In: Statistics and Probability
19) Your friends are at it again. This time they are testing H0 : μ = 14 versus H1 : μ ≠ 14. They have found that x̅ = 13.5, x̅cu = 14.7, n = 64 and they know σ = 2.5 .
a) Do your reject H0 ? Explain.
b) Find alpha.
c) Find the p-value.
d) Find beta if mu is really 14.3.
a) No, Ho is not rejected because Xbar = 13.5 < Xbar critical = 14.7
b)
µ = 14
σ = 2.5
n= 64
X = 14.7
Z = (X - µ )/(σ/√n) = ( 14.7
- 14 ) / ( 2.5 /
√ 64 ) = 2.240
P(X ≥ 14.7 ) = P(Z ≥
2.24 ) = P ( Z <
-2.240 ) = 0.0125
α = 2*0.0125 = 0.025
c)
Ho : µ = 14
Ha : µ ╪ 14
Level of Significance , α = 0.025
population std dev , σ = 2.5000
Sample Size , n = 64
Sample Mean, x̅ = 13.5000
' ' '
Standard Error , SE = σ/√n = 2.5/√64=
0.3125
Z-test statistic= (x̅ - µ )/SE =
(13.5-14)/0.3125= -1.6000
critical z value, z* = ± 2.2400
p-Value = 0.1096
d)
true mean , µ = 14.3
hypothesis mean, µo = 14
significance level, α =
0.025090923
sample size, n = 64
std dev, σ = 2.5000
δ= µ - µo = 0.3
std error of mean=σx = σ/√n = 2.5/√64=
0.3125
(two tailed test) Zα/2 = ±
2.240
We will fail to reject the null (commit a Type II error) if we get
a Z statistic between
-2.240 and 2.240
these Z-critical value corresponds to some X critical values ( X
critical), such that
-2.240 ≤(x̄ - µo)/σx≤
2.240
13.300 ≤ x̄ ≤ 14.700
now, type II error is ,
ß = P ( 13.300 ≤ x̄ ≤
14.700
Z = (x̄-true mean)/σx
Z1=(13.3-14.3)/0.3125=
-3.2000
Z2=(14.7-14.3)/0.3125=
1.2800
P(Z<1.28)-P(Z<-3.2)=
= 0.899727432 - 0.0007
= 0.899027432
(answer)