Question

In January 2011, The Marist Poll published a report stating that 66% of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age. It was also reported that interviews were conducted on 1,018 American adults, and that the margin of error was 3% using a 95% confidence level.


(a) We are 95% confident that 63% to 69% of American adults in this sample think licensed drivers should be required to retake their road test once they turn 65. 
True or false, why?

(b) We are 95% confident that 63% to 69% of American adults think licensed drivers should be required to retake their road test once they turn 65. 
True or false, why?

(c) If we take many random samples of 1018 American adults, and for each sample, calculated the percentage who think licensed drivers should be required to retake their road test once they turn 65. 95% of those sample percentages will be between 63% and 69%. True or false, why?

(d) The margin of error at a 99% confidence level would be higher than 3%. 
True or false, why?

Answer 1

The confidence interval is the range of values in which we think the true value of a population parameter would lie. That is we are trying to estimate the value of the population parameter using the sample value.

Let p be the true value of the proportion of American adults in the population who  think licensed drivers should be required to retake their road test once they turn 65. Here we are trying to establish the value of p using a sample of size 1018. In this sample of American adults, 66% (or )  think licensed drivers should be required to retake their road test once they turn 65.

Now using this sample information we need to find the true value of p.

The margin of error was 3% using a 95% confidence level.

The upper limit of the confidence interval is

The lower limit of the confidence interval is

Hence we can say that we are 95% confident that the population proportion is between 0.63 and 0.69

a) We are 95% confident that 63% to 69% of American adults in this sample think licensed drivers should be required to retake their road test once they turn 65.

ans:False

In this sample we know that exactly 66% think think licensed drivers should be required to retake their road test once they turn 65. But the confidence interval is about the population proportion and it says that we are 95% confident that 63% to 69% of American adults in the population think licensed drivers should be required to retake their road test once they turn 65.

b) We are 95% confident that 63% to 69% of American adults think licensed drivers should be required to retake their road test once they turn 65

Ans: True

We are 95% confident that 63% to 69% of American adults ( in the population ) think licensed drivers should be required to retake their road test once they turn 65.

c) If we take many random samples of 1018 American adults, and for each sample, calculated the percentage who think licensed drivers should be required to retake their road test once they turn 65. 95% of those sample percentages will be between 63% and 69%.

Ans: True

Population parameter is a constant and it is not a random variable. We just are uncertain about its true value. But the sample proportion is a random value as each sample drawn from the population is different. Since we cannot study the whole population (impractical, expensive etc), we use samples to estimate the population parameter. We use the information from one sample and estimate the range in which the population parameter lies. The confidence interval is about long running proportion of samples which would have the population proportion within the confidence interval.

d) The margin of error at a 99% confidence level would be higher than 3%.

ans: True

The margin of error is calculated as

where is the critical value of z for a given level of significance .

Level of significance for 95% confidence interval is and

is

Or

using standard normal tables we get for 95% confidence interval.

Level of significance for 99% confidence interval is

is

Or

using standard normal tables we get for 99% confidence interval.

Hence the margin of error at 99% is

and it is higher than the margin of error at 95%