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Chapter 9: Problem 23

A Rasmussen Reports national survey of 1000 adult Americans found that \(18 \%\) dreaded Valentine's Day. The margin of error for the survey was 4.5 percentage points with \(95 \%\) confidence. Explain what this means.

Short Answer

Expert verified
Between 13.5% and 22.5% of adult Americans dread Valentine's Day with 95% confidence.

Step by step solution

01

Understand the Survey Result

The survey found that 18% of 1000 adult Americans surveyed dreaded Valentine's Day.
02

Define the Margin of Error

The margin of error is a measure of the potential error in the survey result. In this case, the margin of error is 4.5 percentage points.
03

Define the Confidence Level

A 95% confidence level means that if the survey were repeated many times, 95% of the results would fall within the margin of error.
04

Calculate the Confidence Interval

To find the confidence interval, add and subtract the margin of error from the survey result: Lower Bound: 18% - 4.5% = 13.5% Upper Bound: 18% + 4.5% = 22.5%
05

Interpret the Confidence Interval

With 95% confidence, it can be said that between 13.5% and 22.5% of all adult Americans would say they dread Valentine's Day.

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Key Concepts

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

Survey Result Interpretation
When analyzing survey results, it鈥檚 crucial to understand what the numbers truly represent. In our example, the survey reveals that 18% of 1000 adults dread Valentine's Day. But why is this significant? This percentage tells us about the sentiments of a specific group of adults. However, it鈥檚 important to recognize that this is not an absolute figure. It's an estimate derived from the sample to represent the wider population's opinion. Always remember: survey results aim to reflect the population but are never exact.
Margin of Error
The margin of error adds more depth to understanding survey results. It鈥檚 a measure that provides an estimate of the potential error in the reported percentage. For instance, if the survey states an 18% result with a margin of error of 4.5 percentage points, it means the actual percentage could fall anywhere within 4.5 points above or below 18%. So the real value lies between 13.5% and 22.5%. This range shows the survey's precision and helps us understand the extent to which the results might vary.
Confidence Level
Confidence level is another critical aspect of survey results. A 95% confidence level means there is a 95% chance that the true population parameter - in this case, the percentage of adults who dread Valentine's Day - lies within the calculated margin of error. If we repeated the survey many times, 95% of these repetitions would yield results falling within the 13.5% to 22.5% interval. It鈥檚 a way of expressing how sure we can be about the survey result's reliability.
Statistical Analysis
Statistical analysis is the backbone of extracting meaningful insights from data. From our survey, calculating the confidence interval helps summarize the likely opinion of the broader population. Here鈥檚 a simple rundown:
  • Identify the survey result (18%).
  • Apply the margin of error (4.5%).
From this, calculate the confidence interval:
  • Lower Bound: 18% - 4.5% = 13.5%
  • Upper Bound: 18% + 4.5% = 22.5%
This interval, combined with the confidence level, tells us that if we were to repeat the survey, 95 times out of 100, the percentage of those who dread Valentine鈥檚 Day would be between 13.5% and 22.5%. This process underlines the importance of statistical principles in interpreting and validating survey data.

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Most popular questions from this chapter

Explain why quadrupling the sample size causes the margin of error to be cut in half.

In response to the Great Depression, Franklin D. Roosevelt enacted many New Deal policies. One such policy was the enactment of the National Recovery Administration (NRA), which required business to agree to wages and prices within their particular industry. The thought was that this would encourage higher wages among the working class, thereby spurring consumption. In a Gallup survey conducted in 1933 of 2025 adult Americans, \(55 \%\) thought that wages paid to workers in industry were too low. The margin of error was 3 percentage points with \(95 \%\) confidence. Which of the following represents a reasonable interpretation of the survey results? For those that are not reasonable, explain the flaw. (a) We are \(95 \%\) confident \(55 \%\) of adult Americans during the Great Depression felt wages paid to workers in industry were too low. (b) We are \(92 \%\) to \(98 \%\) confident \(55 \%\) of adult Americans during the Great Depression felt wages paid to workers in industry were too low. (c) We are \(95 \%\) confident the proportion of adult Americans during the Great Depression who believed wages paid to workers in industry were too low was between 0.52 and 0.58 . (d) In \(95 \%\) of samples of adult Americans during the Great Depression, the proportion who believed wages paid to workers in industry were too low is between 0.52 and 0.58 ,

Travelers pay taxes for flying, car rentals, and hotels. The following data represent the total travel tax for a 3-day business trip in eight randomly selected cities. Note: Chicago has the highest travel taxes in the country at 101.27 dollar. In Problem 32 from Section \(9.2,\) it was verified that the data are normally distributed and that \(s=12.324\) dollars. Construct and interpret a \(90 \%\) confidence interval for the standard deviation travel tax for a 3 -day business trip. $$ \begin{array}{llll} \hline 67.81 & 78.69 & 68.99 & 84.36 \\ \hline 80.24 & 86.14 & 101.27 & 99.29 \\ \hline \end{array} $$

In March 2014, Harris Interactive conducted a poll of a random sample of 2234 adult Americans 18 years of age or older and asked, "Which is more annoying to you, tailgaters or slow drivers who stay in the passing lane?" Among those surveyed, 1184 were more annoyed by tailgaters. (a) Explain why the variable of interest is qualitative with two possible outcomes. What are the two outcomes? (b) Verify the requirements for constructing a \(90 \%\) confidence interval for the population proportion of all adult Americans who are more annoyed by tailgaters than slow drivers in the passing lane. (c) Construct a \(90 \%\) confidence interval for the population proportion of all adult Americans who are more annoyed by tailgaters than slow drivers in the passing lane.

A school administrator is concerned about the amount of credit-card debt that college students have. She wishes to conduct a poll to estimate the percentage of full-time college students who have credit-card debt of \(\$ 2000\) or more. What size sample should be obtained if she wishes the estimate to be within 2.5 percentage points with \(94 \%\) confidence if (a) a pilot study indicates that the percentage is \(34 \% ?\) (b) no prior estimates are used?
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