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Chapter 9: Problem 70

The Associated Press (December 16,1991 ) reported that in a random sample of 507 people, only 142 correctly described the Bill of Rights as the first 10 amendments to the U.S. Constitution. Calculate a \(95 \%\) confidence interval for the proportion of the entire population that could give a correct description.

Short Answer

Expert verified
The 95% confidence interval for the proportion of the entire population that could correctly describe the Bill of Rights is estimated to be between 0.23642 and 0.32374.

Step by step solution

01

Calculate the Sample Proportion

The sample proportion \(p\) is the ratio of successful trials to the total number of trials. In this case, it will be the number of people who correctly described the Bill of Rights divided by the total sample size. So, \(p = 142 / 507 = 0.28008\).
02

Standard Error of the Sample Proportion

Standard error, an indication of how much our sample proportion is likely to vary from our actual population proportion, can be calculated by the formula \(SE = \sqrt{p(1-p)/n}\), where \(p\) is the sample proportion we computed in Step 1 and \(n\) is the total size of our sample. So, \(SE = \sqrt{0.28008 * (1 - 0.28008) / 507} = 0.02224\).
03

Calculate the 95% Confidence Interval

For a 95% confidence interval, the Z-value (which comes from the standard normal distribution) is approximately 1.96. The formula to calculate the interval is \(p \pm Z*SE\). So the interval will be (0.28008 - 1.96 * 0.02224, 0.28008 + 1.96 * 0.02224) = (0.23642, 0.32374).

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Most popular questions from this chapter

Each person in a random sample of 20 students at a particular university was asked whether he or she is registered to vote. The responses \((\mathrm{R}=\) registered, \(\mathrm{N}=\) not registered) are given here: $$\begin{array}{rrrrr} R & R & N & R & N & N & R & R & R & N & R & R & R & R & R & N & R & R & R & N \end{array}$$ Use these data to estimate \(p\), the proportion of all students at the university who are registered to vote.

The formula used to compute a confidence interval for the mean of a normal population when \(n\) is small is $$\bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}}$$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=17\) b. \(90 \%\) confidence, \(n=12\) c. \(99 \%\) confidence, \(n=24\) d. \(90 \%\) confidence, \(n=25\) e. \(90 \%\) confidence, \(n=13\) f. \(95 \%\) confidence, \(n=10\)

For each of the following choices, explain which would result in a wider large-sample confidence interval for \(p\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

The article "Hospitals Dispute Medtronic Data on Wires" (The Wall Street journal. February 4,2010 ) describes several studies of the failure rate of defibrillators used in the treatment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures were experienced within the first 2 years by 18 of 89 patients under 50 years old and 13 of 362 patients age 50 and older who received a particular type of defibrillator. Assume it is reasonable to regard these two samples as representative of patients in the two age groups who receive this type of defibrillator. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator. b. Construct and interpret a \(99 \%\) confidence interval for the proportion of patients age 50 and older who experience a failure within the first 2 years after receiving this type of defibrillator. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator to within \(.03\) with \(95 \%\) confidence. How large a sample should be used? Use the results of the study as a preliminary estimate of the population proportion.

In spite of the potential safety hazards, some people would like to have an Internet connection in their car. A preliminary survey of adult Americans has estimated this proportion to be somewhere around. 30 (USA Today. May 1,2009 ). a. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within \(.02\) with \(95 \%\) confidence. b. The formula for determining sample size given in this section corresponds to a confidence level of \(95 \%\). How would you modify this formula if a \(99 \%\) confidence level was desired? c. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within \(.02\) with \(99 \%\) confidence.
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