Question

In researching the proportion of Americans who own firearms, you find a BJS survey of 6,500 people which shows that 2,315 own at least one gun. You think the percentage of gun owners in Maryland is less than that, given the relatively strict firearm laws, so you take a sample of 200 Maryland residents and find that 57 own a gun. Conduct a hypothesis test to determine if the proportion of gun owners in Maryland is lower than in the general population. Use an alpha level of .04.

Answer 1

Answer:

$z=\frac{0.285-0.356}{\sqrt{0.354(1-0.354)(\frac{1}{6500}+\frac{1}{200})}}=-2.068$    

Now we can calculate the p value using the alternative hypothesis with this probability:

$p_v =P(Z<-2.068)=0.0193$    

Since the p value is lower than the significance level of 0.04 and we have enough evidence to reject the null hypothesis and we can conclude that the proportion of gun owners in Maryland is significantly lower than in the general population.

Step-by-step explanation:

Information provided

$X_{1}=2315$ represent the number of Americans who own at least one gun

$X_{2}=57$ represent the number of people in Maryland who own at least one gun

$n_{1}=6500$ sample of Americans

$n_{2}=200$ sample from Maryland

$p_{1}=\frac{2315}{6500}=0.356$ represent the proportion estimated of Americans who own at least one gun

$p_{2}=\frac{57}{200}=0.285$ represent the proportion estimated of people in Maryland who own at least one gun

$\hat p$ represent the pooled estimate of p

z would represent the statistic

$p_v$ represent the value for the test

$\alpha=0.04$ significance level given  

Hypothesis to verify

We want to verify if the proportion of gun owners in Maryland is lower than in the general population, the system of hypothesis would be:    

Null hypothesis:$p_{2} \geq p_{1}$    

Alternative hypothesis:$p_{2} < p_{1}$    

The statistic for this case is given by:

$z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$   (1)  

Where $\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{2315+57}{6500+200}=0.354$  

Replacing the info given we got:

$z=\frac{0.285-0.356}{\sqrt{0.354(1-0.354)(\frac{1}{6500}+\frac{1}{200})}}=-2.068$    

Now we can calculate the p value using the alternative hypothesis with this probability:

$p_v =P(Z<-2.068)=0.0193$    

Since the p value is lower than the significance level of 0.04 and we have enough evidence to reject the null hypothesis and we can conclude that the proportion of gun owners in Maryland is significantly lower than in the general population.