Question

The mean of a population is 74 and the standard deviation is 16. 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 32 yielding a sample mean of 76 or more

b. A random sample of size 130 yielding a sample mean of between 72 and 76

c. A random sample of size 220 yielding a sample mean of less than 74.3

Answer 1

Solution:

Given that ,

= 74

= 16

Let be the mean of sample.

The sampling distribution of the is approximately normal with

Mean() =   

SD() =   

a) n = 32

= = 74

= 16/32 =  2.82842712

Find P( > 76)

= P[( - )/ >  (76 - )/]

= P[Z > (76 - 74)/ 2.82842712]

= P[Z > 0.707]

= 1 - P[Z < 0.707]

= 1 - 0.7602 ( use z table)

= 0.2398

b) n = 130

= = 74

= 16/130 =  1.4032928

P(72 < < 76)

= P( < 76) - P( < 72)

= P[( - )/ < (76 - 74)/1.4032928] - P[( - )/ < (72 - 74)/1.4032928]

= P[Z < 1.425] - P[Z < -1.425]

= 0.9229  - 0.0771 (use z table)

= 0.8458

c)n = 220

= = 74

= 16/220 =  1.0787198

P( < 74.3)

= P[( - )/ < (74.3 - 74)/1.0787198]

= P[Z < 0.278]

=  0.6095