A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and a standard deviation of 5 hours. Let 
represent the mean lifetime of the batteries in a simple random sample of size 100.
a. If the claim is true, what is 
?
b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?
c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer’s claim to be plausible? Explain.
d. If the claim is true, what is 
?
e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?
f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer’s claim to be plausible? Explain.
(a) If the claim is true, then 
is approximately normal with mean 
= 40 and
The z-score of 36.7 is (36.7-40)/0.5=-6.6.
The area to the left of z=-6 is neglible.
(b) Yes. Almost none of the samples will have mean lifetimes 36.7 hours or less if the claim is true.
(c) No, because a sample mean lifetime of 36.7 hours is very unusual if the claim is true.
(d) If the claim is true, then 
is approximately normal with mean = 40 and
The z-score of 39.8 is (39.8-40)/0.5=-0.4.
The area to the left of z=-0.4 is 0.3446.
(e) No. More than 1/3 of the samples will have mean lifetimes 39.8 hours or less if the claim is true.
(f) Yes, because a sample mean lifetime of 39.8 hours or less is not unusual if the claim is true.
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