Unit 3 · Topic 3.1 Beta

Estimators

A statistic used to estimate a parameter is an estimator, and its value from one sample is a point estimate.

Practice 2: Collect DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A shipment holds 40,000 phone chargers, and an inspector can test only 200 of them. Grab the first 200 off one shift's line, and the answer comes back too low, again and again. Choose 200 at random, and the answers scatter around the truth. Which method you use decides whether your number can be trusted, long before you do any arithmetic.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A survey finds that 62% of 500 randomly chosen adults in a state own a pet. In this sentence, 62% is

a statistic, because it describes the sample
a parameter, because it describes the state
a parameter, because the sample was random
a statistic, because it is a percent

Show the answer

A statistic describes a sample; a parameter describes the whole population. The 62% comes from the 500 adults surveyed.

Correct: a statistic, because it describes the sample:
a parameter, because it describes the state:
a parameter, because the sample was random:
a statistic, because it is a percent:

2. A computer takes 1,000 random samples of the same size from one population and records the sample proportion from each. The distribution of those 1,000 values approximates

the sampling distribution of the sample proportion
the distribution of the population
the distribution of one sample
the probability that one sample is correct

Show the answer

The values of a statistic over many samples of the same size form (approximately) its sampling distribution.

Correct: the sampling distribution of the sample proportion:
the distribution of the population:
the distribution of one sample:
the probability that one sample is correct:

3. Asking students in the school gym whether they play a sport is likely to give a result that is

too high, because students in the gym are more likely to play
too low, because the gym is a small place
about right, because many students use the gym
random, because nobody chose the students

Show the answer

A convenience sample like this favors students who play sports, so it is biased upward.

Correct: too high, because students in the gym are more likely to play:
too low, because the gym is a small place:
about right, because many students use the gym:
random, because nobody chose the students:

Part 4 · See it

See it first

Three probability histograms on the same scale from 0.1 to 0.8, each with a dashed line at the true proportion p = 0.40. Top: the sample proportion from random samples of 25 is centered at 0.40 and spread widely, from about 0.2 to 0.6: unbiased, high variability. Middle: the sample proportion from random samples of 100 is centered at 0.40 and about half as spread out: unbiased, low variability. Bottom: a biased method whose sample proportions center at 0.55 with low variability: biased, low variability. — The exact sampling distributions of three estimators of p = 0.40. Random samples are centered on the truth; a larger sample is less spread out; a method that favors one group is centered in the wrong place. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

A sample is only part of the populationa statistic such as $\hat{p}$ is used as an estimator of the parameter, and its value is a point estimate
Different samples give different values of the statisticthe statistic has a sampling distribution, with a center and a spread
The center of that sampling distribution equals the parameterthe estimator is unbiased: it is right on average, though single samples still miss
The random sample is made largerthe spread of the sampling distribution shrinks, so a typical estimate lands closer to the parameter
The method favors some members of the populationthe center moves away from the parameter, and a bigger sample of the same kind does not move it back

Part 6 · Key ideas

Key ideas

An estimator is a statistic used to estimate a parameter: $\hat{p}$ for $p$ , $\bar{x}$ for $\mu$ . One sample's value is a point estimate.
Bias is about the center: an unbiased estimator has a sampling distribution centered at the parameter.
Variability is about the spread of the sampling distribution. For random samples, larger samples give less variability.
Bias comes from the method, so a larger sample does not fix it. Variability comes from sample size, so a better method alone does not fix it.
The best estimator has no bias and low variability: random selection plus a large enough sample.

Part 7 · Misconception

A common mistake

The wrong idea: An unbiased estimator gives the right answer, so if a random sample says 0.31, the population proportion is 0.31.

What actually happens: Unbiased means right on average over many samples. One random sample's estimate still misses the parameter by some amount; how far it tends to miss is the variability.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Four ways to estimate the proportion of faulty phone chargers

A shipment holds 40,000 phone chargers, and 8% of them are faulty, so $p = 0.08$ . An inspector simulates each of four sampling methods 1,000 times. Each time, the method gives one sample proportion of faulty chargers. The table summarizes the 1,000 sample proportions from each method.

Summaries of 1,000 simulated sample proportions from each method (invented simulation)
Method	How the chargers are chosen	Mean of the 1,000 sample proportions	Standard deviation of the 1,000 sample proportions
W	Simple random sample of 50 chargers	0.080	0.038
X	Simple random sample of 200 chargers	0.080	0.019
Y	The first 200 chargers packed on one morning shift	0.052	0.016
Z	50 chargers the inspector picks by hand from open boxes	0.110	0.044

1. Which methods appear to give unbiased estimators of $p$ ? Select all that apply.

Method W
Method X
Method Y
Method Z

Show the answer

An estimator is unbiased when the center of its sampling distribution equals the parameter. Methods W and X have means of 0.080, which is $p$ .

