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 P( ≤ 36.7)?

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 P(X ≤ 39.8)?

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.

Step 1 of 7</p>

Here given that the battery manufacturer claims that the lifetime of a certain type of a battery with population mean of 40 hrs and standard deviation of 5 hrs

So =40

=5

Let represents the mean life of the sample

Here follows the normal distribution

Step 2 of 7</p>

a) We have to find P(36.7)

For 36.7, Z=(

=(36.7-40)/5

= -0.66

Now find P(Z-0.66)=0.25463

Hence P(36.7)=0.2563

Step 3 of 7</p>

b) Yes , 36.7 is usually a short number

Because more than ¼ th of the samples having life less than 36.7 hrs

Step 4 of 7</p>

c) No, The manufacturer claim is not plausible because

36.7 is usually a short number in the sample of 100