Which of the following is not a component of a confidence interval for a population mean with an unknown population standard deviation?

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BlackTom AI · Accepted Answer

Question restatement: Which of the following is not a component of a confidence interval for a population mean with an unknown population standard deviation?

Status note: The answer options are not provided in the data, so I cannot evaluate each choice individually. However, the provided answer indicates that the option 'A value for “sigma”' is the one not included. I will justify why that statement makes sense in context.

Why 'A value for sigma' is not a component: In a confidence interval for a population mean when the population standard deviation sigma is unknown, we do not use the true sigma in the calculation. Instead, we replace sigma with an estimate, the sample standard deviation s, and we use the t-distribution with n-1 degrees of freedom. The interval typically has the form x̄ ± t*(s/√n), where t* is the appropriate critical value from the t-distribution. Because sigma is unknown, it cannot appear as a known fixed quantity in the formula; incorporating sigma would require knowing the exact population dispersion, which contradicts the assumption of unknown sigma. Therefore, a true sigma value is not a component of the interval; instead, we rely on the estimate s and the t-critical value.

If additional options were available, the evaluation would proceed like this:
- Any option stating the use of sigma directly in the margin of error would be incorrect, since we replace sigma with s in this scenario.
- Options describing the use of the sample mean x̄, the standard error (s/√n), and a t-distribution critical value would be correct components.
- Any statement that the interval depends on the sample standard deviation s, the sample size n, and the chosen confidence level would be correct, whereas a reliance on the exact known sigma would be incorrect.

Summary reflection: The key misconception to avoid is treating sigma as known and fixed in this unknown-sigma context. The correct approach is to use the sample-based estimate s and the t-distribution, which aligns with standard statistical practice for confidence intervals of a population mean when sigma is unknown.

STAT 1450.02 SP2025 (28132) Checkpoint 20.3

Which of the following is not a component of a confidence interval for a population mean with an unknown population standard deviation?

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