Question

30% of people between 60-70 years old will have at least one visit to a health provider. This proportion will be 5% for people between 10-20 years old, and it will be 15% for kids between 5-10 years old. If we randomly select 75 of people in the age range of 60-70, then answer the following questions:

What is the probability that between 30 and 60 people, inclusive, will visit a health provider?

Answer 1

Let p be the proportion of 60-70 years old who visit health provider then p = 0.3

Let Xi's be persons who we see if they visiting health providers or not in the age range 60-70.

Let distribution of Xi follows Xi = 1 if a person visits a health provider and Xi = 0 if a person does not visit.

Let Y = X1 + X2 + .......+ X75

We are sampling 75 people, so we can say by the central limit theorem that the sample Y is approximately normally distributed.

For p = 0.3

mean = np = 22.5, and standard deviation = = 3.97

then we can approximate this as N(22.5, (3.97)^2)

Now we need to find P(30 < Y < 60) = P( 29.5 < Y < 60.5) due to continuity correction

By central limit theorem, we get P((29.5 - 22.5)/3.97 < Y < (60.5 - 22.5)/3.97)

= P( 1.76 <= Z <= 9.57 )  

= 1-0.96 = 0.04