[NW] Government Waste Gallup News Service conducted a survey of 1017 American adolts aged 18 years or older. The respondents were asked, “Of every tax dollar that goes to the federal government in Washington, D.C., do you believe 51 cents or more are wasted?” Of the 1017 individuals surveyed, 35% indicated that 51 cents or more is wasted. Gallup reported that 35% of all adolt Americans 18 years or older believe the federal government wastes at least 51 cents of each dollar spent, with a margin of error of 4% and a 95% level of confidence.


What can be inferred from this survey?

Verified step by step guidance

1. Identify the sample proportion \( \hat{p} \) from the survey, which is the percentage of respondents who believe 51 cents or more are wasted. Here, \( \hat{p} = 0.35 \) (35%).
2. Understand that the margin of error (ME) is given as 4%, or \( 0.04 \), and the confidence level is 95%. This means the true population proportion \( p \) is estimated to lie within a range around \( \hat{p} \) with 95% confidence.
3. Calculate the confidence interval for the population proportion using the formula: \(\text{Confidence Interval} = \hat{p} \pm \text{ME}\) which translates to: \(\left( \hat{p} - 0.04, \ \hat{p} + 0.04 \right)\)
4. Interpret the confidence interval: This interval estimates the range in which the true proportion of all American adults who believe that 51 cents or more of every tax dollar is wasted lies, with 95% confidence.
5. Conclude that based on this survey and the confidence interval, we can infer that between approximately 31% and 39% of all American adults believe that the federal government wastes at least 51 cents of each dollar spent, acknowledging the uncertainty captured by the margin of error.

Key Concepts

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to lie, given the sample data. In this survey, a 4% margin of error means the actual percentage of adults who believe the government wastes at least 51 cents per dollar is likely between 31% and 39%. It reflects sampling variability and helps assess the precision of the estimate.

Confidence Level

The confidence level indicates the degree of certainty that the population parameter falls within the margin of error. A 95% confidence level means that if the survey were repeated many times, 95% of the calculated intervals would contain the true proportion. It provides a measure of reliability for the survey results.

Sample Proportion and Representativeness

The sample proportion (35%) is the percentage of surveyed individuals with a specific opinion, used to estimate the population proportion. The representativeness of the sample (1017 American adults) ensures that the results can be generalized to the broader population. Proper sampling methods reduce bias and improve the validity of inferences.