Victims of domestic abuse. According to researchers at Dan...

Victims of domestic abuse. According to researchers at Dan Jones \& Associates, 1 in every 3 women has been a victim of domestic abuse (Domestic Violence: Incidence and Prevalence Study, Sept.-Dec., 2005 ). This probability was obtained from a survey of 1,000 adult women residing in Utah. Suppose we randomly sample 10 women and find that 6 have been abused. a. What is the probability of observing 5 or more abused women in a sample of 10 if the proportion $p$ of women who are victims of domestic abuse is really $p=1 / 3 ?$ b. Many experts on domestic violence believe that the proportion of women who are domestically abused is closer to $p=.10 .$ Calculate the probability of observing 5 or more abused women in a sample of 10 if $p=.10$. c. Why might your answers to parts a and b lead you to believe that $p=1 / 3$ ?

Victims of domestic abuse. According to researchers at Dan Jones \& Associates, 1 in every 3 women has been a victim of domestic abuse (Domestic Violence: Incidence and Prevalence Study, Sept.-Dec., 2005 ). This probability was obtained from a survey of 1,000 adult women residing in Utah. Suppose we randomly sample 10 women and find that 6 have been abused.
a. What is the probability of observing 5 or more abused women in a sample of 10 if the proportion $p$ of women who are victims of domestic abuse is really $p=1 / 3 ?$
b. Many experts on domestic violence believe that the proportion of women who are domestically abused is closer to $p=.10 .$ Calculate the probability of observing 5 or more abused women in a sample of 10 if $p=.10$.
c. Why might your answers to parts a and b lead you to believe that $p=1 / 3$ ?

Key Concepts

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent trials with the same probability of success for each trial. It is applicable when each trial results in a binary outcome (success or failure), and the total number of successes in the sample can be modeled using this distribution.

Binomial Probability Formula

This formula calculates the probability of obtaining exactly k successes in n independent Bernoulli trials. It is expressed as P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where p is the probability of success on a single trial. This formula is used to compute probabilities for both individual outcomes and cumulative probabilities over a range of outcomes.

Cumulative Probability

Cumulative probability involves summing the probabilities of individual events to compute the probability of at least (or at most) a certain number of successes. In the context of the problem, it is used to determine the probability of observing 5 or more abused women out of 10 by summing the probabilities for 5, 6, 7, 8, 9, and 10 successes.

Hypothesis Testing

Hypothesis testing in statistics is a method for making inferences about population parameters based on sample data. By comparing the observed sample result to what is expected under different assumptions about the true proportion of abused women, one can assess which hypothesis is more plausible. In this problem, comparing the probabilities under different assumed values of p illustrates the idea of evaluating which hypothesis the data supports.

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