If the population variance is unknown, then a degree of freedom is lost in the calculation of sample variance.

Question

If the population variance is unknown, then a degree of freedom is lost in the calculation of sample variance.

BlackTom AI · Accepted Answer

To analyze the statement, we consider how sample variance is computed when the population variance is unknown.
Option 1: True. In practice, the unbiased estimate of the population variance uses the formula s^2 = [sum (xi - x̄)²] / (n - 1). Here, x̄ is the sample mean, which itself is estimated from the data and consumes one degree of freedom. Because one parameter (the mean) is estimated from the sample, one degree of freedom is effectively lost when calculating the variance estimate.
Option 2: False. If we did not adjust for estimating the mean, we would divide by n and obtain a biased estimator of the population variance. The loss of degrees of freedom arises specifically because we estimate the mean from the sample, not merely because the population variance is unknown. Without this adjustment (using n in the denominator), the estimator would systematically undershoot the true variance, especially for small samples.
Option 3: (If considering alternative formulations) Some contexts discuss degrees of freedom differently, but in standard statistical practice for a single sample variance with unknown population variance, the loss of one degree of freedom due to using the sample mean is the reason for the n − 1 denominator, reinforcing that a df loss occurs.
Option 4: (Clarifying common misconception) It is not the ignorance of the population variance alone that creates df loss; it is the combination of estimating the mean from the same data used to compute the variance, which reduces the available information by one degree of freedom.

If the population variance is unknown, then a degree of freedom is lost in the calculation of sample variance.Single choice

Similar Questions

The degrees of freedom of a 4 x 5 table is:

What happens as the df of a dataset is increased?

A researcher reports df = 18 for a repeated-measures t statistic. This research study used a total of n = 19 participants.

A repeated-measures study using a sample of n = 20 participants would produce a t statistic with df = ____.

In a contingency table, what does the degrees of freedom indicate?

Suppose that a study compares the means of two groups. There are 40 males in Group 1 and 35 females in Group 2. The degrees of freedom for this study when using the t-test for the difference between means is:

Based on the information in Question 1, what were the degrees of freedom (DF) for the interaction of color and style, C?

What concept do "degrees of freedom" represent in statistical analysis?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler