Question

In 2013, the Pew Research Foundation reported that "42% of U.S. adults report that they live with one or more chronic conditions". However, this value was based on a sample, so it may not be a perfect estimate for the population parameter of interest on its own. The study reported a standard error of about 2.2%, and a normal model may reasonably be used in this setting.

1. Create a 95% confidence interval for the proportion of U.S. adults who live with one or more chronic conditions. Round to four decimal places.

_____ % to _____ % ???

2. Identify each of the following statements as true or false. Provide an explanation to justify each of your answers.

? True False  a. We can say with certainty that the confidence interval from part 1 contains the true percentage of U.S. adults who suffer from a chronic illness.

? True False  b. If we repeated this study 1,000 times and constructed a 95% confidence interval for each study, then approximately 950 of those confidence intervals would contain the true fraction of U.S. adults who suffer from chronic illnesses.

? True False  c. The poll provides statistically significant evidence (at the α=0.05α=0.05 level) that the percentage of U.S. adults who suffer from chronic illnesses is below 50%.We can say with certainty that the confidence interval from part 1 contains the true percentage of U.S. adults who suffer from a chronic illness.

? True False  d. Since the standard error is 2.2%, only 2.2% of people in the study communicated uncertainty about their answer.

Answer 1

1)

Here we have

P^=0.42 , SE=0.022

At 95% confidence level

=1-.95=0.05

Now Z_crit= Z_0.05/2= 1.96

Now 95%CI

P^ +/- Zcrit*SE

0.42 +/- 1.96*0.022

=0.42 +/- 0.0431

=0.3769 ,0.4631

i.e. 37.69% to 46.31%

2)

True False  a) We can say with certainty that the confidence interval from part 1 contains the true percentage of U.S. adults who suffer from a chronic illness.

This statement is True

We are 95% confident that true proportion lies in the above interval.

True False  b) If we repeated this study 1,000 times and constructed a 95% confidence interval for each study, then approximately 950 of those confidence intervals would contain the true fraction of U.S. adults who suffer from chronic illnesses.

This statement is false

The above interval just says about the how confident we are about the true proportion interval it does not say anything about how many times we can be succeed .

True False  c. The poll provides statistically significant evidence (at the α=0.05α=0.05 level) that the percentage of U.S. adults who suffer from chronic illnesses is below 50%.We can say with certainty that the confidence interval from part 1 contains the true percentage of U.S. adults who suffer from a chronic illness.

This statement is true.

As we can see both the upper and lower limits are less than 50%

True False  d. Since the standard error is 2.2%, only 2.2% of people in the study communicated uncertainty about their answer.

This statement is False . It standard error does not say about the uncertainity about the answer, It is for sample data only.