Question

3. 28% of the population has high blood pressure. in a random sample of 200 people, what is the exact probability (to 7 decimal places) that

A. less then 65 people have high blood pressure

B. more then 30, but less then 54, of the people selected have high blood pressure

C. more then 140 people do not have high blood pressure?

D. answer B using normal approximation to the binomial, remembering to use the continuity correction. use the pnorm function described above and include your R code and output in your lab report. How well did the approximaton perform?

Please include R code and output for all parts.

Answer 1

Solution:-

p = 0.28, n = 200

A) The exact probability that less then 65 people have high blood pressure is 0.9083.

x = 65

By applying binomial distribution

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x < 65) = 0.9083

B) The exact probability that more then 30, but less then 54, of the people selected have high blood pressure is 0.3505.

x₁ = 30

x₂ = 54

By applying binomial distribution

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(30 < x < 54) = P(x > 30) - P(x > 54)

P(30 < x < 54) = 0.9999 - 0.6494

P(30 < x < 54) = 0.3505

C) The exact probability that more then 140 people do not have high blood pressure is 0.7120

x = 140

p = 1 - 0.28 = 0.72

n =200

By applying binomial distribution

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x > 40) = 0.7120

D) The answer B using normal approximation to the binomial is 0.3467.

Mean = n*p

Mean = 200*0.28

Mean = 56

After continuity correction

x₁ = 30.5

x₂ = 53.5

By applying normal distribution:-

z₁ = - 4.02

z₂ = - 0.394

P( -4.02 < z < -0.394) = P(z > - 4.02) - P(z > -0.394)

P( -4.02 < z < -0.394) = 0.9999 - 0.6532

P( -4.02 < z < -0.394) = 0.3467