An automobile battery manufacturer offers a 39/50 warranty on its batteries. The first number in the warranty code is the free-replacement period; the second number is the prorated-credit period. Under this warranty, if a battery fails within 39 months of purchase, the manufacturer replaces the battery at no charge to the consumer. If the battery fails after 39 months but within 50 months, the manufacturer provides a prorated credit toward the purchase of a new battery. The manufacturer assumes that x, the lifetime of its auto batteries, is normally distributed with a mean of 44 months and a standard deviation of 3.6 months. 1. If the maufacturer's assumptions are correct, it would need to replace of its batteries free of charge.

a. 8.23%
b. 4.75%
c. 91.77%
d. 95.25%

2. The company finds that it s replacing 9.34% of its batteries free of charge. It suspects that its assumption about the standard deviation of the life of its batteries is incorrect. A standard deviation of results in a 9.34% replacement rate.

a. 3.8
b. 5.4
c. 4.1
d. 4.2

1)

If the manufacturer's assumptions are correct, it would need to replace 8.23% of its batteries free of charge.

Option a) 8.23% is the correct option

2)

the required standard deviation is 3.8

Option a) 3.8 is the correct answer

Explanation:

Given the data in the question;

mean μ = 44

standard deviation σ = 3.6

if a battery fails within 39 months of purchase, the manufacturer replaces the battery at no charge to the consumer. If the battery fails after 39 months but within 50 months

1)

If the manufacturer's assumptions are correct, it would need to replace of its batteries free of charge

p( X < 39 ) = P( X-μ/σ < 39-μ/σ )

we substitute

= p( Z < ((39-44)/3.6 ))

= p( Z < -1.39 )

from standard normal table; p( Z < -1.39 ) = 0.08226 ≈ 0.0823

p( X < 39 ) = 0.0823 or 8.23%

Therefore, If the manufacturer's assumptions are correct, it would need to replace 8.23% of its batteries free of charge.

Option a) 8.23% is the correct option

2)

The company finds that it s replacing 9.34% of its batteries free of charge. It suspects that its assumption about the standard deviation of the life of its batteries is incorrect. A standard deviation of results in a 9.34% replacement rate.

given that;

P( Z < x-μ/σ ) = 9.34%

⇒ P( Z < 39-44/σ ) = 0.0934 let this be equation 1

now, from standard normal tables

∅( -1.32 ) = 0.0934   let this equation 2

so from equation 1 and 2

39-44/σ = -132

-5/σ = -1.32

σ  = -5 / - 1.32

σ  = 3.7879 ≈ 3.8

Therefore, the required standard deviation is 3.8

Option a) 3.8 is the correct answer

there are so many examples for that in different areas, like biology experiment carried out in our lab recently.

here's one link :