Help please

Question

In October 2004, the Gallup Organization surveyed 1134 American adults and found that 431 owned a gun. In February 1999, the Gallup Organization had 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? Use the P-value method. Construct the 99% confidence interval for the true difference between two population proportions (you can use calculator to build the interval, 2PropZ Int). Do the two methods result in the same conclusion?

Answer 1

No, the headline is not accurate. There is no statistically significant evidence to support the claim that the percentage of American gun owners is on the rise. Both the hypothesis test and the confidence interval lead to this same conclusion.

1. Hypothesis Test (P-Value Method)

We want to test if the proportion of gun owners in October 2004 (\(p_{1}\)) is greater than in February 1999 (\(p_{2}\)).

1. Hypotheses:
2. \(H_0: p_1 \le p_2\) (The proportion has not increased)
3. \(H_1: p_1 > p_2\) (The proportion is on the rise)
4. Sample Proportions:
5. \(\hat{p}_1 = \frac{431}{1134} \approx 0.3801\) (38.01%)
6. \(\hat{p}_2 = \frac{408}{1134} \approx 0.3598\) (35.98%)
7. Test Statistic (\(Z\)): Using a pooled sample proportion (\(\bar{p} = \frac{431+408}{1134+1134} \approx 0.3699\)), the test statistic is \(Z = 1.00\).
8. P-value: The probability of getting a \(Z\)-statistic this extreme or greater by chance is \(0.1586\).

Conclusion: Using any standard significance level (like \(\alpha = 0.01\) or \(0.05\)), the P-value (\(0.1586\)) is greater than \(\alpha \). We fail to reject the null hypothesis (\(H_{0}\)). The slight increase in the sample data is not statistically significant.

2. 99% Confidence Interval (2PropZInt)

Using a standard graphing calculator or mathematical software to calculate the 99% confidence interval for the true difference between the two proportions (\(p_1 - p_2\)):

1. 99% Confidence Interval: \((-0.0319, 0.0725)\) or \((-3.19\%, 7.25\%)\)

Conclusion: Because the interval contains 0, it is entirely plausible that there is zero difference between the two population proportions. It is even possible that the true proportion decreased (since the interval includes negative numbers). Therefore, we cannot confidently claim the percentage is rising.

3. Comparison of Methods

Yes, both methods result in the exact same conclusion.

1. The Hypothesis Test tells us that the observed difference is likely due to random sampling variation (\(P > \alpha\)).
2. The Confidence Interval confirms this by showing that a difference of \(0\) is a highly plausible value within our 99% confidence limits.

The headline is misleading because it states a statistical certainty ("on the rise") based on a minor fluctuation that can be explained by pure chance.