our model is mis-specified.
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