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?

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.