Sam computed a 90 % confidence interval for μfrom a specific...

Sam computed a $90 \%$ confidence interval for $\mu$ from a specific random sample of size $n$. He claims that at the $90 \%$ confidence level, his confidence interval contains $\mu$. Is his claim correct? Explain.

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Key Concepts

Confidence Intervals

A confidence interval is a range of values computed from sample data that, under repeated sampling, would contain the true population parameter a certain proportion of the time (e.g., 90% for a 90% confidence interval). It is a method used in inferential statistics to estimate unknown parameters and quantify uncertainty.

Interpretation of Confidence Levels

The confidence level (90% in this case) refers to the long-run proportion of confidence intervals that would successfully capture the true parameter if the study were repeated numerous times. It does not imply that there is a 90% probability that any one computed interval contains the parameter, because the parameter is fixed and the interval is random.

Frequentist Perspective on Parameters

From a frequentist viewpoint, the true population mean is considered fixed, and the randomness comes from the sampling process. Once a specific interval is computed, it either contains the parameter or it does not, which means it is incorrect to state that the interval 'contains' the parameter with a 90% probability.

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