Answer:
The probability that more than 22 of the women sampled have not been the victim of domestic abuse is P=0.537.
Step-by-step 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:
[tex]p=1-\dfrac{X}{n}=1-\dfrac{1}{10}=1-0.1=0.9[/tex]
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:
[tex]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}\\\\\\[/tex]
Then, the probability that more than 22 of the women sampled have not been the victim of domestic abuse is:
[tex]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[/tex]
