Question

Least Squares Regression: A group of asthma patients has an average age of 42 years with a standard deviation of 16.05 years. They experience asthma attacks with a mean duration of 23.8 minutes with a standard deviation of 5.85 minutes. a) Using the raw data below and the summary data in the problem statement, find the least squares regression line to predict duration of an asthma attack from the age of the patient. Age Duration of Attack 30 25 65 50 30 15 28 30 22 24 b) If a patient is 30 years old, how long would you expect their asthma attack to last? c) That patient's asthma attack actually lasted 15 minutes. What is the residual for this patient? Did the model under or over predict the duration?​

Answer 1

a) The equation of the least squares line is

y = 0.139x + 18.23

b) The patient of 30 years old would expect their asthma attack to last for 22.41 minutes.

c) The model over predicted the duration.

How the equation of least squares line is derived.

Given

Age: 30 25 65 50 30

Duration of attack: 15 28 30 22 24

Mean age: 42 yrs

S.D is 16.05 yrs

Mean duration of attack: 23.8 mins

S.D of attack is 23.8 mins.

a) The least squares regression line is

y = b1x + b

where

y = predicted value

b1 = slope

b = y-intercept

x = age

From the given values

x = (30+25+65+50+30)/5

= 200/5

= 40

y = (15+28+30+22+24)/5

= 119/5

= 23.8

( xi - xbar)

30 - 40 = -10

25 - 40 = -15

65 - 40 = 25

50 - 40 = 10

30 - 40 = -10

(yi - ybar)

15 - 23.8 = -8.8

28 - 23.8 = 4.2

30 - 23.8 = 6.2

22 - 23.8 = -1.8

24 - 23.8 = 0.2

(xi - xbar)²

30 - 40 = 100

25 - 40 = 225

65 - 40 = 625

50 - 40 = 100

30 - 40 = 100

b1 = Σ(xi - xbar)(yi - ybar)/(xi - xbar)²

= 88 + (-63) + 155 + (-18)+(-2)/1150

= 160/1150

= 0.139

b = ybar - b1(xbar)

= 23.8 - 0.139(40)

= 18.23

y = 0.139x + 18.23

The equation of the least squares line is

y = 0.139x + 18.23

b) when x = 30

y = 0.139(30) + 18.23

= 22.41

c) If the attack lasted for 15 minutes

The residual for the patient

Residual = actual - predicted

= 15 - 22.41

= -7.41

From the calculated residual time it can be deducted that the model over predict the dduration. This is indicated by the minus sign in the residual time.