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Gun homicide in United States and Britain According to recent United Nations figures, the annual gun homicide rate is 62.4 per one million residents in the United States and 1.3 per one million residents in Britain. a. Show how to compare the proportion of residents of the two countries killed annually by guns using the difference of proportions. Show how the results differ according to whether the United States or Britain is identified as Group \(1 .\) b. Show how to compare the proportion of residents of the two countries using the relative risk. Show how the results differ according to whether the United States or Britain is identified as Group 1. c. When both proportions are very close to 0 , as in this example, which measure do you think is more useful for describing the strength of association? Why?

Step by step solution

01

Calculating the Difference of Proportions (US as Group 1)

Let's consider the United States as Group 1 and Britain as Group 2. We define the proportion of gun homicides for each group:- Proportion in the US, \( p_1 = \frac{62.4}{1,000,000} = 0.0000624 \)- Proportion in Britain, \( p_2 = \frac{1.3}{1,000,000} = 0.0000013 \)The difference in proportions, \( p_1 - p_2 \), is calculated as follows:\[p_1 - p_2 = 0.0000624 - 0.0000013 = 0.0000611\]

02

Calculating the Difference of Proportions (Britain as Group 1)

Now, consider Britain as Group 1 and the United States as Group 2. The proportions remain the same, but their order is reversed:- Proportion in Britain, \( p_1 = 0.0000013 \)- Proportion in the US, \( p_2 = 0.0000624 \)The difference in proportions, \( p_1 - p_2 \), is then:\[p_1 - p_2 = 0.0000013 - 0.0000624 = -0.0000611\]

03

Calculating the Relative Risk (US as Group 1)

Relative risk compares the likelihood of an event between two groups. Using the US as Group 1, the formula for relative risk (RR) is:\[RR = \frac{p_1}{p_2} = \frac{0.0000624}{0.0000013} \approx 48\]This means the gun homicide rate in the US is 48 times higher than in Britain when the US is considered Group 1.

04

Calculating the Relative Risk (Britain as Group 1)

Now let's calculate the relative risk using Britain as Group 1:\[RR = \frac{p_1}{p_2} = \frac{0.0000013}{0.0000624} \approx 0.0208\]Here, the relative risk indicates that the gun homicide rate in Britain is about 0.021 times that of the US, meaning significantly lower.

05

Choosing a Measure for Comparison

When both proportions are very close to 0, as in this example, relative risk is more useful for describing the strength of association. This is because the relative risk provides a multiplicative comparison that highlights how much more (or less) likely an event is to occur between groups, especially when the actual proportions are small values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Proportions

When we analyze gun homicide rates between the United States and Britain, one effective way is to look at the **difference of proportions**. The difference of proportions tells us how much higher or lower the rate of an event (like gun homicides) is between two groups.
To compute this difference, we first establish the proportions of gun homicides in each country. For the United States, the proportion \( p_1 = 0.0000624 \), and for Britain, \( p_2 = 0.0000013 \). The difference of proportions is simply \( p_1 - p_2 \).

• Difference when the US is Group 1: \( 0.0000624 - 0.0000013 = 0.0000611 \).
• Difference when Britain is Group 1: \( 0.0000013 - 0.0000624 = -0.0000611 \).

These calculations show that gun homicides are notably higher in the US compared to Britain, regardless of which country we assign as Group 1.

Relative Risk

Relative risk (RR) provides a ratio that describes how much more likely an event is to occur in one group compared to another. It is particularly helpful in comprehending the multiplicative difference in occurrence rates.
When viewed from the United States' perspective (as Group 1), the relative risk is calculated as:

• RR = \( \frac{0.0000624}{0.0000013} \approx 48 \). This indicates the gun homicide rate in the US is 48 times higher than Britain.

Alternatively, from Britain's perspective (as Group 1):

• RR = \( \frac{0.0000013}{0.0000624} \approx 0.0208 \). This means Britain's gun homicide rate is about 0.021 times that of the US, essentially highlighting that it is much lower.
  

Relative risk is a powerful statistic to see how starkly different risks are between these two countries.

Gun Homicide Rates

Gun homicide rates offer a specific look into the violent crime landscape of different countries. By evaluating these rates through statistical methods, we can uncover meaningful insights about safety and crime.
The annual rates are often standardized per million people to allow for straightforward comparison between countries of different population sizes. As noted:

• In the United States: 62.4 gun homicides per million.
• In Britain: 1.3 gun homicides per million.
  

Such dramatic differences highlight differing national circumstances, from gun laws to social factors, that contribute to these statistics.

Statistical Comparison Methods

Statistical comparison methods, like the difference of proportions and relative risk, are essential tools in the analysis of social science data. They offer perspectives on how one rate or probability stacks up against another.
When proportions are small, as with gun homicide rates, relative risk becomes more informative than the difference in proportions. It highlights the magnitude of difference in a more intuitive way.

• Difference of proportions gives an absolute comparison, useful in understanding exact differences.
• Relative risk shows multiplication, helping us understand relative changes in rare events.

Using a balance of these methods ensures solid, reliable insights into statistics like gun homicide rates.