The 112th Congress of the United States of America has 535 members, of which 87 are women. An alien lands near the US Capital and treats members of Congress as a random sample of the human race. He reports to his superiors the 95% confidence interval for the proportion of the human race that is female.
A) Calculate the confidence interval that the alien reports?
***Is it 13% to 19%????
B) What is wrong with the alien's approach to estimating the proportion of the human race that is female?
***It is not a randomized sample (although appears to be). Congress does not accurately reflect the overall population of women as Congress does not have many female members for one, and the sample does not include women from around the world.
Am I right? Thanks!
A) To calculate the confidence interval for the proportion of the human race that is female based on the given information, you can use the formula for a proportion confidence interval:
CI = p̂ ± z * √((p̂(1 - p̂))/n)
Where:
CI is the confidence interval
p̂ is the proportion of females in the sample (87/535)
z is the z-score corresponding to the desired confidence level (e.g., for 95% confidence, z = 1.96)
n is the sample size (535)
Plugging in the values, we have:
CI = (87/535) ± 1.96 * √(((87/535)(1 - (87/535))) / 535)
Calculating this expression will give you the confidence interval reported by the alien.
B) You are correct, there are issues with the alien's approach to estimating the proportion of the human race that is female based on members of Congress. The main issue is that Congress is not a representative and randomized sample of the human race. Congress members are elected from specific regions and may not accurately reflect the overall population in terms of demographics, including gender.
Furthermore, the sample only includes members of the United States Congress, which does not include women from around the world. To estimate the proportion of the human race that is female, a truly representative and randomized sample from a diverse range of countries and regions would be needed.
The alien's report:
The population of Earth is adequately large, we assume this is an SRS, and with p-hat being approximately .16, n*p-hat and n*(1 - p-hat) are both greater than 10. Therefore we can use the normal curve.
We will use the confidence interval LaTeX: p-hat\:\pm z^{\cdot}\sqrt{\frac{p-hat\left(1-p-hat\right)}{n}} p − h a t ± z ⋅ p − h a t ( 1 − p − h a t ) n where p-hat = .1626, n = 535, and LaTeX: z^{\cdot} z ⋅ is 1.96.
Our interval, then, is (.1314, .1939).
We are 95% confident that the true proportion of Earth people who are women lies between 13% and 19%.