Question 1 
The worksheet ‘Chi-square’ of file contains data on 350 orders placed to the Elect-Mart company over the period of several months. For each order, the file lists the time of day, the type of credit card used, gender of the customer, order size of purchases (buy Category), the region of the country where the customer resides, and others. 

. a) Describe the variable type and the measurement scale of variables Gender and Buy Category. [1 mark] 

. b) Use Excel to obtain a Pivot (or cross-tab/ contingency) table that provides frequency counts for cells created on the basis of categories of variables ‘Gender’ and ‘Buy Category’. [2 marks] 

. c) If the interest is to find out whether purchasing behaviour (Buy Category) of customers is related to their gender, which test would you apply? What distribution does the test statistic follow, and how many degrees of freedom are associated with the test you suggest? [2 marks] 

. d) Use the critical value approach to test whether the variables Gender and Buy Category are independent at the 1% level of significance. State the null and the alternative hypotheses and show all relevant details. All calculations must be correct to 4 decimal places. 
[5 marks] [Total 10 marks] 


Question 2
Fitting percentage of Body Fat to body measurements using simple linear regression.
Body fat, a measure of health, is estimated through an underwater weighing technique. A medical researcher was asked by her research team leader to determine the body measurements that can predict body fat of young men. The body fat can be measured by two equation mainly percentage body fat using Brozek’s equation and percentage body fat using Siri’s equation.
Her research assistant collected data on variables that might be related to body fat. Data for this question can be found in the excel file Bodyfat.
A sample of 252 young men was selected at random. Fitting body fat to body measurements using regression provides a convenient way of estimating body fat for young men using only a scale and a measuring tape. The variables needed for this study are ; Percentage of body fat,age, weight and eleven body circumference measurements (e.g. abdomen). These measurements are recorded for 252 men.
The variables are listed below:
PercentFat1 = Percentage of Body Fat using Brozek’s equation (Model 1) PercentFat2 = Percentage of Body Fat using Siri’s equation (Model 2)
Age_Years = Age in Years
Weight = Weight in Pounds
Height = Height in inches
Neck = Neck circumference (cm)
Chest = Chest circumference (cm)
Abdomen = Abdomen circumference (cm) "at the umbilicus and level with the
iliac crest"
Hip = Hip circumference (cm)
Thigh = Thigh circumference (cm)
Biceps =Extended biceps circumference (cm)
Forearm = Forearm circumference (cm)
Wrist = Wrist circumference (cm) "distal to the styloid processes"
a. Use scatterplots to investigate the relationship between percentage of body fat using Brozek’s equation (Model 1) with age, weight, height, neck and chest. [5 marks] 

b. Calculate the correlation matrix for all the explanatory variables and comment on mutlicollinearity. [5 marks] 

c. Develop a multiple linear regression models (Model 1 and Model 2) for the explanatory (except chest, hip, weight and height) variables to predict percentage of body fat in men. Perform a test of significance for the regression slope coefficients. Are the signs of the estimated coefficients consistent with your expectations? Explain. 
[14 marks] 

d. Interpret the model R^2 for both models. [2 marks] 

e. Which model is better? Support your answer with appropriate information. (Note: it may be useful to present information in a tabular form for comparing of models) 
[3 marks] 


3
Model R2 Standard Error F-statistics t--ratios
f. Perform a test to determine whether use of ALL explanatory variables for both models is reasonable. You may use ether a critical value or a p-value approach at the 5% level of significance. Be sure to state your null and alternative hypotheses, and ALL other relevant information. 
[3.5 marks] 

g. Analyze the standardized residuals of the regression model. Do there appear to be any outliers. If so, state which observation appears to be outliers? 
[2.5 marks] 

h. Discuss why the weight and height was not included in both the models.
[3 marks] 

i. Discuss why age in years (Age_Years) and age in days (Age_Days) were not use to together in the explanatory variable in both the models.

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