What Is Meant By The Term 90% Confident When Constructing A
What Is Meant By The Term 90% Confident When Constructing a Con
What is meant by the term “90% confident” when constructing a confidence interval for a mean?
a. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
b. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
c. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
d. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.
Paper For Above instruction
The concept of confidence levels in statistical inference is fundamental to understanding how we estimate parameters such as the population mean. When we say we are “90% confident” in a confidence interval, we are referring to the frequency with which such intervals, constructed from repeated samples, will contain the true population parameter, in this case, the mean. Specifically, the correct interpretation is that if we were to take many repeated samples from the same population and construct a confidence interval from each sample, approximately 90% of those intervals would contain the true population mean. Thus, the most accurate choice among the options provided is option c, which states: “If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.”
This interpretation hinges on the concept of the confidence level (90% in this case), which is a property of the method used rather than a particular interval. The confidence level does not imply that any single calculated interval has a 90% probability of containing the true mean after it is computed. Instead, it reflects the long-term success rate if the same sampling process were repeated numerous times. Each confidence interval either contains or does not contain the true mean; the 90% confidence level indicates that, over many such repetitions, approximately 90% of those intervals will encompass the true population parameter.
This understanding is crucial in statistical inference because it emphasizes the reliability of the method rather than the probability associated with a specific interval once it has been computed. The other options either misinterpret the meaning of confidence levels or confuse the properties of the sample and the population. For instance, option a incorrectly states that repeated samples yield the same interval, which is not the case—the intervals will vary because they are based on different samples. Option b confuses the sample mean with the population mean and the concept of interval coverage. Lastly, option d incorrectly suggests that the sample mean would often equal the true mean in 90% of samples, which is not guaranteed or implied by the confidence level.
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