What Is Meant By The Term 90 Confident When Constructing A C

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.

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A confidence interval provides a range of values within which we expect the true population parameter, such as the mean, to lie with a certain level of confidence. When we specify a 90% confidence level, it means that if we were to take many samples and construct a confidence interval from each sample using the same procedure, approximately 90% of those intervals would capture the true population mean. This interpretation reflects the reliability of the method rather than the probability that any one particular interval contains the parameter.

Specifically, the statement “90% confident” indicates that the confidence interval calculation is based on repeated sampling procedures. Out of all possible samples, about 90% of the confidence intervals formed using this method are expected to contain the actual population mean. It is crucial to understand that this does not mean that there is a 90% probability that a specific calculated interval contains the true mean after the data is observed, as the true mean is fixed but unknown. Instead, the confidence level refers to the long-run success rate of the method over numerous repetitions.

This concept stems from the principles of statistical inference, where confidence intervals are used to estimate population parameters under uncertainty. The width of the interval depends on the sample size, variability in the data, and the desired confidence level. A higher confidence level, such as 95%, would lead to a wider interval to maintain the same level of confidence, whereas a lower confidence level, like 90%, results in a narrower range.

In conclusion, when a confidence level of 90% is used, it means that over many repeated samples and intervals, approximately 90% of these intervals would contain the true population mean. This interpretation is fundamental to understanding the purpose and proper usage of confidence intervals in statistical analysis.

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