Question

The Senator of Azenator State, is worried about the rising numbers in high blood pressure related deaths in his Jurisdiction and wants an end to this canker. A reputable medical research officer has claimed that, the situation is probably as a result of the ageing population of his State. The blood pressures, Y (mmHg), and Ages, X (years) of 10 hospital patients were sampled from Azenator State and summarized below. Patient A B C D E F G H I J Age (X) in Years 20 25 50 30 45 60 10 15 35 70 BP(Y) in (mmHg) 80 85 125 90 100 135 80 70 100 140 NB: Approximate to 2 decimal places Use the table to answer the questions that follow; i) Calculate the product moment correlation coefficient for the data and interpret your result. (3 Marks) ii) If the Senator decides to purchase and distribute Norvasc (a medicine that reduces blood pressure), based on your results in (i), which age group (youth or old adults) should be given priority? Briefly explain your answer. (3 Marks) iii) Give a reason to support fitting a regression model of the form = + + , where (a) is the y-intercept or constant, (b) is the slope of the function, (y) is the blood pressure of a patient, (X) is the age of a patient and is the error term. (3 Marks) iv) From the model specified in (iii), estimate the values of (a) and (b) and interpret the values (4 Marks) v) What is the blood pressure of an infant at birth? (2 Marks) vi) Predict the blood pressure of a patient who is 90 years old. (2 Marks) vii) Estimate the coefficient of determination and interpret your result.

Answer 1

Explanation:

Hello!

Given the variables:

Y: Blood pressure of a hospital patient (mmHg)

X: Age of a patients (years)

You have to analyze the claim that the rising numbers in high blood pressure related deaths is associated with the aging of the State's population.

i)

`r= \frac{sumXY-((sumX)(sumY))/(n) }{\sqrt{[sumX^2((sumX)^2)/(n) ]+[sumY^2-((sumY)^2)/(n)] } }`

To calculate the correlation coefficient you need to do several additional calculations.

n= 10

∑Y= 1005 mm Hg; ∑Y²= 106475 mmHg²

∑X= 360 years; ∑X²= 16500 years²

∑XY= 40425 years*mmHg

`r= \frac{40425-(360*1005)/(10) }{\sqrt{[16500((360)^2)/(10) ]+[106475-((1005)^2)/(10)] } }= 0.96`

The coefficient of correlation is close to 1 and positive, this means that there is a strong positive correlation between these two variables. This means the blood pressure increases as age does.

ii)

If blood pressure increases with age and so does the risk of dying due to a high blood pressure, then the most logical decision to take is to give priority to the old adults.

iii)

The estimated regression is ^Y= a + bXi + ei

a= y-intercept: is the estimated average blood pressure of the patients when age is 0 years.

b= slope: in the modification of the estimated average blood pressure of the patients when age increases 1 year.

ei= residues

The estimated regression model will allow you to analyze and predict the values of blood pressure of patients within the determined range of age.

iv)

`b= (sumXY-((sumX)(sumY))/(n) )/(sumX^2-((sumX)^2)/(n) ) = (40425-(360*1005)/(10) )/(16500-((360)^2)/(10) ) = 1.20 (mmHg)/(years)`

X[bar]= 36 years

Y[bar]= 100.50 mmHg

`a= Y[bar] - bX[bar]= 100.50-1.20*36= 57.33 mmHg`

The estimated regression equation is ^Y= 57.33 + 1.20Xi

57.33mmHg in the estimated average blood pressure of a patient age zero.

1.20 mmHg/years is the modification on the estimated average blood pressure when the age of the patients increases one year.

v)

At birth an infant is zero years old, so according to the regression model, it's blood pressure will be 57.33 mmHg

According to medical parameters 64.41 mmHg is the average blood pressure of a newborn child, as you can see this model is a little off the average but still within acceptable parameters.

vi)

Using the estimated regression equation you can predict the value of blood pressure for a determined age by replacing the value of X in it:

^Y= 57.33 + 1.20*90= 165.33 mmHg

It is predicted that a 90-year-old patient will have a blood pressure of mmHg.

vii)

`R^2= (b^2[sumX^2-((sumX)^2)/(n) ])/(sumY^2-((sumY)^2)/(n) ) = (1.20^2[16500-((360)^2)/(n) ])/([106475-((1005)^2)/(10) ]) = 0.93`

93% of the variability of the patients blood pressure is explained by their age under the model: ^Y= 57.33 + 1.20Xi

I hope this helps!