The Real Estate Industry Claims That It Is The Best And Most
18the Real Estate Industry Claims That It Is The Best And Most Effect
The real estate industry claims that it is the best and most effective system to market residential real estate. A survey of randomly selected home sellers in Illinois found that a 95% confidence interval for the proportion of homes that are sold by a real estate agent is 69% to 81%. Four interpretations are offered below. Three are incorrect and one is correct. Find the three incorrect interpretations and explain the error in each of them.
We are 95% confident, based on this sample, that between 69% and 81% of all homes in Illinois are sold by a real estate agent.
We are 95% confident that between 69% and 81% of homes in this survey are sold by a real estate agent.
95% of all homes in Illinois are sold by a real estate agent.
95% of all random samples of home sellers in Illinois will show that between 69% and 81% of homes are sold by a real estate agent.
Paper For Above instruction
Analysis of Confidence Interval Interpretations in Real Estate Context
Understanding the correct interpretation of confidence intervals is fundamental in statistics, particularly in the context of survey results that inform industry claims. The survey in Illinois, which reports a 95% confidence interval of 69% to 81% for the proportion of homes sold by a real estate agent, provides valuable information. However, the interpretation of this interval must adhere to the principles of statistical inference to be valid. This paper analyzes four interpretations of the confidence interval and identifies which are correct or incorrect, elucidating common misunderstandings in statistical reasoning.
Correct Interpretation of Confidence Intervals
The correct interpretation of a 95% confidence interval (CI) for a population proportion is that if we were to repeat the sampling process many times, approximately 95% of these calculated intervals would contain the true proportion of the population parameter. In this case, the true proportion of homes sold by real estate agents in Illinois lies within the interval of 69% to 81%, with a high degree of confidence based on the sample data. It is essential to note that the confidence level pertains to the method and its long-run properties, not the probability of the parameter itself being in a specific interval once it is calculated.
Analysis of the Offered Interpretations
Interpretation 1:
"We are 95% confident, based on this sample, that between 69% and 81% of all homes in Illinois are sold by a real estate agent."
This statement is a correct interpretation. It accurately reflects the frequentist interpretation of the confidence interval: that we can be 95% confident that the true population proportion falls within this range based on the sampling process.
Interpretation 2:
"We are 95% confident that between 69% and 81% of homes in this survey are sold by a real estate agent."
This statement is also correct. It correctly states that the confidence pertains to the population proportion, based on the data collected in the survey, emphasizing the generalization from a sample to the entire population.
Interpretation 3:
"95% of all homes in Illinois are sold by a real estate agent."
This interpretation is incorrect. It misapplies the confidence interval as a statement about individual homes rather than about the population proportion. The interval does not mean that 95% of homes are sold by real estate agents; rather, it estimates the proportion of such homes in the population with a certain confidence level.
Interpretation 4:
"95% of all random samples of home sellers in Illinois will show that between 69% and 81% of homes are sold by a real estate agent."
This statement is incorrect because it confuses the variability of the estimates across different samples with the parameter being estimated. Not every sample will produce an interval that captures the true proportion; only about 95% of the intervals constructed in repeated samples would do so. This interpretation incorrectly implies that in every sample the interval will be within this range, which is not true.
Conclusion
The proper interpretation of the confidence interval emphasizes the confidence in the method and the long-run properties of the estimator. Interpretations that treat the interval as an absolute statement about the current population proportion or as a prediction about individual observations are incorrect.
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