91Ó°ÊÓ

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 ,

Step by step solution

01

Title - Understand the Margin of Error

The margin of error for the survey is given as 3 percentage points. This means the actual proportion is within 3 percentage points of the reported percentage, which is 55%.

02

Title - Calculate the Confidence Interval

To find the range of the confidence interval, subtract and add the margin of error to the sample proportion (0.55): Confidence interval = 0.55 - 0.03 = 0.52 (52%) 0.55 + 0.03 = 0.58 (58%) So, the confidence interval is between 0.52 and 0.58.

03

Title - Evaluate Option A

Option (a) says: 'We are 95% confident 55% of adult Americans during the Great Depression felt wages paid to workers in industry were too low.'This is incorrect because it does not account for the margin of error around 55%.

04

Title - Evaluate Option B

Option (b) says: 'We are 92% to 98% confident 55% of adult Americans during the Great Depression felt wages paid to workers in industry were too low.'This is incorrect because the confidence level should be precisely 95%, not a range of percentages.

05

Title - Evaluate Option C

Option (c) says: '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.'This is correct as it accurately reflects the confidence interval.

06

Title - Evaluate Option D

Option (d) says: '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.'This is a correct interpretation of the confidence interval.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

margin of error

The margin of error is a crucial concept when interpreting survey results. It represents the range within which we expect the true population parameter to lie. In the given exercise, the margin of error is 3 percentage points. This means that the reported 55% opinion could actually vary from 52% to 58%.
To find this range, you subtract and add the margin of error to the sample proportion.
For this example:

• Lower bound: 55% - 3% = 52%
• Upper bound: 55% + 3% = 58%

So, the confidence interval ranges from 0.52 to 0.58.
This range helps account for sampling variability and provides a buffer to show where the true proportion likely falls.

sample proportion

The sample proportion is the percentage of survey respondents who gave a particular response. In our case, the sample proportion is 55%. This means that out of the 2025 surveyed adult Americans in 1933, 55% felt that wages in industry were too low.
Sample proportion is a point estimate which means it is an estimate of the population proportion. It’s important to remember that the sample proportion alone does not tell the full story. This is where the margin of error and confidence interval come in, providing a more complete picture.

confidence level

Confidence level indicates the probability that the confidence interval contains the true population parameter. In our example, the confidence level is 95%. This means that if we were to take 100 different samples and build confidence intervals for each of them, we would expect about 95 of those intervals to contain the true proportion of the population.
Some important points about the 95% confidence level:

• It shows a high level of certainty, making it a robust measure.
• It does not mean there is a 95% chance that the true parameter is within the interval for any single sample—it’s about long-run performance over multiple samples.

By understanding these concepts, you can more accurately interpret survey results and the likelihood that they reflect the broader population’s views.