Question

3. Recent research has addressed the association between BMI (kg/m²) and sleep apnea. The latter has received extensive public attention in recent years as scientists have been discovering details about how we breathe while we sleep, and how that changes with age. Currently, obstructive sleep apnea is defined as an apnea hypoxia index (AHI) over 30, a measurement that interprets how many times the body is deprived of oxygen while you sleep. Suppose a researcher gathers a SRS of patients and records their BMI and AHI and displays it in the table below. (2 pt)

BMI (kg/m2)	AHI (count)
28.9	34
32.5	35
31.5	42
36.4	46
33.5	43
29.7	37
30.4	48

Suppose a computer software program created the following regression equation relating AHI(X) to BMI(Y).

BMI=22.565+0.228(AHI)

A) Calculate basic descriptive statistics for your Y variable, BMI.

B) Using the information presented in the table above and from the regression equation, fill in the following ANOVA table

	Sum of Squares	df	Mean Square (MS)
Regression
Residual
Total

C) Calculate an F statistic for an ANOVA test with the null hypothesis that there is no relationship between the two variables.

D) Determine your pvalue and interpret your conclusions about the relationship between AHI and BMI.

Answer 1

a)

slope ,    ß1 = SSxy/SSxx =   0.228
      
intercept,   ß0 = y̅-ß1* x̄ =   22.565

ΣY=222.9
n=7

ȳ =Σy/n = 31.843

std dev of BMI(Y) =  2.562

b)

Anova table
variation	SS	df	MS
regression	9.316	1	9.316
error,	30.081	5	6.016
total	39.397	6

c)

F-stat =MS regression / MS error = 9.316/6.016 = 1.549

d)

pvalue = 0.2685

p-value >α=0.05, fail to reject Ho

so, there is no relationship between the two variables at α=0.05