Respuesta :
Answer:
a. False. The sample is 46%. We are 95% that the proportion of the population is between 43% and 49%.
b. True, this is the correct interpretation of a confidence interval of proportions.
c. False. The confidence interval does not say anything about the proportion of samples, just about the population.
d. False. As the confidence level decreases, so does z, which means that the margin of error decreases.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
So the confidence interval can be written as:
[tex]\pi \pm M[/tex]
Interpreting a confidence interval:
x% confidence interval between (a,b) means that we are 95% sure that the true proportion of the population is between a and b.
In this problem, we have that:
95% confidence interval between (43%, 49%).
a. We are 95% confident that between 43% and 49% of Americans in this sample support the decision of the U.S. Supreme Court on the 2010 healthcare law.
False. The sample is 46%. We are 95% that the proportion of the population is between 43% and 49%.
b. We are 95% confident that between 43% and 49% of Americans support the decision of the U.S. Supreme Court on the 2010 healthcare law.
True, this is the correct interpretation of a confidence interval of proportions.
c. If we considered many random samples of 1,012 Americans, and we calculated the sample proportions of those who support the decision of the U.S. Supreme Court, 95% of those sample proportions will be between 43% and 49%.
False. The confidence interval does not say anything about the proportion of samples, just about the population.
d. The margin of error at a 90% confidence level would be higher than 3%.
False
As the confidence level decreases, so does z, which means that the margin of error decreases.
