A USU Today survey found that of the gun owners surveyed 275 favor stricter gun laws. The survey
Involved 500 gun owners. Test the claim that a majority (more than 50%) of gun owners favor stricter gun laws. Use a .05 significance level.
To test the claim that a majority of gun owners favor stricter gun laws, we can use a hypothesis test.
Let's state the null and alternative hypotheses:
- Null Hypothesis (H0): The proportion of gun owners in favor of stricter gun laws is equal to or less than 50%.
- Alternative Hypothesis (H1): The proportion of gun owners in favor of stricter gun laws is greater than 50%.
Next, we need to determine the critical value for the test. Since the significance level is 0.05, we need to find the critical z-value at alpha = 0.05 for a one-tailed test.
Using a standard normal distribution table, the critical z-value is approximately 1.645.
Now, we can calculate the test statistic, which is the z-score for the sample proportion. The formula for the z-score is:
z = (p - P) / √(P(1 - P) / n)
where:
- p is the sample proportion in favor of stricter gun laws (275/500 = 0.55)
- P is the hypothesized proportion (0.50)
- n is the sample size (500)
Using these values, we can calculate the test statistic:
z = (0.55 - 0.50) / √(0.50 * (1 - 0.50) / 500)
z = 0.05 / √(0.25 / 500)
z = 0.05 / √0.0005
z ≈ 3.16
Finally, we compare the test statistic (z = 3.16) to the critical value (z = 1.645) to make a decision.
Since the test statistic is greater than the critical value, we reject the null hypothesis. This means we have evidence to support the claim that a majority of gun owners favor stricter gun laws.
To test the claim that a majority of gun owners favor stricter gun laws, we can use a hypothesis test.
Hypotheses:
Null hypothesis (H0): The proportion of gun owners favoring stricter gun laws is equal to or less than 50%.
Alternative hypothesis (Ha): The proportion of gun owners favoring stricter gun laws is greater than 50%.
Significance level: .05 (or 5%)
To test this, we can use a one-sample proportion test. Since we have sample data, we can calculate the test statistic and compare it to the critical value.
Let's calculate the test statistic using the formula:
z = (p̂ - p0) / sqrt(p0(1 - p0)/n)
Where:
p̂ is the sample proportion (275/500),
p0 is the hypothesized proportion (0.5 or 50%),
n is the sample size (500).
Calculating the test statistic:
p̂ = 275/500 = 0.55
p0 = 0.5
n = 500
z = (0.55 - 0.5) / sqrt(0.5 * (1 - 0.5) / 500)
z = 0.05 / sqrt(0.25 / 500)
z = 0.05 / sqrt(0.0005)
z ≈ 14.14
Next, we need to find the critical value for a one-tailed z-test with a significance level of 0.05. We can look up this value in the z-table or use statistical software.
Since the alternative hypothesis is that the proportion is greater than 50%, we need to find the z-score cutoff that corresponds to the area under the normal curve to the left of 0.05 (95% confidence interval).
Looking up the critical value in the z-table, we find that the z-score corresponding to a cumulative probability of 0.95 is approximately 1.645.
Comparison:
Since the calculated test statistic (14.14) is greater than the critical value (1.645), we reject the null hypothesis. This indicates that there is strong evidence to support the alternative hypothesis that a majority of gun owners (more than 50%) favor stricter gun laws.
Therefore, based on the sample data and the hypothesis test, we can conclude that a significant majority of gun owners favor stricter gun laws.