Question

Problem:

The distribution of micro-strokes per year in the general population can be modeled as a random variable M=N(u=40, σ=4). A research study found 80 out of 100 patients with M>48 (that is more than 48 micro-strokes) suffered a major stroke within a year whereas only 30 out of 100 patients with M≤48 did. What is the probability a person picked randomly from the same population suffers a major stroke within a year?

Answer 1

M is the random variable represents number of micro strokes per year.

M follows approximately normal distribution with mean µ = 40 and standard deviation σ = 4

We are given that around 80% of the patients has more than 48 micro stokes.

So probability that patients has more than 48 micro stokes. = 0.8*P( M > 48 )

We can find P( M > 48 ) using normal distribution.

P( M > 48 ) =

= P( z > 2 ) = 1 - P( z ≤ 2 ) = 1 - 0.9773 --- ( 0.9773 is area below z score 2 , from z score table )

P( M > 48 ) = 0.0227

0.8*P( M > 48 ) = 0.8*0.0227 = 0.0182

We are also given that around 30% of the patients has less than or 48 micro stokes.

So probability that patients has less than or equal to 48 micro stokes. = 0.3*P( M ≤ 48 )

P( M ≤ 48 ) = 1 - P( M > 48 ) = 1 - 0.0227 = 0.9773

Therefore 0.3*P( M ≤ 48 ) = 0.3*0.9773 = 0.2932

So probability a person picked randomly from the same population suffers a major stroke within a year =

0.8*P( M > 48 ) + 0.3*P( M ≤ 48 ) = 0.0182 + 0.2932

= 0.3114