Question

Use the Central Limit Theorem to calculate the following probability. Assume that the distribution of the population data is normally distributed. A person with “normal” blood pressure has a diastolic measurement of 75 mmHg, and a standard deviation of 4.5 mmHg.

i) What is the probability that a person with “normal” blood pressure will get a diastolic result of over 80 mmHg, indicating the possibility of pre-hypertension?

ii) If a patient takes their blood pressure every day for 10 days, what is the probability of getting an average diastolic blood pressure result of over 80 mmHg, assuming the patient has normal blood pressure.

iii) What can we conclude about such a result (based on preceding calculation)?

Answer 1

Solution :

Let X be a random variable which represents the diastolic measurements of people witt normal blood pressure.

Given that, X ~ N(75, 4.5²)

μ = 75 mmHg and  σ = 4.5 mmHg

i) We have to find P(X > 80 mmHg).

We know that, if X ~ N(μ, σ²) then,

Using "pnorm" function of R we get, P(Z > 1.1111) = 0.1333

Hence, the probability that a person with “normal” blood pressure will get a diastolic result of over 80 mmHg is 0.1333.

(ii) We have to find P(x̅ > 80).

(Where, x̅ is average blood pressure measurements of 10 days.)

If X ~ N(μ, σ²) then, x̅ ~ N(μ, σ²/n).

And if x̅ ~ N(μ, σ²/n) then,

We have, n = 10

Using "pnorm" function of R we get, P(Z > 3.5136) = 0.0002

Hence, the required probability is 0.0002.

(iii) We can conclude that 13.33% of people with normal blood pressure have diastolic measurements over 80 mmHg.