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Refer to Exercise 8.a. Find a 99% upper confidence bound

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 15E Chapter 5.1

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 15E

Problem 15E

Refer to Exercise 8.

a. Find a 99% upper confidence bound for the mean temperature.

b. The claim is made that the mean temperature is less than 349.5°F. With what level of confidence can this statement be made?

Exercise 8

Oven thermostats were tested by setting them to 350°F and measuring the actual temperature of the oven. In a sample of 67 thermostats, the average temperature was 348.2°F and the standard deviation was 5.1°F.

a. Find a 90% confidence interval for the mean oven temperature.

b. Find a 95% confidence interval for the mean oven temperature.

c. What is the confidence level of the interval (347.5, 348.9)?

d. How many thermostats must be sampled so that a 90% confidence interval specifies the mean to within ±0.8°F?

e. How many thermostats must be sampled so that a 95% confidence interval specifies the mean to within ±0.8°F?

Step-by-Step Solution:
Step 1 of 3

Solution 15E

Step1 of 3:

Let us consider a random variable X it represents the oven temperature. Here random variable X follows normal distribution with mean standard deviation and n = 67.

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 99% upper confidence bound for the mean temperature.

b). We need to find level of confidence can this statement be made by taking the mean temperature is less than 349.5°F.


Step2 of 3:

a).

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

Now,

 

   = 0.01

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 0.99 value falls, it falls in row 2.3 under column 0.03.

Hence,

99% upper confidence bound for the mean temperature is given by

   

     

         

                                                (349.6517)

Hence, 99% upper confidence bound for the mean temperature is 349.6517.


Step3 of 3:

b).

We have and n = 67

In a given information we have upper bound 349.5

Now,

                 

                           

       349.5 - 348.2 = (0.6230)

                         1.3 =  (0.6230)

                                                       =

            = 2.09

Hence, = 2.09.

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 2.0 under column 0.09.

 = 0.9817.


Step 2 of 3

Chapter 5.1, Problem 15E is Solved
Step 3 of 3

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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Refer to Exercise 8.a. Find a 99% upper confidence bound