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A manufacturer of floor wax has developed two new brands,

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer ISBN: 9780495110811 47

Solution for problem 49E Chapter 3

Mathematical Statistics with Applications | 7th Edition

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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Mathematical Statistics with Applications | 7th Edition

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Problem 49E

Problem 49E

A manufacturer of floor wax has developed two new brands, A and B, which she wishes to subject to homeowners’ evaluation to determine which of the two is superior. Both waxes, A and B, are applied to floor surfaces in each of 15 homes. Assume that there is actually no difference in the quality of the brands. What is the probability that ten or more homeowners would state a preference for

a brand A?

b either brand A or brand B?

Step-by-Step Solution:
Step 1 of 3

Solution 49E

Step1 of 3:

Here a A manufacturer of floor wax has developed two new brands, A and B, which she wishes to subject to homeowners evaluation to determine which of the two is superior. Also we have

n = 15, p = 0.5.

Here our goal is:

a). We need to find the probability that ten or more homeowners would state a preference for brand A.

b). We need to find the probability that ten or more homeowners would state a preference for either brand A or brand B.


Step2 of 3:

a).

Let ‘X’ be the random variable it presents number of homeowners who would state a preference.

Here X follows binomial distribution with parameter ‘n and p’.

That is XB(n, p)

                 XB(15, 0.5)

Then the probability mass function of binomial distribution is given by:

Where,

x = random variable

p = probability of success

n = sample size

 

It is obtained from Excel by using the function “=Binomdist(x,n,p,false)” 

X

P(X10)

10

0.091644

11

0.041656

12

0.013885

13

0.003204

14

0.000458

15

3.05E-05

Total

0.150879

   Therefore, the probability that ten or more homeowners would state a preference for brand A is 0.1508.


Step3 of 3:

b).

Here they have given the quality of both brands is same since, the probability of any selecting brand is same hence, brand A and brand B are mutually Exclusive events.

From part (a), we have P() = 0.1508

Similarly,  P() = 0.1508.

Therefore, the probability that ten or more homeowners would state a preference for either brand A or brand B is given by:

                       

      = 0.3016

Hence, The probability that ten or more homeowners would state a preference for either brand A or brand B is 0.3016.


Step 2 of 3

Chapter 3, Problem 49E is Solved
Step 3 of 3

Textbook: Mathematical Statistics with Applications
Edition: 7
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
ISBN: 9780495110811

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A manufacturer of floor wax has developed two new brands,