Problem 16E

Refer to Exercise 9.

a. Find a 90% upper confidence bound for the mean weight.

b. Someone says that the mean weight is less than 1.585 g. With what level of confidence can this statement be made?

Solution 16E

Step1 of 3:

Let us consider a random variable X it represents the weight of the penny nail. Here random variable X follows normal distribution with mean standard deviation and

n = 80.

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 90% upper confidence bound for the mean weight.

b). We need to find level of confidence can this statement be made by taking the mean weight is less than 1.585 g.

Step2 of 3:

a).

Here we have to find 90% CI, let us take .

Now,

= 0.10

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 0.90 value falls, it falls in row 1.2 under column 0.08.

Hence,

90% upper confidence bound for the mean weight is given by

(1.5743)

Hence, 90% upper confidence bound for the mean weight is 1.5743.

Step3 of 3:

b).

We have and n = 80.

In a given information we have upper bound 1.585

Now,

1.585 - 1.56 = (0.0111)

0.025 = (0.0111)

=

= 2.2522

Hence, = 2.25.

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 2.2 under column 0.05.

= 0.9878

Area to the right of Z is 1 - 0.9878

= 0.0122

Hence level is 0.0122.