Question

2. According to researchers at Dan&Jones Associates, 1 on every 3 women has been a victim of domestic abuse. This probability was obtained from a survey of 1000 adult women residing in Utah. Suppose we randomly sample 15 women and find that 3 have been abused.

a) What's the probability of observing 3 or more abused women in a sample of 15 if the proportion of women who are victims of domestic abuse is really 1/3?

b) Many experts on domestic violence believe that the proportion of women who are domestically abused is closer to 0.1. Calculate the probability of observing 3 or more domestic abused women in a sample of 15 if the proportion of abused women is 0.1.

c) Based on the calculation result of parts a and b, what's your guess about the proportion of domestic abused women?

Answer 1

The scenario resembles that of binomial distribution,

P(X=x) = ⁿC_x p^x (1-p)^n-x

a)

p = 1/33 = 0.33

n = 15

P(0) = ¹⁵C₀ 0.33⁰ (1-0.33)^15-0 = 0.0023

Similarly, P(1) = 0.0171

And P(2) = 0.0599

Hence P(x>=3) = 1 – P(0) – P(1) – P(2) = 0.9206

b)

p = 0.1

n = 15

P(0) = ¹⁵C₀ 0.1⁰ (1-0.1)^15-0 = 0.2059

Similarly, P(1) = 0.3432

And P(2) = 0.2669

Hence P(x>=3) = 1 – P(0) – P(1) – P(2) = 0.1841

c)

Our guess is that the proportion is 0.1