Question

The mean of a population is 75 and the standard deviation is 12. 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 35 yielding a sample mean of 78 or more b. A random sample of size 150 yielding a sample mean of between 73 and 76 c. A random sample of size 219 yielding a sample mean of less than 75.8

Answer 1

= 75

= 12

According to Central Limit Theorem, the sampling distribution of sample mean will be approximately normal for sample size of at least 30. Also, P( < A) = P(Z < (A - )/)

a) Sample size, n = 35

= = 75

=

=

= 2.03

P( 78) = 1 - P( < 78)

= 1 - P(Z < (78 - 75)/2.03)

= 1 - P(Z < 1.48)

= 1 - 0.9306

= 0.0694

b) Sample size, n = 150

= = 75

=

=

= 0.98

P(73 < < 76) = P( < 76) - P( < 73)

= P(Z < (76 - 75)/0.98) - P(Z < (73 - 75)/0.98)

= P(Z < 1.02) - P(Z < -2.04)

= 0.8461 - 0.0228

= 0.8233

c) Sample size, n = 219

= = 75

=

=

= 0.81

P( < 75.8) = P(Z < (75.8 - 75)/0.81)

= P(Z < 0.99)

= 0.8389