(a) Explanation Given information: X is the number of owners who greet their dogs first. Sample size = 12 Percentage of dog owners who greet their dogs first = 66 % Concept used: 1 0 % condition n < 0. 1 0 N The rule says that if the sample is less than 1 0 % of the population, then it is reasonable to assume that the trials are independent and can be modelled by binomial distribution irrespective of the without replacement sample. Here, the sample size ( = 12 ) is far less than 1 0 % of all dog owners. Moreover, here the trial of each adult is independent. Whether one dog greets their dogs first does not give any idea about the other dog owner. So, X ~ Bin ( 12 , 0. 66 ) Hence, X can be modelled by a binomial distribution. (b) To determine To calculate: Probability that exactly 6 owners greet their owners first. Answer Probability that exactly 6 owners greet their owners first is 0.117980. Explanation Calculation: P ( X = 6 ) = 12 C 6 × 0.66 6 × ( 1 − 0.66 ) 12 − 6 P ( X = 6 ) = 0. 11798 0 (c) To determine To check: Whether there is any evidence against the Ladies Home Journal claim or not. Answer There seems to be strong evidence against the Ladies Home Journal claim. Explanation Given information: 4 owners in reality greeted their digs first. Calculation: Finding the probability of the above fact according to Ladies Home Journal claim P ( X ≤ 4 ) = P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 ) So, P ( X ≤ 4 ) = C 0 12 × 0.66 0 × ( 1 − 0.66 ) 12 − 0 + C 1 12 × 0.66 1 × ( 1 − 0.66 ) 12 − 1 + C 2 12 × 0.66 2 × ( 1 − 0.66 ) 12 − 2 + C 3 12 × 0.66 3 × ( 1 − 0.66 ) 12 − 3 + C 4 12 × 0.66 4 × ( 1 − 0.66 ) 12 − 4 P ( X ≤ 4) = 0 .021265 It can be seen that the probability according to Ladies Home Journal is very low giving evidence to strongly reject the claim.

6th Edition

ISBN: 9781319287573

Author: Starnes

Publisher: MPS PUB

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Question

Chapter 6, Problem T6.12SPT

(a)

To determine

To explain: X can be modelled by a binomial distribution.

(a)

Expert Solution

Explanation of Solution

Given information:

X is the number of owners who greet their dogs first.

Sample size $= 12$

Percentage of dog owners who greet their dogs first $= 66%$

Concept used:

$10% condition$

$n < 0.10N$

The rule says that if the sample is less than $10%$ of the population, then it is reasonable to assume that the trials are independent and can be modelled by binomial distribution irrespective of the without replacement sample.

Here, the sample size ( $= 12$ ) is far less than $10%$ of all dog owners.

Moreover, here the trial of each adult is independent. Whether one dog greets their dogs first does not give any idea about the other dog owner.

So, $X ~ Bin (12, 0.66)$

Hence, X can be modelled by a binomial distribution.

(b)

To determine

To calculate: Probability that exactly 6 owners greet their owners first.

(b)

Expert Solution

Answer to Problem T6.12SPT

Probability that exactly 6 owners greet their owners first is 0.117980.

Explanation of Solution

Calculation:

$P (X = 6) = 12C6 × 0.666 × (1−0.66)12−6$

$P (X = 6) = 0.117980$

(c)

To determine

To check: Whether there is any evidence against the Ladies Home Journal claim or not.

(c)

Expert Solution

Answer to Problem T6.12SPT

There seems to be strong evidence against the Ladies Home Journal claim.

Explanation of Solution

Given information:

4 owners in reality greeted their digs first.

Calculation:

Finding the probability of the above fact according to Ladies Home Journal claim

$P(X≤4)= P (X = 0) + P (X = 1) + P (X = 2) + P(X = 3) + P (X = 4)$

So,

$P (X ≤4)=$

$C012× 0.660 × (1−0.66)12−0+C112× 0.661 × (1−0.66)12−1+C212× 0.662 × (1−0.66)12−2+C312× 0.663 × (1−0.66)12−3+C412× 0.664 × (1−0.66)12−4$

$P (X ≤ 4) = 0.021265$

It can be seen that the probability according to Ladies Home Journal is very low giving evidence to strongly reject the claim.

Chapter 6, Problem T6.11SPT

Chapter 6, Problem T6.13SPT

Chapter 6 Solutions

EBK PRACTICE OF STAT.F/AP EXAM,UPDATED

Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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