Answer:
a. the probability of a type II error is 0.5319
b. the required sample size to satisfy and the type II error probability is 59.4441
Step-by-step explanation:
From the information given; we have:
sample size n = 25
Population standard deviation = 4
true average lifetime = Sample Mean = 62
We can state our null hypothesis and alternative hypothesis as follows:
Null hypothesis:
Alternative hypothesis
Where ;
∝ = 0.01
From the standard normal tables at critical value ∝ = 0.01 ; the level of significance is -2.575 lower limit and 2.575 upper limit
The z statistics for the lower limit is:
The z statistics for the upper limit is:
Thus; the probability of a type II error is determined as follows:
β = P ( )
= P ( -5.08 < Z < 0.08 )
= P ( Z < 0.08) - P ( Z < - 5.08)
Using Excel Function: [ (=NORMDIST (0.08)) - (=NORMDIST(-5.08)) ] ; we have:
= 0.531881372 - 0.00000001887
= 0.531881183
≅ 0.5319
b.
What is the required sample size to satisfy and the type II error probability of b(62) = 0.1
Recall that:
The critical value of ∝ = 2.575 ( i. e )
Now ;
the critical value of β is :
The required sample size to satisfy and the type II error probability is therefore determined as :
n = 7.71 ²
n= 59.4441
Thus; the required sample size to satisfy and the type II error probability is 59.4441