Question

A report states that the total cholesterol level of children between the ages of 2 and 6 years is approximately normally distributed with a mean of 192 mg/dL (milligrams per deciliter) and a standard deviation of 27 mg/dL. Please answer all the questions.

a) What is the probability that one randomly selected child between the ages of 2 and 6 years has a total cholesterol level between 150 mg/dL and 200 mg/dL?

b) What is the probability that of 7 children between the ages of 2 and 6 years, exactly 4 have total cholesterol levels between 150 mg/dL and 200 mg/dL? Hint: Think of the binomial distribution!

c) What is the 30th percentile for total cholesterol level for children between the ages of 2 and 6 years? NO EXCEL

d) Between what values are the middle 70% of total cholesterol levels for children between the ages of 2 and 6 years? NO EXCEL

Answer 1

a) P(150 < X < 200)

= P((150 - )/ < (X - )/ < (200 - )/)

= P((150 - 192)/27 < Z < (200 - 192)/27)

= P(-1.56 < Z < 0.30)

= P(Z < 0.30) - P(Z < -1.56)

= 0.6179 - 0.0594

= 0.5585

b) n = 7

P = 0.5585

It is a binomial distribution.

P(X = x) = nCx * p^x * (1 - p)^{n - x}

P(X = 4) = 7C4 * (0.5585)^4 * (1 - 0.5585)^3 = 0.2931

c) P(X < x) = 0.3

Or, P((X - )/ < (x - )/) = 0.3

Or, P(Z < (x - 192)/27) = 0.3

Or, (x - 192)/27 = -0.52

Or, x = -0.52 * 27 + 192

Or, x = 177.96

d) P(X < x) = 0.175

Or, P((X - )/ < (x - )/) = 0.175

Or, P(Z < (x - 192)/27) = 0.175

Or, (x - 192)/27 = -0.935

Or, x = -0.935 * 27 + 192

Or, x = 166.755

P(X > x) = 0.175

Or, P((X - )/ > (x - )/) = 0.175

Or, P(Z > (x - 192)/27) = 0.175

Or, P(Z < (x - 192)/27) = 0.825

Or, (x - 192)/27 = 0.935

Or, x = 0.935 * 27 + 192

Or, x = 217.245

So the two values are 166.755 and 217.245