Self-InjuryAccording to the article “Self-injurious Behaviors in a College Population,” 17% of undergraduate or graduate students have had at least one incidence of self-injurious behavior. The researchers conducted a survey of 40 college students who reported a history of emotional abuse and found that 12 of them have had at least one incidence of self-injurious behavior. What do the results of this survey tell you about college students who report a history of emotional abuse?

Verified step by step guidance

1. Step 1: Identify the population proportion given in the problem. The article states that 17% (or 0.17) of all undergraduate or graduate students have had at least one incidence of self-injurious behavior. This is the baseline proportion for the general college student population.
2. Step 2: Note the sample data from the survey. Among 40 college students who reported a history of emotional abuse, 12 have had at least one incidence of self-injurious behavior. Calculate the sample proportion \(\hat{p}\) as \(\hat{p} = \frac{12}{40}\).
3. Step 3: Compare the sample proportion \(\hat{p}\) to the population proportion \(p = 0.17\). This comparison will help determine if students with a history of emotional abuse have a higher incidence of self-injurious behavior than the general student population.
4. Step 4: To assess whether the difference is statistically significant, consider performing a hypothesis test for a population proportion. Set up the null hypothesis \(H_0: p = 0.17\) (the proportion is the same as the general population) and the alternative hypothesis \(H_a: p > 0.17\) (the proportion is higher for students with emotional abuse history).
5. Step 5: Calculate the test statistic using the formula for a one-proportion z-test: \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\), where \(n=40\) is the sample size. Then, use the z-value to find the p-value and decide whether to reject the null hypothesis, which will tell you if the incidence is significantly higher in this group.

Key Concepts

Proportion and Percentage

Proportion represents a part of a whole expressed as a fraction or decimal, while percentage is that proportion multiplied by 100. In this question, understanding how to calculate and interpret the proportion of students with self-injurious behavior (e.g., 12 out of 40) is essential to compare it with the general 17% rate.

Comparative Analysis

Comparative analysis involves comparing two or more groups to identify differences or similarities. Here, comparing the 30% (12/40) incidence in students with emotional abuse history to the overall 17% helps assess whether emotional abuse is associated with higher self-injurious behavior.

Sampling and Generalizability

Sampling refers to selecting a subset of individuals from a population to make inferences. Understanding the sample size (40 students) and its representativeness is crucial to evaluate how confidently we can generalize the findings about emotional abuse and self-injury to the broader college population.