In: Statistics and Probability
A quality expert inspects 430 items to test whether the population proportion of defectives exceeds .02, using a right-tailed test at α = .10. (a) What is the power of this test if the true proportion of defectives is π = .03? (Round your intermediate calculations and final answer to 4 decimal places.) (b) What is the power of this test if the true proportion is π = .04? (Round your intermediate calculations and final answer to 4 decimal places.) (c) What is the power of this test if the true proportion of defectives is π = .05? (Round your intermediate calculations and final answer to 4 decimal places.)
a)
true proportion, p= 0.03
hypothesis proportion, po=
0.02
significance level, α = 0.100
sample size, n = 430
std error of sampling distribution, σpo =
√(po*(1-po)/n) = √ ( 0.020 *
0.980 / 430 ) =
0.0068
std error of true proportion, σp = √(p(1-p)/n) = √
( 0.03 * 0.97
/ 430 ) = 0.0082
Zα = 1.282 (right
tailed test)
We will fail to reject the null (commit a Type II error) if we get
a Z statistic <
1.282
this Z-critical value corresponds to X critical value( X critical),
such that
(p^ - po)/σpo ≤ Zα
p^ ≤ Zα*σpo + po
p^ ≤ 1.282*0.0068+0.02
= 0.0287
now, type II error is ,ß = P( p^ ≤
0.0287 given that p =
0.03
= P ( Z < (p^ - p)/σp )=
P(Z < (0.0287-0.03) / 0.0082)
= P ( Z < ( -0.164
)
ß = 0.434933
power = 1 - ß = 0.5651
b)
std error of true proportion, σp = √(p(1-p)/n) = √
( 0.04 * 0.96
/ 430 ) = 0.0094
now, type II error is ,ß = P( p^ ≤
0.0287 given that p = 0.04
= P ( Z < (p^ - p)/σp )=
P(Z < (0.0287-0.04) /
0.0094)
= P ( Z < ( -1.201
)
ß = 0.114911
power = 1 - ß = 0.8851
c)
std error of true proportion, σp = √(p(1-p)/n) = √
( 0.05 * 0.95
/ 430 ) = 0.0105
now, type II error is ,ß = P( p^ ≤
0.0287 given that p =
0.05
= P ( Z < (p^ - p)/σp )=
P(Z < (0.0287-0.05) / 0.0105)
= P ( Z < ( -2.031
)
ß = 0.021121
power = 1 - ß = 0.9789