Question

A medical study, a doctor is interested in determining what factors affect forced expired volume values (FEV). The doctor randomly samples 15 patients and records their smoking status and age. Using the data, create the regression output (including the y-intercept and both independent variables) then answer the questions below. Note that smoking status and age are recorded as categorical variables.

• Write out the estimated regression line. To receive full credit, you must label all variables/categories.

• Run an overall F-test to determine if the multiple regression model is significant at a 5% significance level. To receive full credit, you must give me:

Test Statistic
P-Value
Is the model significant? (Circle your answer)	Yes                    No

• Predict the FEV value for a smoker over 50 years old.

• Estimate the average FEV for non-smokers between 30-40 years old.

FEV	Smoking Status	Age
0.79	smoker	< 30
0.81	smoker	30 - 40
0.93	non-smoker	30 - 40
0.59	smoker	40 - 50
0.77	non-smoker	> 50
0.61	smoker	40 - 50
0.95	non-smoker	< 30
0.87	non-smoker	< 30
0.63	smoker	40 - 50
0.88	non-smoker	30 - 40
0.61	smoker	> 50
0.64	smoker	< 30
0.67	smoker	30 - 40
0.87	non-smoker	40 - 50
0.93	non-smoker	< 30

Answer 1

Let y = Forced Expired Volume (FEV)

     x1 = Smoking status

    x2 = Age

For calculation purpose lets categorize x1 and x2 in numeric formats.

if smoker then x1 = 1, if non-smoker then x1 = 0.

If age < 30 then x2 = 1, if 30 < age < 40 then x2 = 2

if 40 < age < 50 then x2 = 3, if age > 50 then x2 = 4

so the regression equation is as below;

y = m1x1 + m2x2 + b

where m1 and m2 are coefficients for each of x1 and x2 respectively and b is constant (intercept).

                                                                                                                           ............................ (1)

Find attached image for the detailed regression analysis output by using the Regression Data analysis pack from excel.

From above image we get below values:

Test Statistic	F-value in above table = 32.28562325
P-value	For intercept = 0.0000000000035370710
Is the model significant?	No

From above output we get the regression equation as:

y = (-0.199808774)x1 + (-0.045748031)x2 + 0.977210349

                                                                                                                                          .......................(2)

Now we want to predict FEV(y) for smoker(x1) above 50 years(x2).

Here x1 = 1 and x2 = 4 by using categorization defined in (1) above.

so from given values regression equation will be as:

y = (-0.199808774)*1 + (-0.045748031)*4 + 0.977210349

y = 0.594409 = 0.59(2 decimals)

Hence the FEV for Smoker of Age above 50 years = 0.59

Now we want to estimate average FEV for smokers between 30-40 years old.

From given data there are total 8 smoker patients. Of this 8 there are 2 patients of age between 30-40 years as below

FEV	Smoking Status	Age
0.81	smoker	30 - 40
0.67	smoker	30 - 40

Hence in given situation the average FEV = (0.81 + 0.67)/2 = 0.74

Average FEV for Smoker between 30-40 years old = 0.74