Frank Is Studying The Relationship Between Juvenile Crime
Frank Is Studying The Relationship Between Juvenile Crime Rates And Co
Frank is examining whether there is a significant difference in juvenile crime rates between neighborhoods with high collective efficacy (HCE) and low collective efficacy (LCE). Specifically, he hypothesizes that the mean juvenile crime rates differ significantly between these two groups. He collected data from 100 neighborhoods in each group, finding an average juvenile crime rate of 37.52 with a standard deviation of 8.20 for HCE neighborhoods, and an average of 52.44 with a standard deviation of 14.25 for LCE neighborhoods. The goal is to assess whether the observed difference is statistically significant using the six-step hypothesis testing process.
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Introduction
The relationship between community characteristics and juvenile crime rates has been a focal point in criminological and sociological research. Collective efficacy, which refers to a community's social cohesion and its ability to exert social control, has been seen as a protective factor against crime (Sampson, Raudenbush, & Earls, 1997). Frank's study aims to determine whether differences in collective efficacy levels are associated with changes in juvenile crime rates. This analysis employs a hypothesis testing framework to evaluate whether the observed difference in crime rates between high and low collective efficacy neighborhoods is statistically significant.
Step 1: State the hypotheses
The first step in hypothesis testing involves formulating the null (H0) and alternative (H1) hypotheses.
- Null hypothesis (H0): There is no significant difference in the mean juvenile crime rates between high collective efficacy (HCE) and low collective efficacy (LCE) neighborhoods. Mathematically, H0: μ_HCE = μ_LCE
- Alternative hypothesis (H1): There is a significant difference in the mean juvenile crime rates between the two groups. Mathematically, H1: μ_HCE ≠ μ_LCE
Step 2: Set the significance level (α)
The significance level, α, is commonly set at 0.05, representing a 5% risk of rejecting the null hypothesis when it is actually true. This threshold will determine whether the observed difference is statistically significant.
Step 3: Collect data and calculate the test statistic
From the problem, the summary statistics are:
- HCE: N₁ = 100, mean₁ = 37.52, SD₁ = 8.20
- LCE: N₂ = 100, mean₂ = 52.44, SD₂ = 14.25
Since the sample sizes are large and the population standard deviations are unknown but the sample sizes are equal, a two-sample z-test is appropriate. We calculate the standard error (SE) of the difference between means:
SE = √[(SD₁² / N₁) + (SD₂² / N₂)]
SE = √[(8.20² / 100) + (14.25² / 100)]
SE = √[(67.24 / 100) + (203.06 / 100)]
SE = √[0.6724 + 2.0306]
SE = √[2.703] ≈ 1.642
Next, compute the z-statistic:
z = (mean₁ - mean₂) / SE
z = (37.52 - 52.44) / 1.642
z = (-14.92) / 1.642 ≈ -9.09
Note: The large magnitude of z suggests a highly significant difference.
Step 4: Determine the critical value and make a decision
For a two-tailed test at α = 0.05, the critical z-values are ±1.96.
Since |z| ≈ 9.09 > 1.96, we reject H0.
Step 5: Interpret the results
Rejecting the null hypothesis indicates that there is a statistically significant difference in juvenile crime rates between high and low collective efficacy neighborhoods. The data support the hypothesis that collective efficacy levels are associated with juvenile crime rates.
Step 6: Draw conclusions
Based on the hypothesis testing, Frank’s hypothesis that neighborhoods with greater levels of collective efficacy (HCE) have different juvenile crime rates than LCE neighborhoods is supported. Specifically, the mean juvenile crime rate in HCE neighborhoods is significantly lower than in LCE neighborhoods, suggesting that higher collective efficacy may play a protective role against juvenile crime.
Discussion
The findings align with previous research indicating that collective efficacy contributes to reductions in crime through mechanisms such as informal social control and social cohesion (Sampson et al., 1997). The substantial difference in average crime rates underscores the importance of community-level interventions aimed at increasing social cohesion to mitigate juvenile delinquency. However, it is essential to consider other variables that may influence crime rates, such as socioeconomic factors, availability of resources, and law enforcement practices, which were not controlled in this analysis. Future research could incorporate multivariate analyses to better understand these complex relationships.
References
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