Question

Gun ownership Concerned about recent incidence of gun violence, a public interest group conducts a poll of 850 randomly selected American adults and finds that 44% of those surveyed have a gun in their home.

a. Construct and interpret a 95% confidence interval for the proportion of all American adults who have a gun in their home.
b. Explain what is meant by 95% confidence in this context.

Answer 1

a.

The 95% confidence interval for the proportion of all American adults who have a gun in their home is (0.4066, 0.4734). This means that we are 95% sure that the true population proportions of American adults who have a gun in their home is between these two values.

b.

The confidence level is the probability of the confidence interval containing the population proportion.

Explanation:

Question a:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

Poll of 850 randomly selected American adults and finds that 44% of those surveyed have a gun in their home.

This means that
`n = 850, \pi = 0.44`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.44 - 1.96\sqrt{(0.44*0.56)/(850)} = 0.4066`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.44 + 1.96\sqrt{(0.44*0.56)/(850)} = 0.4734`

The 95% confidence interval for the proportion of all American adults who have a gun in their home is (0.4066, 0.4734). This means that we are 95% sure that the true population proportions of American adults who have a gun in their home is between these two values.

b. Explain what is meant by 95% confidence in this context.

The confidence level is the probability of the confidence interval containing the population proportion.