If an estimate of the slope is unbiased, then when we calculate that estimate from a sample, it means:

Question

BlackTom AI · Accepted Answer

Unbiasedness of an estimator means its expected value equals the true population parameter. For a slope estimator β̂ in a regression context, unbiasedness is formally written as E[β̂] = β (where β is the true population slope and the expectation is taken over the sampling distribution of β̂).

Relevant concepts:
- Estimator vs parameter: β̂ is a random variable computed from a sample; β is the fixed but unknown population slope.
- Sampling distribution: repeated samples produce different β̂ values; their distribution has a mean (expected value) and variance.
- Unbiasedness: the mean of the sampling distribution equals the true parameter.

Step-by-step reasoning:
1. Define the estimator β̂ obtained from one sample. Because it depends on sample data, β̂ varies across repeated samples.
2. Compute the expectation across the sampling distribution: E[β̂] = integral or sum of β̂ times its probability. Unbiasedness requires E[β̂] = β.
3. Interpret practically: if you take many independent random samples of the same size and compute β̂ for each, the average of these β̂ values will approach β.

Illustrative calculation:
- Suppose the true slope β = 2. If the estimator is unbiased, then after N large repeated samples the sample mean of estimates (1/N)Σβ̂_i → 2. For example, with estimates 1.9, 2.1, 2.0, 2.05, 1.95 the average is 2.0 ≈ β.

Why the correct answer is correct:
The provided statement exactly matches the formal definition: unbiased means the expected value over all possible random samples equals the true slope. It does not require every single sample estimate to equal β, nor does it imply low variance or consistency by itself.

Why other interpretations are incorrect:
- Saying an individual sample estimate equals the true slope is false (random sampling noise).
- Confusing unbiasedness with small variance or precision is incorrect—an estimator can be unbiased yet highly variable.
- Equating unbiasedness with consistency or correctness for a single sample is also wrong.

Conclusion: The correct interpretation is that the expected value of the estimator over many random samples equals the true population slope, which matches the provided answer.

If an estimate of the slope is unbiased, then when we calculate that estimate from a sample, it means:单项选择题

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