[NW] You Explain It! New Deal Policies 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 businesses 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.


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.

Verified step by step guidance

1. Step 1: Understand the meaning of the confidence interval and confidence level. The survey reports a 55% proportion with a margin of error of 3 percentage points at 95% confidence. This means the 95% confidence interval is from 52% to 58%.
2. Step 2: Recognize that the confidence level (95%) indicates how confident we are that the interval contains the true population proportion, not the probability that the proportion itself lies within a certain range after the data is collected.
3. Step 3: Analyze the statement: 'We are 92% to 98% confident 55% of adult Americans felt wages were too low.' This statement incorrectly reverses the roles of the sample proportion and the population proportion. The 55% is the sample estimate, not the unknown population parameter.
4. Step 4: Identify the flaw: Confidence intervals provide a range for the population parameter based on the sample data, not a confidence range for the sample statistic itself. Also, the confidence level is fixed (95%), not a range (92% to 98%).
5. Step 5: Conclude that a reasonable interpretation would state that we are 95% confident that the true proportion of adult Americans who thought wages were too low lies between 52% and 58%, rather than expressing confidence about the sample proportion.

Key Concepts

Confidence Interval and Confidence Level

A confidence interval provides a range of values within which the true population parameter is expected to lie, based on sample data. The confidence level (e.g., 95%) indicates the proportion of such intervals that would contain the true parameter if the survey were repeated many times. It does not mean the probability that the parameter lies within the interval for a single sample.

Margin of Error

The margin of error quantifies the maximum expected difference between the sample statistic and the true population parameter, reflecting sampling variability. For example, a 3 percentage point margin of error means the true proportion is likely within 3 points above or below the sample estimate, assuming the confidence level is met.

Interpretation of Survey Results and Confidence Statements

Interpreting survey results requires understanding that the confidence level applies to the method, not the specific sample. Saying 'we are 92% to 98% confident' is incorrect if the stated confidence level is 95%. Confidence intervals do not provide a probability range for the parameter but rather a range constructed from the sample that likely contains the parameter.