Question

Too much cholesterol in the blood increases the risk of heart disease. The cholesterol levels for women aged 20 to 34 follow the Normal distribution with mean ?? =185 milligram per deciliter (mg/dl) and standard deviation ?? =39 mg/dl. a) Cholesterol level above 240 mg/dl demand medical attention. If a young woman is randomly selected, what is the probability that her cholesterol level is above 240 mg/dl? b) Find the lowest cholesterol level for the women in the top 30% of the distribution. c) If a group of five (5) young women is randomly selected, what is the probability that the sample mean cholesterol level of a group of five women is above 240 mg/dl?

Answer 1

Let X be the random variable denoting the cholesterol level

of women between age 20-34.

Thus, X ~ N(185, 39) i.e. (X - 185)/39 ~ N(0,1)

a) Required probability = P(X > 240)

= P[(X - 185)/39 > (240 - 185)/39] = P[(X - 185)/39 > 1.4103]

= 1 - P[(X - 185)/39 < 1.4103] = 1 - (1.4103) = 1 - 0.9208

[(.) is the cdf of N(0,1)]

= 0.0792. (Ans).

b) Let the lowest cholesterol level of top 30% be a.

Thus, P(X > a) = 0.3 i.e. P(X < a) = 1 - 0.3 = 0.7

i.e. P[(X - 185)/39 < (a - 185)/39] = 0.7

i.e. [(a - 185)/39] = 0.7 i.e. (a - 185)/39 = (0.7) = 0.524

i.e. a = 205.436 mg/dl. (Ans).

c) Let M be the sample mean of cholesterol level of 5:women.

Thus, E(M) = 185 and s.d.(M) = 39 / = 17.4413.

Hence, required probability = P(M > 240)

= P[(M - 185)/17.4413 > (240 - 185)/17.4413]

= P[(M - 185)/17.4413 > 3.1534]

= 1 - P[(M - 185)/17.4413 < 3.1534]

= 1 - (3.1534) = 1 - 0.9992 = 0.0008. (Ans).