1) The 90% confidence interval shows that between 0.444 and 0.581 proportion of people support new gun regulations.
2) Similarly, the 95% confidence interval shows that between 0.444 and 0.581 proportion of people support new gun regulations.
How the confidence intervals are constructed:
We use the confidence interval formula as follows:
CI = p ± z*(sqrt(p*(1-p)/n))
Where:
The confidence interval = CI
The sample proportion = p
The sample size = n
The z-score corresponding to the desired confidence level = z.
1) For a 90% confidence level, the z-score is 1.645. Given that 82 people interviewed out of the 160 interviews supported new gun regulations, we can calculate the sample proportion as follows:
p = 82/160
= 0.5125
Substituting the value of p, n = 160, and z = 1.645 in the confidence interval formula, we can calculate the confidence interval as follows:
CI = 0.5125 ± 1.645*(sqrt(0.5125*(1-0.5125)/160))
≈ (0.444, 0.581)
2) At 95% confidence interval, the sample proportion is:
p = 82/160
= 0.5125
Substituting the value of p, n = 160, and z = 1.96 in the confidence interval formula, we can calculate the confidence interval as follows:
CI = 0.5125 ± 1.96*(sqrt(0.5125*(1-0.5125)/160))
≈ (0.444, 0.581)
Therefore, we can be both 90% and 95% confident that the true proportion of people who support new gun regulations falls between 0.444 and 0.581.