Question

A company makes batteries with an average life span of 300 hours with a standard deviation of 75 hours. Assuming the distribution is approximated by a normal curve fine the probability that the battery will last:(give 4 decimal places for each answer)

1. Less than 250 hours ________

2. Between 225 and 375 hours _________

3. More than 400 hours _________

Answer 1

1= 33.5 percent
2=51 percent
3=20.2 percent

Answer 2

1.
`P(X<250)=0.2546`

2.
`P(225<X<375)=0.6826`

3.
`P(X>400)=0.0918`

Explanation:

Given information: Population mean = 300 hours, Standard deviation=75

Let X be the life of battery in hours.

`Z=(X-\mu)/(\sigma)`

(1) The probability that the battery will last less than 250 hours is

`P(X<250)=P((X-300)/(75)<(250-300)/(75))`

`P(X<250)=P(Z<-0.667)`

`P(X<250)=0.2546`

Therefore the probability that the battery will last less than 250 hours is 0.2546.

(2) The probability that the battery will last between 225 and 375 hours is

`P(225<X<375)=P((225-300)/(75)<(X-300)/(75)<(375-300)/(75))`

`P(225<X<375)=P(-1<Z<1)`

`P(225<X<375)=P(Z<1)-P(Z<-1)`

`P(225<X<375)=0.8413-0.1587`

`P(225<X<375)=0.6826`

Therefore the probability that the battery will last between 225 and 375 hours is 0.6826.

(3) The probability that the battery will last more than 400 hours is

`P(X>400)=P((X-300)/(75)<(400-300)/(75))`

`P(X>400)=P(Z>1.33)`

`P(X>400)=1-P(Z<1.33)`

`P(X>400)=1-0.9082`

`P(X>400)=0.0918`

Therefore the probability that the battery will last more than 400 hours is 0.0918.