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The Gallup poll reported that \(45 \%\) of Americans have tried marijuana. This was based on a survey of 1021 Americans and had a margin of error of plus or minus 5 percentage points with a \(95 \%\) level of confidence. a. State the survey results in confidence interval form and interpret the interval. b. If the Gallup Poll was to conduct 100 such surveys of 1021 Americans, how many of them would result in confidence intervals that did not include the true population proportion? c. Suppose a student wrote this interpretation of the interval: "We are \(95 \%\) confident that the percentage of Americans who have tried marijuana is between \(40 \%\) and \(50 \% .\) " What, if anything, is incorrect in this interpretation?

Step by step solution

01

Confidence Intervals Formulation

The first step toward answering the given problem is to write the survey results in confidence interval form. Since the Gallup poll reported that \(45 \% \) of Americans have tried marijuana with a margin of error of plus or minus \(5 \% \) points, we just need to add and subtract this margin of error to/from the reported population proportion to get the confidence interval. Hence, the confidence interval is \( [45 - 5 , 45 + 5] \)% or \( [40, 50] \% \).

02

Determine Percentage of Surveys Outside of the True Population Proportion

A 95% confidence interval means that we are 95% confident that the true population proportion lies within our set interval. In this case, the confidence interval is from 40% to 50%. Thus, in 100 surveys, we would expect \(100 - 95 = 5 \) of those surveys to result in confidence intervals that did not include the true population proportion.

03

Interpretation of the Confidence Interval

The given interpretation of the confidence interval states, 'We are 95% confident that the percentage of Americans who have tried marijuana is between 40% and 50%.' This is slightly wrong. The correct interpretation of a confidence interval should be 'We are 95% confident that our interval of [40, 50]% includes the true population proportion of Americans who have tried marijuana.' It's about the interval capturing the true parameter, not about the true parameter falling within the interval.

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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 concept often used in statistics to indicate the range within which we can expect the true population parameter to lie, given our sample results. It represents the uncertainty in your survey's results. For instance, in the Gallup poll reporting that 45% of Americans have tried marijuana, the margin of error is ±5 percentage points.

• This means the true percentage could reasonably be expected to be between 40% and 50%.
• It provides a buffer around the sample statistics, accounting for sampling variability.

The margin of error is crucial because, without it, you wouldn't have a sense of how precise your estimates from the sample data are in reflecting the whole population.

Population Proportion

Population proportion is a statistical term that describes the fraction of the population that has a certain characteristic. In this exercise, we talk about the percentage of Americans who have tried marijuana.

• The report states that 45% of the survey respondents have tried marijuana.
• This proportion can be used to estimate the population proportion, which is the ultimate goal.

Importantly, differences between the sample and true population proportions can arise from random sampling errors or the margin of error. The sample proportion is just a snapshot that provides an estimate of the larger population characteristic.

Survey Sampling

Survey sampling is the process of selecting a subset of individuals from a population to estimate the characteristics of the whole population. This method helps in collecting data when it's impractical to survey an entire population.

• In the Gallup poll, a sample of 1021 Americans was selected to represent the population.
• Survey sampling reduces costs and limitations associated with large data collection.

Nonetheless, it's essential that the sample is randomly chosen and large enough to ensure that it's representative of the larger population, minimizing bias and errors.

Confidence Level

The confidence level indicates how confident you are that the population parameter lies within the computed confidence interval. In this context, a 95% confidence level means we are 95% sure that the true proportion of Americans who have tried marijuana falls within our interval of 40% to 50%.

• A high confidence level like 95% provides more assurance in the results.
• This also means that if the same survey were repeated multiple times, 95 out of 100 resultant confidence intervals would contain the true population proportion.

Thus, the confidence level is a measure of reliability in the estimate provided by the confidence interval, largely impacting our interpretation and trust in the survey results.