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Chapter 11: Problem 21

In October \(2004,\) the Gallup Organization surveyed 1134 American adults and found that 431 owned a gun. In February \(1999,\) the Gallup Organization had surveyed 1134 American adults and found that 408 owned a gun. Suppose that a newspaper article has a headline that reads, "Percentage of American Gun Owners on the Rise." Is this an accurate headline? Why?

Short Answer

Expert verified
Yes, the headline is accurate because the percentage of gun owners increased from 36.0% in 1999 to 38.0% in 2004.

Step by step solution

01

- Calculate the Percentage of Gun Owners in 2004

Determine the percentage of American adults who owned a gun in October 2004. This is done by dividing the number of gun owners by the total surveyed population and then multiplying by 100.\[\text{Percentage in 2004} = \left( \frac{431}{1134} \right) \times 100\]Let's calculate that:\[\frac{431}{1134} \approx 0.38\] so,\[\text{Percentage in 2004} \approx 38.0\%\]
02

- Calculate the Percentage of Gun Owners in 1999

Determine the percentage of American adults who owned a gun in February 1999 using a similar calculation:\[\text{Percentage in 1999} = \left( \frac{408}{1134} \right) \times 100\]Let's calculate that:\[\frac{408}{1134} \approx 0.36\] so,\[\text{Percentage in 1999} \approx 36.0\%\]
03

- Compare the Percentages

Compare the percentages calculated for 2004 and 1999 to determine if there was a rise in the percentage of gun owners:\[38.0\% \text{ (2004)} > 36.0\% \text{ (1999)}\]Since the percentage for 2004 is indeed higher than that for 1999, there was an increase in the percentage of American gun owners.
04

- Evaluate the Accuracy of the Headline

Determine if the headline 'Percentage of American Gun Owners on the Rise' is accurate based on the comparison. Given that there was an increase from 36.0% in 1999 to 38.0% in 2004, the headline accurately reflects the data.

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Key Concepts

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

Percentage Calculation
Percentage calculation is a fundamental concept in statistics. It answers the question, 'What part of the whole is this number?'. In this example, we used percentage to determine the proportion of American adults who owned a gun in specific years. To find the percentage, follow these steps:
  • Divide the number of interest (e.g., gun owners) by the total number surveyed.
  • Multiply the result by 100 to get the percentage.
Here's the formula we used: \( \text{Percentage} = \left( \frac{\text{Number of Gun Owners}}{\text{Total Surveyed Population}} \right) \times 100 \). For example, in 2004, the formula looks like this: \( \left( \frac{431}{1134} \right) \times 100 \), which gives approximately 38%. Always remember to multiply by 100 to convert a fraction to a percentage.
Data Comparison
Data comparison involves evaluating two or more datasets to determine similarities, differences, or trends. In our example, we compared the percentages of gun owners in 1999 and 2004. The steps are simple:
  • Calculate the percentage for each dataset (as shown in the previous section).
  • Directly compare the percentages.
In 1999, 36% of those surveyed owned a gun, while in 2004, 38% did. The increase from 36% to 38% shows a rise in gun ownership. This comparison helped us verify whether the newspaper headline 'Percentage of American Gun Owners on the Rise' was accurate. Always ensure that you compare similar data to draw meaningful and valid conclusions.
Statistical Accuracy
Statistical accuracy ensures the reliability and correctness of conclusions drawn from data. In practice, you must consider several aspects:
  • The sample size should be large enough to represent the entire population.
  • Verify calculations to ensure they are done correctly.
  • Context matters; a small percentage change can be statistically insignificant.
In our example, both surveys had the same sample size of 1134 adults, making it easier to trust the comparison. The increase from 36% to 38%, although minor, is statistically correct as per the available data. But remember, accuracy does not necessarily mean that the headline captures the complexity of societal trends convincingly. Always look carefully at the data and see whether the change is significant enough to support the claims being made.

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