ALGM-Assignment 1 Solved

Task 1                                                                                                         3 points
A multiple linear regression model has been estimated to study the relationship between Y = violent crime rate (per 100,000 people), X1 = poverty rate (percentage with income below the poverty line) and X2 = percentage living in urban area. Data are collected in 51 cities in the U.S.

The table below reports the output.

 
                Est.            s.e.
t-value
p-value
SST

SSReg

SSR
1841257.15
 
 
Fill in the missing information.
(Please report the formula and computation).

Interpret the coefficient of determination R2.
Compute and interpret the overall F-test.
5 points

The table below shows the scores of the first test (maximum score 10 points) in a beginning German course. Students in the course are grouped as follows:

Group A:
Never studied foreign language before, but have good English skills

Group B:
Never studied foreign language before, have poor English skills

Group C:
Studied other foreign language

Group A
Group B
Group C
4
1
9
6
5
10
8
 
5
Use an adequate method to compare the mean scores of the groups. Specify the assumptions, hypotheses, test statistics and p-values.
Suppose that the first observation in the second group was actually 9, not 1. Then, the standard deviations are the same, but the sample means are 6, 7 and 8, rather than 6, 3 and 8. Do you think that the F-test statistic would be larger, the same or smaller? Explain your reasoning without doing any computation.
Suppose you have the same means as these data, but the sample standard deviations were 1.0, 1.8 and 1.6, instead of the actual 2.0, 2.8 and 2.6. Do you think that the F-test statistic would be larger, the same or smaller? Explain your reasoning without doing any computation.
Suppose you have the same means and standard deviations as these data, but the sample size were 30, 20 and 30, instead of 3, 2 and 3. Do you think that the F-test statistic would be larger, the same or smaller? Explain your reasoning without doing any computation.
In (1), (2), (3) and (4), would the p-values of the F-tests be larger, the same or smaller? Why?
4 points

The compressive strength of concrete is being studied, and four different mixing techniques are being investigated. The following data have been collected. For each mixing technique, 4 compressive strength measurements (in pounds per square inch) have been recorded.

 
Compressive strength
Mixing
1
2
3
4
1
3129
3000
2865
2890
2
3200
3300
2975
3150
3
2800
2900
2985
3050
4
2600
2700
2600
2765
Does the technique affect the compressive strength?

To answer this question draw comparative box plots and perform an analysis of variance using R.

8 points

Consider the data set munich.csv. The data set contains information on the rent prices of apartments in Munich. The variables in the data set are:

rent: net rent per month (in Euro)
area: living area in square meters
yearc: year of construction
location: quality of location according to an expert assessment
(0 = average location, 1 = good location, 2 = top location)

Read the data into R and tell R that location is a categorical variable.
Discuss whether a linear regression model would be adequate to investigate the relationship between rent, area, yearc and location.
Estimate the model
Y = β0+β1Xarea+β2Xloc:good+β3Xloc:top+β4Xloc:goodXarea+β5Xloc:topXarea+β6Xyearc .

Run and comment on the model diagnostics. Hint: use the Rcode mod <- lm(rent ∼ location+area+area:location+yearc, data=munich)
Interpret the model results.
Compare the fit of the model in (3) with the model including only the variable area.