Question

For each of these problems, conduct a significance test. Remember there are four steps after
confirming the conditions are met for inference:
1) state the hypotheses
2) calculate test statistic
3) find P-value. Remember that we are using α = 0.05 as our guideline for statistical
significance. If P-value ≤ 0.05, reject Ho. If P-value > 0.05, do not reject Ho.
4) state the conclusion in plain English in the context of the problem, not just “reject Ho” or “do
not reject Ho.” Look at the statements for Problem 1 and use them as a template for your
conclusions. If you decide to reject Ho give the P-value.
Read the entire problem before answering the questions.
Problem 1) The reputations (and hence sales) of many businesses can be severely damaged by
shipments of manufactured items that contain a large percentage of defectives. For example, a
manufacturer of alkaline batteries may want to be reasonably certain that fewer than 6% of its
batteries are defective. Suppose a random sample of 800 batteries are selected from a very large
shipment; each is tested and 31 defective batteries are found. (The manufacturer is analyzing the
proportion of DEFECTIVE batteries.)
a) Consider just one experimental unit – that is, one battery. What is the response variable for
that one battery? Categorical or quantitative?
b) The manufacturer wants to conduct a significance test to decide if there is sufficient evidence
for the manufacturer to conclude that the fraction defective in the entire shipment is less than
6 percent. Verify the conditions for using the normal approximation for the sample proportion.
c) Conduct the significance test. Keep two nonzero digits in your calculation for ??̂ and for the
standard deviation. Sketch the distribution for ??̂ showing mean and area for P-value. State
your conclusion in plain English in the context of the problem: choose one of the statements
for your conclusion, depending on your P-value:
• We have evidence to show that the fraction defective in the entire shipment is less than
6 percent (P-value = _______)
• We do not have evidence to show that the fraction defective in the entire shipment is
less than 6 percent.

Answer 1

a) Response variable is that the battery is defective or not, it is Categorical variable

b) Test for single proportion:

(1) Null and Alternative Hypothesis:

H0:The defective proportion of alkaline batteries not fewer than 6%

H1:The defective proportion of alkaline batteries fewer than 6%

i.e.

Thus we conclude that the defective proportion of alkaline batteries fewer than 6%