Judging an estimator

Mathematics · Data & Statistics · ages 20-21

Name ______________________ Date ____________

A sample of three readings is 4, 6 and 8. Compute the sample mean.
Answer: ______________
A sample of three readings is 4, 6 and 8. Compute the sample mean.
Answer: ______________
For the readings 4, 6 and 8 with mean 6, the squared deviations total 8. With the n minus 1 divisor, what is the sample variance?
- 4
- 8 over 3
- 8
Priya says the sample mean tends to sit above the population mean. Is Priya right?
Circle one: True False
Why does the sample variance divide by n - 1 instead of n?
- It corrects the downward bias from dividing by n
- It makes the arithmetic easier
- It only matters for very large samples
- It removes outliers automatically
For the same sample 4, 6 and 8, compute the sample variance with the n - 1 divisor.
Answer: ______________
An estimator whose bias and variance both drain away as the sample grows is called consistent.
Circle one: True False
Why does the sample variance divide by n minus 1 instead of n?
- It makes the arithmetic easier
- It corrects the downward bias from dividing by n
- It removes outliers automatically
Estimator A is unbiased with variance 4. Estimator B has bias 0.5 and variance 1. Which has the smaller mean squared error?
- Estimator B
- Estimator A
- They tie
- The sample size is needed first
A class takes one sample with mean 6 and declares the population mean is exactly 6 with no uncertainty. What is wrong?
- Nothing, one sample settles it
- Unbiasedness describes the average over all samples, not the exactness of one sample
- The mean should have used n minus 1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Judging an estimator W1-mt_I374r2UU_i-s1

6 · The average of 4, 6 and 8 is 18/3 = 6, which estimates the population mean without bias.
6 · Adding the readings and dividing by 3 gives the unbiased estimate.
4 · With n equal to 3, the divisor is 2, and 8 over 2 is 4.
False · The sample mean is unbiased, so it leans neither high nor low.
It corrects the downward bias from dividing by n · Deviations from the sample mean run smaller than deviations from the true mean, so dividing by n underestimates spread. The n - 1 divisor fixes that bias exactly.
4 · Squared deviations are 4, 0 and 4, totalling 8. Dividing by n - 1 = 2 gives 4.
True · That draining away toward the target is exactly what consistency means.
It corrects the downward bias from dividing by n · Deviations from the sample mean run short, and the n minus 1 divisor fixes that bias exactly.
Estimator B · Mean squared error is variance plus squared bias: A scores 4, B scores 1 + 0.25 = 1.25. B wins despite its bias.
Unbiasedness describes the average over all samples, not the exactness of one sample · Unbiasedness is about the average over all samples, while one sample still varies.