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You Explain It! Hours Worked In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a \(95 \%\) confidence interval for mean number of hours worked was lower bound: 42.7 and upper bound: \(44.5 .\) Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) There is a \(95 \%\) probability the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours. (b) We are \(95 \%\) confident that the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours. (c) \(95 \%\) of adult Americans worked between 42.7 hours and 44.5 hours last week. (d) We are \(95 \%\) confident that the mean number of hours worked by adults in Idaho in the previous week was between 42.7 hours and 44.5 hours.

Step by step solution

01

- Review Option (a)

Interpret the statement: 'There is a 95% probability the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours.' This statement is not reasonable because a confidence interval does not give the probability of the parameter being within an interval. Once the interval is calculated, the mean is either within this interval or it is not.

02

- Review Option (b)

Interpret the statement: 'We are 95% confident that the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours.' This statement is reasonable. A 95% confidence interval means that if we were to take many samples and create intervals in the same way, 95% of these intervals would contain the true population mean.

03

- Review Option (c)

Interpret the statement: '95% of adult Americans worked between 42.7 hours and 44.5 hours last week.' This statement is not reasonable because a confidence interval refers to the mean number of hours worked, not individual values. It does not make a statement about the proportion of individuals within the interval.

04

- Review Option (d)

Interpret the statement: 'We are 95% confident that the mean number of hours worked by adults in Idaho in the previous week was between 42.7 hours and 44.5 hours.' This is not reasonable because the confidence interval was established for adult Americans, not for the subset of adults in Idaho.

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Key Concepts

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

Survey Analysis

In survey analysis, data is collected from a subset of a population to draw conclusions about the entire population. The Gallup Organization’s survey of 1100 adult Americans about their working hours is a classic example. Here, 1100 individuals represent a sample from the much larger group of adult Americans. The goal is to estimate the population parameters, like the mean number of hours worked in a week, based on this sample data.

Effective survey analysis involves ensuring the sample is representative. This means the sample should be random and sufficiently large to reflect the broader population's characteristics. Survey results can then be generalized to the whole population with a certain level of confidence.

Interpretation of Statistical Results

Understanding how to interpret statistical results, especially confidence intervals, is crucial. Confidence intervals provide a range within which we expect the true population parameter to lie.

For instance, from the given data, a 95% confidence interval for the mean number of hours worked by adult Americans ranges from 42.7 to 44.5 hours. This does not mean there's a 95% chance the true mean is within this interval. Instead, it implies that if repeated samples were taken and intervals computed, 95% of those intervals would contain the true mean. Misinterpretations, like thinking it applies to individual hours worked or a specific subgroup, are common but incorrect.

Mean Estimation

Mean estimation involves calculating the average of a sample to infer the population mean. In the Gallup survey, the objective was to estimate the mean number of hours worked in a week by adult Americans.

Using the sample data, statisticians compute the sample mean and use it to estimate the population mean. However, since this estimate can vary depending on the sample, confidence intervals are used to provide a range for this estimate. In this case, the confidence interval was from 42.7 to 44.5 hours. This range gives a sense of the reliability of the mean estimate, reflecting the variability inherent in sampling processes.

Gallup Organization

The Gallup Organization is known for conducting extensive surveys and public opinion polls. Their methods are well-established and designed to gather reliable data on various topics, including work hours, political views, and social behavior.

Gallup's surveys are based on carefully designed sampling methods to ensure that the collected data is representative of the target population. This process involves selecting random samples and ensuring a sufficient sample size. Gallup’s surveys, like the one mentioned, are instrumental in providing insights into population trends and behaviors, backed by statistically sound analyses.