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Chapter 8: Problem 134

In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as likely or very likely. Use the plus four鈥� method to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely. Explain what this confidence interval means in the context of the problem.

Short Answer

Expert verified
The confidence interval for the proportion is between 8.59% and 33.71%.

Step by step solution

01

Understand the Plus Four Method

The 'plus four' confidence interval is a technique used to estimate a population proportion. It adjusts the sample by adding four imaginary observations: two success and two failures. This is particularly useful when the real sample size is small.
02

Adjust Sample Proportion

First, adjust the original data to account for the 'plus four' method. Originally, there are 9 successes out of 48. By adding two successes and two failures, we get 11 successes out of 52 total observations.
03

Calculate Sample Proportion

Calculate the adjusted sample proportion \( \hat{p} \) using the formula \( \hat{p} = \frac{x+2}{n+4} \), where \( x \) is the number of successes and \( n \) is the sample size. For this problem, \( \hat{p} = \frac{11}{52} \approx 0.2115 \).
04

Find the Z-score for 97% Confidence Level

The Z-score for a 97% confidence level is found using a statistical table or calculator and is approximately 2.17.
05

Compute the Standard Error

The standard error (SE) is calculated using the formula \( SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n+4} } \). Substituting the values, \( SE = \sqrt{ \frac{0.2115 \times (1 - 0.2115)}{52} } \approx 0.0584 \).
06

Calculate Confidence Interval

The confidence interval is \( \hat{p} \pm Z \times SE \). Substituting the known values, we find the interval to be \( 0.2115 \pm 2.17 \times 0.0584 \) which results in \( [0.0859, 0.3371] \).
07

Interpret the Confidence Interval

The confidence interval suggests that we are 97% confident that the true proportion of American adults who believe that a terrorist attack in their community is likely or very likely falls between 8.59% and 33.71%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
A confidence interval is a range of values used to estimate a population parameter, in this case, a proportion. It's built around a sample statistic and incorporates the level of uncertainty or randomness present when you draw samples from a population. When we talk about a 97% confidence interval, it means we expect that 97% of the intervals calculated from several samples would contain the true population parameter. In the context of our problem, the confidence interval tells us about the expected range for the proportion of American adults who think a terrorist attack in their community is likely or very likely. This interval was calculated using the 'plus four' method, resulting in a range of 0.0859 to 0.3371, indicating that the true proportion is likely between 8.59% and 33.71%. To summarize, confidence intervals provide a way to express the reliability of our estimates.
Sample Proportion
The sample proportion is a statistic that estimates the proportion of a particular characteristic in a population based on a sample. In this example, the sample proportion is calculated from those who believe a terrorist attack is likely or very likely. Without any adjustments, the sample proportion ( 1) would simply be the number of people who responded positively out of the total number surveyed. However, the 'plus four' method adjusts the sample to improve the accuracy of our confidence intervals, especially with smaller samples. After adjusting with two additional successes and two additional failures, we have 11 successes out of 52 observations, leading to a sample proportion of approximately 0.2115. This adjusted sample proportion gives us a more reliable starting point for estimating the actual population proportion.
Z-score
A Z-score is a statistical metric that quantifies how much a data point (or, in this case, a statistic) deviates from the mean. It is crucial when constructing confidence intervals, as it determines how far from the mean we should search to set up our interval. For a 97% confidence interval, the Z-score is approximately 2.17. This means that in 97% of cases, the true population parameter will fall within about 2.17 standard deviations of the sample mean. In our problem, applying the Z-score helps determine how wide our confidence interval will be, which is essential for estimating the range for our adjusted sample proportion.
Standard Error
Standard error (SE) is a measure of the variability of a sampling distribution. It helps us understand how much the sample proportion is expected to vary from the true population proportion.The formula for calculating the standard error of the sample proportion is: \[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}\]where \( \hat{p} \) is the sample proportion, and \( n+4 \) accounts for the 'plus four' adjustment.In our scenario, after substituting the values, we find that the standard error is approximately 0.0584. This value indicates how much we expect the sample proportion to differ from the true population proportion, helping us to construct the confidence interval by multiplying it with the Z-score.Therefore, the SE is an important part of determining how confident we can be in the interval we have calculated.

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Most popular questions from this chapter

Suppose that an accounting firm does a study to determine the time needed to complete one person's tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal. a. i. \(\overline{x}=\) _____ ii. \(\sigma=\) _____ iii. \(n=\) _____ b. In words, define the random variables \(X\) and \(\overline{X}\) . c. Which distribution should you use for this problem? Explain your choice. d. Construct a 90\(\%\) confidence interval for the population mean time to complete the tax forms. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. e. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make? f. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why? g. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?

Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months with a sample standard deviation of three months. Assume that the underlying population distribution is normal. a. i. \(\overline{x}=\) _____ ii. \(s_{x}=\) _____ iii. \(n=\) _____ iv. \(n-1=\) _____ b. Define the random variable \(X\) in words. c. Define the random variable \(X\) in words. d. Which distribution should you use for this problem? Explain your choice. e. Construct a 99\(\%\) confidence interval for the population mean length of time using training wheels. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. f. Why would the error bound change if the confidence level were lowered to 90\(\%\)

Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55. Assume the underlying population is normally distributed. What is the error bound? a. 0.87 b. 1.98 c. 0.99 d. 1.74

Forbes magazine published data on the best small firms in 2012 . These were firms that had been publicly traded for at least a year, have a stock price of at least \(\$ 5\) per share, and have reported annual revenue between \(\$ 5\) million and \(\$ 1\) billion. The Table 8.13 shows the ages of the corporate \(\mathrm{CEOs}\) for a random sample of these firms. $$\begin{array}{|c|c|c|c|c|}\hline 48 & {58} & {51} & {61} & {56} \\ \hline 59 & {74} & {63} & {53} & {50} \\ \hline 59 & {60} & {60} & {57} & {46} \\\ \hline 55 & {63} & {57} & {47} & {55} \\ \hline 57 & {43} & {61} & {62} & {49} \\ \hline 67 & {67} & {55} & {55} & {49} \\ \hline\end{array}$$ Use this sample data to construct a 90% confidence interval for the mean age of CEO鈥檚 for these top small firms. Use the Student's t-distribution.

Use the following information to answer the next five exercises: A poll of \(1,200\) voters asked what the most significant issue was in the upcoming election. Sixty-five percent answered the economy. We are interested in the population proportion of voters who feel the economy is the most important. Construct a 90\(\%\) confidence interval, and state the confidence interval and the error bound.
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