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Chapter 8: Problem 78

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. What would happen to the confidence interval if the level of confidence were 95\(\% ?\)

Short Answer

Expert verified
The confidence interval becomes wider at 95\(\%\) confidence.

Step by step solution

01

Understand the Given Information

The problem states that 65% of 1,200 voters believe the economy is the most important issue. This is our sample proportion \(\hat{p} = 0.65\). The population proportion \(p\) is unknown and what we're estimating using confidence intervals.
02

Identify the Current Confidence Level

Currently, a specific confidence level hasn't been mentioned for the 65% statistic. However, the question asks what happens if we use a 95\(\%\) confidence level.
03

Prepare Confidence Interval Formula

The formula for a confidence interval for a proportion is given by \(\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(z^*\) is the z-score corresponding to the desired confidence level, \(\hat{p}\) is the sample proportion, and \(n\) is the sample size.
04

Calculate Margin of Error at 95\(\%\) Confidence Level

For a 95\(\%\) confidence level, the \(z^*\) value is approximately 1.96. The margin of error (ME) is calculated as \(ME = 1.96 \cdot \sqrt{\frac{0.65(1-0.65)}{1200}}\). Substitute the values to compute the ME.
05

Understand the Effect on Confidence Interval

Increasing the confidence level to 95\(\%\) means the confidence interval will likely become wider. This is because a higher confidence level requires capturing more possible values, thus increasing the margin of error.

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Key Concepts

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

Sample Proportion
In statistics, the sample proportion refers to the portion of a sample that expresses a particular characteristic. In the context of the given exercise, we are examining the proportion of the population that considers the economy the most significant issue in the upcoming election. Here, the sample proportion is denoted by \(\hat{p}\) and is calculated from the sample data gathered. In our scenario, the sample proportion \(\hat{p}\) is calculated as \(0.65\) since 65% of the 1,200 voters surveyed prioritized the economy. This value, 0.65, serves as our best estimate from the sample of the "true" proportion in the broader population.
Population Proportion
The population proportion represents the proportion of all voters that believe the economy is the most important issue in the election. This is what statisticians ultimately wish to estimate, though typically, it cannot be known exactly. This unknown population proportion is denoted as \(p\). While we use the sample proportion \(\hat{p}\) as an estimate of the population proportion, confidence intervals help us to express the reliability of this estimate and provide a range of values in which we believe the true population proportion lies, with a specified level of confidence.
Margin of Error
The margin of error is vital in determining how precise our estimate of the population proportion is likely to be. It is the range within which we expect the true population proportion to fall and depends on both the size of the sample and the inherent variability in the data.
  • In our problem, the margin of error for a 95% confidence level is calculated using the formula for the confidence interval for a proportion: \(\text{ME} = z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\).
  • For a confidence level of 95%, the z-score (\(z^*\)) is approximately 1.96, which reflects how many standard deviations we cover on either side of the sample proportion to achieve our confidence level.
Therefore, calculating the margin of error informs us how much the sample proportion may differ from the population proportion in either direction. This will widen or narrow the confidence interval, impacting the certainty we have in our estimate.
Z-Score
The z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed as the number of standard deviations a data point is from the mean. In the context of confidence intervals, the z-score \(z^*\) specifically refers to the critical value that corresponds to a desired confidence level.
  • Common confidence levels include 90%, 95%, and 99%, with the corresponding z-scores commonly being 1.645, 1.96, and 2.576 respectively.
  • A 95% confidence level, for example, makes use of a z-score of 1.96 because we want to account for 95% of the data around our sample proportion, meaning approximately 2.5% is placed in each tail of the normal distribution.
The choice of z-score affects the width of our confidence interval and, by extension, the margin of error, determining how confident we can be in our estimations.

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Most popular questions from this chapter

Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours. Define the random variables \(X\) and \(\overline{X}\) in words.

Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. a. Define the random variables \(X\) and \(P^{\prime}\) in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 97\(\%\) confidence interval for the population proportion of people over 50 who ran and died in the same eight-year period. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. d. Explain what a 97\% confidence interval means for this study.

Use the following information to answer the next five exercises. Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population of households where women make the majority of the purchasing decisions. List two difficulties the company might have in obtaining random results, if this survey were done by email.

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.

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?
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