Question

12. Suppose the probability distribution of the lifetimes of the participants in a plan is approximately a normal distribution with mean of 68 years, standard deviation of 3.5 years. a) What proportion of the plan participants would receive payments beyond age 70? b) What proportion of the plan participants would receive payments below age 75? c) What proportion of the plan participants would receive payments between 69 and 96 years?

Answer 1

Let M be the mean = 68 and S be the std dev = 3.5

Part (a)

The proportion of the plan participants that would receive payments beyond age 70

= P(X > 70)

= P (Z > (X - M)/ S = (70 - 68) / 3.5 = 0.5714)

= P (Z > 0.5714)

= 1 - P (Z < 0.5714)

= 1 - 0.7161

= 0.2839

Part (b)

The proportion of the plan participants that would receive payments below age 75

= P(X < 75)

= P (Z < (X - M)/ S = (75 - 68) / 3.5 = 2)

= P (Z < 2)

= 0.9772

Part c)

The proportion of the plan participants who would receive payments between 69 and 96 years

= P (69 < X < 76)

= P [(69 - 68)/3.5 < Z < (76 - 68) / 3.5)

= P (0.2857 < Z < 2.2857)

= P (Z < 2.2857) - P( Z < 0.2857)

= 0.9889 - 0.6125

= 0.3764