The capacities (in ampere-hours) were measured for a sample of 120 batteries. The average was 178 and the standard deviation was 14.

a. Find a 95% confidence interval for the mean capacity of batteries produced by this method.

b. Find a 99% confidence interval for the mean capacity of batteries produced by this method.

c. An engineer claims that the mean capacity is between 176 and 180 ampere-hours. With what level of confidence can this statement be made?

d. Approximately how many batteries must be sampled so that a 95% confidence interval will specify the mean to within ±2 ampere-hours?

e. Approximately how many batteries must be sampled so that a 99% confidence interval will specify the mean to within ±2 ampere-hours?

Solution 7E

Step1 of 6:

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 95% confidence interval for the mean capacity of batteries produced by this method.

b). We need to find 99% confidence interval for the mean capacity of batteries produced by this method.

c). An engineer claims that the mean capacity is between 176 and 180 ampere-hours. We need to check what level of confidence can this statement be made.

d). We need to find how many batteries must be sampled so that a 95% confidence interval will specify the mean to within ±2 ampere-hours.

e). We need to find how many batteries must be sampled so that a 99% confidence interval will specify the mean to within ±2 ampere-hours.

Step2 of 6:

a).

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

Now,

= 0.025

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 0.9750 value falls, it falls in row 1.9 under column 0.06.

Hence,

95% confidence interval for the mean capacity of batteries produced by this method is given by

(175.4950, 180.5049)

Hence, 95% confidence interval for the mean capacity of batteries produced by this method is

(175.4950, 180.5049).

Step3 of 6:

b).

Here we have to find 99% CI, let us take .

Now,

= 0.005

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 0.9950 value falls, it falls in row 2.5 under column 0.08.

Hence,

99% confidence interval...