In: Statistics and Probability
A sample size 5 will be drawn from a normal population with mean 60 and standard deviation 12.
a. Is it appropriate to use the normal distribution to find probability for ?̅? Explain why?
b. If appropriate find the probability that ?̅will be between 50 and 70.
c. If appropriate find the 80th percentile of ?̅?
a) yes, It appropriate to use the normal distribution to find probability for ?̅
because population is normally distributed, so, sampling distribution will also be normally distributed
b)
µ = 60
σ = 12
n= 5
we need to calculate probability for ,
50 ≤ X ≤ 70
X1 = 50 , X2 =
70
Z1 = (X1 - µ )/(σ/√n) = ( 50
- 60 ) / ( 12 /
√ 5 ) = -1.86
Z2 = (X2 - µ )/(σ/√n) = ( 70
- 60 ) / ( 12 /
√ 5 ) = 1.86
P ( 50 < X <
70 ) = P ( -1.86
< Z < 1.86 )
= P ( Z < 1.86 ) - P ( Z
< -1.86 ) =
0.9688 - 0.0312 =
0.9376
(answer)
c)
µ = 60
σ = 12
n= 5
proportion= 0.8000
Z value at 0.8 =
0.842 (excel formula =NORMSINV(
0.80 ) )
z=(x-µ)/(σ/√n)
so, X=z * σ/√n +µ= 0.842 *
12 / √ 5 +
60 = 64.52