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Consider the endothermic reactions in Exercise 3-32. A

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger ISBN: 9781118539712 55

Solution for problem 114E Chapter 3.6

Applied Statistics and Probability for Engineers | 6th Edition

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Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers | 6th Edition

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Problem 114E

Problem 114E

Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.

(a) What is the probability that exactly 12 reactions result in a final temperature less than 272 K?

(b) What is the probability that at least 19 reactions result in a final temperature less than 272 K?

(c) What is the probability that at least 18 reactions result in a final temperature less than 272 K?

(d) What is the expected number of reactions that result in a final temperature of less than 272 K?

3-32. The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:

Final Temperature Conditions

Number of Reactions

266 K

48

271 K

60

274 K

92

Calculate the probability mass function of final temperature.

Step-by-Step Solution:

Answer

Step 1 of 4

(a)

Consider the endothermic reactions in given below the table.

A total of  independent reactions is to be conducted.

Final Temperature Conditions

Number of Reactions

266 K

48

271 K

60

274 K

92

We are asked to find the probability that exactly  reactions result in a final temperature less than .

Hence the probability mass function PMF is according to the definition of PMF present in the table given below.

Final Temperature Conditions

Number of Reactions

PMF

266 K

48

271 K

60

274 K

92

Let  denote the number of the reactions result in a final temperature less than .

We need to find

Hence  follows a binomial distribution which we can write,

The random variable  that equals the number of trials that result in a success is a binomial random variable with parameters

The probability mass function (PMF) of  is,

………………(1)

The probability  that the reactions result in a final temperature less than  is from the table,

We have given

Hence the substitute the above values into equation (1),

Hence the probability that exactly  reactions result in a final temperature less than  is


Step 2 of 4

Chapter 3.6, Problem 114E is Solved
Step 3 of 4

Textbook: Applied Statistics and Probability for Engineers
Edition: 6
Author: Douglas C. Montgomery, George C. Runger
ISBN: 9781118539712

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Consider the endothermic reactions in Exercise 3-32. A