Applied Linear Models Assignment -

EXERCISE 1: Analysis of Paper Planes Data

Ever made a paper plane? Generally the aim is to make one that travels as far as possible. Two Australian students conducted an experiment to find the best strategy for making planes stay airborne for as long as possible. They decided to test whether different types of paper and different angles of release would have any effect on the distance travelled. To eliminate the effect of wind to some degree the experiment took place in a long hallway where all doors had been closed.

One student folded and threw all planes, while the second student measured all the distances, checked that the angles of flight were right, and checked that the plane release was the same each time. The second student also calculated the order in which the trials were carried out using the random number function of a calculator.

You can read these data from the file planes.txt into R, and store them as a data frame plane.

The variables that appear in the data frame are as follows:

Distance - Distance travelled in mm

Paper - A4 paper weighing either 80gms = 1, or 50gms = 2

Angle - Horizontal = 1, 45 degrees upward = 2

Design - High-performance Dual Glider = 1, Incredibly Simple Glider = 2

(a) Is this a balanced experimental design?

(b) What type of format are each of the variables stored as? You can use str() to find out. We would like all variables apart from Distance (which is meant to be numeric) to be treated as factors. Alter the format if necessary.

(c) Fit a linear model with Distance as a numeric response, including Paper, Angle and Design as factors, and the Paper: Design factor interaction. Write down the R code that you use and the ANOVA table for your fitted model.

(d) Interpret the ANOVA output. Specifically, indicate whether Distance depends on Paper and Design, and whether the effect of Paper depends on the level of Design.

(e) For the fitted model, write down the fitted value for a plane made of 50gms paper, thrown at 45 degrees upwards and designed as a high-performance dual glider.

(f) What combination of Paper and Design should be used in order to maximize the distance the plane travels?

EXERCISE 2: Analysis of CEO data

The data set ceo2012.csv contains information on Chief Executive Officer (CEO) compensation payments, share ownership and age for the year 2012 in relation to their corresponding company performances as approximated by efficiency levels.

The data includes the following five variables:

Company - Name of company headed by CEO

Salary - Salary in US dollars

Age - Age in years

Gender - Gender, M (male) and F (female)

Efficiency - Proxy measure of company's performance

Note: Please provide the R code used in your answers.

We consider two regression models. We define M₀ by E[Efficiency] = β₀ + β₁Age + β₂Gender

and model M₁ by E[Efficiency] =  β₀ + β₁Age + β₂Gender + β₃Salary.

(a) Using an appropriate plot of the data, comment on any issues you foresee with a regression analysis.

(b) Fit M₀ to the dataset and report the effect of a CEO's salary and age on the corresponding company's efficiency.

(c) Fit M₁ to the data and report if the effect of age in this model is significant.

(d) Test if the improvement in goodness of fit of M₁ over M₀ is worth the 'cost'. Write down the hypotheses, the p-value and your conclusion.

(e) Use a stepwise variable selection algorithm to find a linear regression model for the data set. Write down the final model.

EXERCISE 3: Analysis of American companies data

The data stored in companies1985.csv contains information on various performance indicators of 47 American companies from the year 1985.

The data includes the following variables:

Variable - Description

Company - Name of company

Sales - Sales revenue in US millions

Market.Value - Company's worth in terms of market capitalisation in US millions

Profits - Company's profit in US millions

Cash.Flow - Company's operating cash flow minus capital expenditures in US thousands

Sector - Sector company operates within.

Fortune500 - Indicates whether company was on Fortune500 list (yes=1, no=0)

Aims of Analysis: Find a suitable a linear model for predicting Profits in terms of the other variables.

What to Hand In:

For this exercise you should hand in a mini-report no longer than a single side of A4 paper (excluding any relevant Figures, Tables or computer output, which can be attached as an appendix). You will be marked down for exceeding this page limit. The aim of the mini-report is to convey the aims, methodology and results of your data analysis in a concise, readable fashion. It is strongly recommended that you structure your report into sections, Introduction, Methodology, Results and Discussion.

(a) Introduction: Summarise the data and the aims of the analysis.

(b) Methodology: Describe the statistical methods that you use (technical de-tails not required). They should be appropriate considering the data used.

(c) Results: Describe the results of your analysis and their correct interpretations.

(d) Discussion: Draw conclusions (based on your results) as necessary. Discuss any interesting issues arising from your analysis.

(e) Exposition Your mini-report should be well organized. You should aim to write in a concise, yet readable, manner.

Attachment:- Assignment Files.rar