OVERVIEW

The primary aim of this assignment is to give you some practice on various concepts covered during the            lectures. Its secondary aim is to familiarize you with R programming. There is only one part to this assignment:

1.   Four statistical analysis questions (5 marks each).

This assignment is graded out of 20 marks. It contributes 10% to your final course grade.

SUBMISSIONS INSTRUCTIONS

Submit your completed response document through the LEARN portal. There is a Dropbox folder for this    assignment, under Submit ➔ Dropbox ➔ Problem Set #3. The deadline for submission is  10pm on July 25, 2022 (Monday).

You are required to submit one R notebook file (.Rmd) which contains the codes and your responses. Please    make sure your code will be able to run without any modifications; dependent libraries should be loaded at the beginning of the R notebook. You should also comment your code to clearly explain each of your steps in case your code does not compile properly.

Statistical Analysis

(20 marks in total, 5 marks each)

1.   Richard has developed a new non-invasive measurement tool for brain blood flow (BBF). The research hypothesis that he has is that the new measurement tool performs the same as the current standard. To test this research hypothesis, he designs a study that measures the BBF using MRI, transcranial Doppler, and the new tool. The BBF derived from MRI serves as a reference, whereas transcranial Doppler is the current standard. The collected data are included in Q1.csv where BBF is the reference value for each subject, MCA is the flow rate measured using transcranial Doppler and CCA is the flow rate measured using the new tool. Richard decides to conduct a correlation analysis to statistically test the hypothesis. The null hypothesis is that the correlation coefficient between BBF and CCA is the same as the one between BBF and MCA. With α = 0.05

(i)   Show that flow rate measured using the new tool (CCA) is statistically correlated with the reference flow rate (BBF)

(ii)  Show that flow rate measured using transcranial Doppler (MCA) correlates with the reference flow

rate (BBF) as well

(iii) By comparing the correlation coefficients derived in (i) and (ii), show that the new tool performs the same as the current standard (i.e. RCCA-BBF = RMCA-BBF are the same).

2.  A company is developing a new tool that can non-invasively quantify arterial elasticity in terms of stiffness. To calibrate this new device, the company researchers measure the arterial stiffness from 20 cadavers, and their corresponding arterial elasticity was measured using the tensile test. The raw data can be found in Q2.csv. Simple linear regression is to be performed to calibrate the device and the null hypothesis for the regression is that the regression coefficient between arterial stiffness from the new device (non-destructive) and elasticity from the destructive test is zero. With α = 0.05

(i)   Perform simple linear regression and show that the relationship between the arterial stiffness           (nondestructive as the dependent variable) and elasticity (destructive as the independent variable) is linear

(ii)  Check the following model assumptions: 1. Linearity; 2. Equal residual spreads; and 3. Normally

distributed residuals

(iii) Derive the confidence and prediction interval for an artery that has an elasticity of 30.

3.   Jason wants to know the relationship between the amount of body fat, triceps skinfold thickness, thigh circumference, and midarm circumference. To achieve this objective, he randomly recruits a group of 20 subjects and measures their body fat content and other parameters as mentioned. The file Q3.csv contains the raw data for all 20 subjects. Try to construct a model that best describes the relationship between body fat content and measured parameters. The null hypothesis is that there is no relationship between body fat (fat), triceps skinfold thickness (thickness), thigh circumference (thigh), and midarm circumference (midarm). With α = 0.05

(i)   Regress a multiple regression model withfat as the dependent variable and the rest as independent variables (i.e., full model); provide a summary table on statistics

(ii)  Regress a reduced model with only triceps skinfold thickness and thigh circumference as

independent variables; is this reduced model statistically different from the full model? (iii) Check all the model assumptions for the reduced model in (ii)

4.   To meet the increasing demand for hay, an agricultural company wants to improve its hay yield. The company first needs to better understand the relationship between hay yield and other soil parameters. The company researchers have collected various soil parameters and hay yields and the parameters are summarized in the table below: