Problem 14E

Refer to Exercise 7.

a. Find a 95% lower confidence bound for the mean capacity of this type of battery.

b.An engineer claims that the mean capacity is greater than 175 ampere-hours. With what level of confidence can this statement be made?

Solution 14E

Step1 of 3:

Let us consider a random variable X it represents the capacity of batteries. Here random variable X follows normal distribution with mean standard deviation and n = 120.

That is

The probability density function of normal distribution is given by

, .

Where,

x = random variable

= mean of X

= variance of X

= standard deviation os X

= mathematical constant and its value is 3.14

Here our goal is:

a). We need to find a 95% lower confidence bound for the mean capacity of this type of battery.

b). We need to find level of confidence can this statement be made, by taking the mean capacity is greater than 175 ampere-hours.

Step2 of 3:

a).

Here we have to find 95% CI, let us take .

Now,

= 0.05

Z-scores(are to the right) is given by

is obtained from standard normal table(area under normal curve). In standard normal table we have to see where 0.95 value falls, it falls in row 1.6 under column 0.04.

Hence,

95% lower confidence bound for the mean capacity of this type of battery is given by

(175.9040)

Hence, 95% lower confidence bound for the mean capacity of this type of battery is 175.9040.

Step3 of 3:

b).

We have and n = 120.

In a given information we have lower bound 175

Now,

175 - 178 = (1.2780)

-3 = (1.2780)

=

= -2.3474

Hence, = -2.34.

Z-scores is given by , and this value is obtained from standard normal table(area under normal curve). In standard normal table we have to see in row -2.3 under column 0.04.

= 0.0096

Area to the right of Z is 1 - 0.0096

= 0.9904.

Hence level is 0.9904.