Question

The ages of a group of 50 women are approximately normally distributed with a mean of 51 years and a standard deviation of 6 years. One woman is randomly selected from the​ group, and her age is observed.

a. Find the probability that her age will fall between 56 and 61 years.

b. Find the probability that her age will fall between 48 and 51 years.

c. Find the probability that her age will be less than 35 years

. d. Find the probability that her age will exceed 40 years.

Answer 1

Solution :

Given that ,

a.

P(56 < x < 61) = P[(56 - 51)/ 6) < (x - ) /  < (61 - 57) / 6) ]

= P(0.83 < z < 1.67)

= P(z < 1.67) - P(z < 0.83)

= 0.9525 - 0.7967

= 0.1558

Probability = 0.1558

b.

P(48 < x < 51) = P[(48 - 51)/ 6) < (x - ) /  < (51 - 51) / 6) ]

= P(-0.5 < z < 0)

= P(z < 0) - P(z < -0.5)

= 0.5 - 0.3085

= 0.1915

Probability = 0.1515

c.

P(x < 35) = P[(x - ) / < (35 - 51) / 6]

= P(z < -2.67)

= 0.0038

Probability = 0.0038

d.

P(x > 40) = 1 - P(x < 40)

= 1 - P[(x - ) / < (40 - 51) / 6)

= 1 - P(z < -1.83)

= 1 - 0.0336

= 0.9664

Probability = 0.9664