bA known population has an average height of 682 inches The

bA known population has an average height of 68.2 inches. The population distribution is normal and it is known that the middle 50% of the population have heights between 66.5 and 69.9 inches. A sample of 36 individuals is drawn independently and at random from this population. (NOTE: The first step will be to calculate the population standard deviation. Round this value to the nearest tenth of an inch).

a) How likely is it that their mean height would exceed 69 inches? (Carry answer to 4 decimal places if appropriate).

b) How likely is it that their mean height would be less than 67 inches? (Carry your answer to 4 decimal places).

c) Beyond what mean height should only 5% of the sample means fall? (Round your answer to the nearest tenth of an inch.)

Solution

A known population has an average height of 68.2 inches. The population distribution is normal and it is known that the middle 50% of the population have heights between 66.5 and 69.9 inches. A sample of 36 individuals is drawn independently and at random from this population. (NOTE: The first step will be to calculate the population standard deviation. Round this value to the nearest tenth of an inch).

a) How likely is it that their mean height would exceed 69 inches? (Carry answer to 4 decimal places if appropriate).

Z Value for middle 50% =1.15

Mean+z*sd

69.9=68.2+1.15*sd

Sd=1.4782

Sd=1.5 ( 1 decimal)

Standard error =1.5/sqrt(36) = 0.25

Z value for 69, ( 69-68.2)/0.25 =3.2

P( mean x >69) = p( z >3.2)

=0.0007

b) How likely is it that their mean height would be less than 67 inches? (Carry your answer to 4 decimal places).

Z value for 67, ( 67-68.2)/0.25 =-4.8

P( mean x <67) = p( z <-4.8)

=0.0000

c) Beyond what mean height should only 5% of the sample means fall? (Round your answer to the nearest tenth of an inch.)

z value for top 5% =1.645

required value =68.2+1.645*0.25

=68.6