An industrial firm making small battery-powered toys periodically purchases a la
ID: 3309982 • Letter: A
Question
An industrial firm making small battery-powered toys periodically
purchases a large number of batteries for use in the toys.
To protect itself,
the company, by default, believes a new vendor is no good.
The policy of the company is never to buy from a vendor
unless it has enough evidence at the 0.0075 significance level
to indicate that the batteries have a true mean life larger than 70 hours.
Historically, the standard deviation of the batteries' life is 3 hours.
A decision is made using evidence from a random sample
of 100 batteries from the vendor.
[(a)]
What are the appropriate Ho and Ha for this situation?
[(b)]
What are the consequences of a Type I Error
in the context of this situation?
[(c)]
What are the consequences of a Type II Error
in the context of this situation?
[(d)]
Should the company buy from a vendor if the random sample of 100 batteries
from the vendor results in a sample mean life, ybar, = 70.9 hours?
[(e)]
Suppose 100 batteries are randomly selected from a vendor,
what is the minimum sample mean life, ybar,
at which the company decides to buy?
Explanation / Answer
a)
Below are the hull and alternate hypothesis
H0: mu = 70
H1: mu > 70
b)
Type I error is the case where null hypothesis will be rejected though true mean is not greater than 70.
Its probability is 0.0075
c)
Type II error will lead to the fail of null hypothesis though it is true. In the context of this problem, we will conclude that that the mean life of battery is greater than 70 though actual mean is less than or equal to 70.
d)
test statistics, z = (70.9 - 70)/(3/sqrt(100)) = 3
p-value = 0.001349898
As p-value is less than significance level of 0.0075, fail to reject null hypothesis.
Hence there is significant evidence to conclude that true mean is larger than 70 hours.
e)
z-value = 2.432379059
2.4324 = (ybar - 70)/(3/sqrt(100))
ybar = 2.4324*(3/sqrt(100)) + 70
ybar = 70.7297
Hence minimum sample mean life is 70.7297 hours