32.2k views
0 votes
According to a recent study, 1 in every 10 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse

User Aelphaeis
by
8.4k points

1 Answer

2 votes

Answer:

The probability that more than 22 of the women sampled have not been the victim of domestic abuse is P=0.537.

Explanation:

This problem can be modeled by a binomial random variable.

The probability p can be calculated as the proportion of women that has not been a victim of domestic abuse at some point in her life:


p=1-(X)/(n)=1-(1)/(10)=1-0.1=0.9

The sample size is n=25.

We want to calculate the probability that more than 22 of the women of the sample have not been victim of domestic abuse.

The probability that exactly k women have not been victim of domestic abuse can be calculated as:


P(x=k) = \dbinom{n}{k} p^(k)(1-p)^(n-k)\\\\\\P(x=k) = \dbinom{25}{k} 0.9^(k) 0.1^(25-k)\\\\\\

Then, the probability that more than 22 of the women sampled have not been the victim of domestic abuse is:


P(x>22)=P(x=23)+P(x=24)+P(x=25)\\\\\\P(x=23) = \dbinom{25}{23} p^(23)(1-p)^(2)=300*0.089*0.01=0.266\\\\\\P(x=24) = \dbinom{25}{24} p^(24)(1-p)^(1)=25*0.08*0.1=0.199\\\\\\P(x=25) = \dbinom{25}{25} p^(25)(1-p)^(0)=1*0.072*1=0.072\\\\\\\\P(x>22)=0.266+0.199+0.072=0.537

User Desmond Lua
by
8.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories