Bias, variability and the 10% condition

About this lesson

Statistics (AP-aligned) › Sampling distributions

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Bias, variability and the 10% condition

Warm-up

Three quick questions from earlier lessons.

Discover

A scale always reads 2 kg too heavy, but very consistently. Is it a good scale?

  1. No, it is biased
  2. Yes, it is very consistent

Key idea

Good estimators are unbiased and precise.

An estimator is unbiased if the mean of its sampling distribution equals the parameter (p^\hat{p} and xˉ\bar{x} are unbiased). Its variability shrinks as nn grows. The SD formulas assume independence; when sampling without replacement, keep nn at most 10% of the population.

Worked example

A town of 6,000 households.

Largest sample: 600 households
10% of 6,000.
A larger random sample
To shrink variability.
A larger sample does not fix a biased method
Bias comes from the method.

Your turn, guided

Your turn: population 2,500. The largest sample meeting the 10% condition.

250250
10% of 2,500.

Practice ★

  1. What does it mean that p̂ is an unbiased estimator of p (samples of 40)?
    1. The mean of its sampling distribution equals p
    2. Every sample gives p̂ = p
    3. Its variability is small
  1. A school has 1200 students. Sampling without replacement, what is the largest sample size that meets the 10% condition?
  2. A school has 5500 students. Sampling without replacement, what is the largest sample size that meets the 10% condition?

Watch out!

Practice ★★

  1. A survey method gives estimates that cluster tightly, but far from the true value. How would you describe it (samples of 116)?
    1. Biased, with low variability
    2. Unbiased, with low variability
    3. Unbiased, with high variability

Exit ticket

Three questions to check you have got it.

  1. What does it mean that p̂ is an unbiased estimator of p (samples of 40)?
    1. The mean of its sampling distribution equals p
    2. Every sample gives p̂ = p
    3. Its variability is small
  2. What does it mean that p̂ is an unbiased estimator of p (samples of 187)?
    1. The mean of its sampling distribution equals p
    2. Every sample gives p̂ = p
    3. Its variability is small
  3. What does it mean that p̂ is an unbiased estimator of p (samples of 106)?
    1. The mean of its sampling distribution equals p
    2. Every sample gives p̂ = p
    3. Its variability is small

Summary card

Bias and variability

  • Unbiased: centered on the parameter.
  • Variability: shrinks with nn.
  • 10% condition for independence.

Watch out: large does not mean unbiased.

End of the lesson

Back to the chapter

Answers: Bias, variability and the 10% condition

A

  1. The mean of its sampling distribution equals p

B

  1. 120120
  2. 550550

C

  1. Biased, with low variability

D

  1. The mean of its sampling distribution equals p
  2. The mean of its sampling distribution equals p
  3. The mean of its sampling distribution equals p

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