[SOLVED] math statistic QUESTION 1. Multiple linear regression [20 marks]

QUESTION 1. Multiple linear regression [20 marks]

Below is a table of some quantiles from the relevant Students T distribution.

-2.69
-2.41
-2.01
-1.68
-1.30
1.30
1.68
2.01
2.41
2.69

Using an appropriate hypothesis test with 0.05 significance level, determine if there is enough evidence to conclude that the beta coefficient foris less 1750. Write down the hypotheses [1 mark], calculate the test statistic [1 mark], report the test result with reason [2 marks] and write a conclusion using a minimum of mathematical language [1 mark].

Using appropriate plots, perform a visual analysis of the standardised residuals in terms of the assumptions made about the error terms in the model [3 marks].

Using an appropriate hypothesis test with 0.05 significance level, determine if there is enough evidence to conclude that the residuals are not normally-distributed. Write down the hypotheses [1 mark], the test statistic and p-values [1 mark], the result of the test [1 mark] and a conclusion using a minimum of mathematical language [1 mark].

QUESTION 2. Regression with categorical predictor [20 marks]

In this question we extend the model built in Question 2. The variables we now consider are summarised in the table below.

Name
Type
Description

response
state fuel consumption

predictor (continuous)
miles of paved highway

predictor (continuous)
proportion of population with drivers license

predictor (continuous)
per capita income

predictor (categorical)
petrol tax bracket: low (1), medium (2), high (3)

Construct a linear regression model withas response and with ,andas continuous predictors. Also includeas categorical predictor (via appropriate dummy variables withas reference category) and interaction betweenand(via appropriate dummy variables). Write down the general regression equation [1 mark].

Provide interpretations of the beta coefficients of the two binary dummy variables [2 marks] and beta coefficients of the two interaction terms [2 marks].

Using SPSS to make the calculations (i.e. without using the regression equation directly), find predicted fuel consumption and 95% individual confidence interval associated with this prediction when
,andfor low petrol tax bracket states [2 marks];
,andfor high petrol tax bracket states [2 marks];

Determine if there is any statistical evidence against the assumption of independence of the error terms [2 marks].

By performing an appropriate regression and including the Model Summary and Coefficients tables with your answer [1 mark], show how to calculate the VIF for the predictorin the model considered above [2 marks].

Using appropriate plots, perform a visual analysis of the standardised residuals in terms of the assumptions made about the error terms in the model [3 marks].

Citing appropriate statistical evidence [1 mark], identify any potentially influential points [2 marks].