Question

The mean of a population is 75 and the standard deviation is 14. The shape of the population is unknown. Determine the probability of each of the following occurring from this population.

a.A random sample of size 34 yielding a sample mean of 80 or more
b.A random sample of size 120 yielding a sample mean of between 72 and 78
c.A random sample of size 220 yielding a sample mean of less than 75.3

Answer 1

A) P(> 80)

= P(( - )/() > (80 -  )/())

= P(Z > (80 - 75)/(14/))

= P(Z > 2.08)

= 1 - P(Z < 2.08)

= 1 - 0.9812

= 0.0188

B) P(72 < < 78)

= P((72 - )/() < ( -   )/() < (78 -  )/())

= P((72 - 75)/(14/) < Z < (78 - 75)/(14/))

= P(-2.35 < Z < 2.35)

= P(Z < 2.35) - P(Z < -2.35)

= 0.9906 - 0.0094

= 0.9812

C) P( < 75.3)

= P(( - )/() < (75.3 -  )/())

= P(Z < (75.3 - 75)/(14/))

= P(Z < 0.32)

= 0.6255