Correct: Method W: Right: its 1,000 estimates center at 0.080, the true value of $p$ .
Correct: Method X: Right: its 1,000 estimates also center at 0.080.
Method Y: Its estimates center at 0.052, well below 0.08, so it underestimates $p$ : biased.
Method Z: Its estimates center at 0.110, above 0.08, so it overestimates $p$ : biased.

2. Which method gives estimates that vary the least from one sample to the next?

Method X, because it is unbiased and uses 200 chargers
Method Y, with a standard deviation of 0.016
Method W, because its mean equals the true proportion
Method Z, because its mean of 0.110 is the largest

Show the answer

Variability is the spread of the sampling distribution, measured here by the standard deviation of the 1,000 estimates. Method Y has the smallest, 0.016, even though it is biased.

Method X, because it is unbiased and uses 200 chargers: X is the best method overall, but Y has a slightly smaller standard deviation (0.016 against 0.019). Low variability and no bias are separate properties.
Correct: Method Y, with a standard deviation of 0.016: Right: the smallest spread of the four, so Y varies least, even though it misses $p$ .
Method W, because its mean equals the true proportion: A mean equal to $p$ says W is unbiased. It says nothing about spread, and W's spread, 0.038, is large.
Method Z, because its mean of 0.110 is the largest: The mean tells you where the estimates center, not how much they vary. Z has the largest spread.

3. The inspector can afford to check 200 chargers. Which method should the inspector use, and why?

Method Y, because its estimates have the smallest spread
Method X: its estimates center at $p$ with little spread
Method Z, because a person checks each box before choosing
Method W, because a smaller sample has less chance of error

Show the answer

A good estimator has no bias and low variability. Method X centers on $p$ and has a small spread; Method Y has a slightly smaller spread but misses $p$ by about 0.03 on average.

Method Y, because its estimates have the smallest spread: Precision does not help when the estimates are centered in the wrong place: Y's estimates cluster around 0.052, not 0.08.
Correct: Method X: its estimates center at $p$ with little spread: Right: no bias and low variability together.
Method Z, because a person checks each box before choosing: Choosing by hand is not random; Z centers at 0.110 and has the largest spread.
Method W, because a smaller sample has less chance of error: A smaller random sample gives more variable estimates: W's standard deviation is twice X's.

4. A website poll asks visitors whether a new stadium should be built, and 1,200 visitors respond. A second poll with the same method gets 12,000 responses. What does the larger poll improve?

Both its bias and its variability
Its bias, but not its variability
Its variability, but not its bias
Neither, because website polls give no information

Show the answer

A larger sample reduces variability, so the second poll's result would vary less from poll to poll. Voluntary response bias comes from who chooses to answer, and a bigger sample of the same kind of people keeps that bias.

Both its bias and its variability: Bias comes from the method (people choose to respond). More responses of the same kind do not remove it.
Its bias, but not its variability: It is the other way round: sample size works on variability, and method works on bias.
Correct: Its variability, but not its bias: Right: more responses means less sample to sample variation, but the same voluntary response bias.
Neither, because website polls give no information: The larger poll does reduce variability; it still estimates the wrong thing because of bias.

5. An archer's arrows land in a tight group well to the left of the bullseye. If the bullseye is the parameter and each arrow is an estimate from one sample, the archer is like an estimator with

high bias and high variability
no bias and low variability
no bias and high variability
high bias and low variability

Show the answer

Tight grouping means low variability; a group centered away from the bullseye means bias.

high bias and high variability: The arrows are close together, so variability is low.
no bias and low variability: The group is centered to the left of the bullseye, so the estimator is biased.
no bias and high variability: This would be arrows scattered widely around the bullseye.
Correct: high bias and low variability: Right: consistent, but consistently off target.

6. A store manager wants the mean time customers wait in line. Each day for 100 days he times 25 randomly chosen customers and records their mean. The mean of his 100 sample means is very close to the true mean wait. This shows that the sample mean is

an unbiased estimator of the mean wait for all customers
equal to the true mean wait on each of the 100 days
a biased estimator, because the wait times vary from day to day
an estimator with no variability once 100 days are averaged

Show the answer

When the center of the distribution of a statistic matches the parameter, the statistic is an unbiased estimator. Each single sample mean still differs from the true mean.

Correct: an unbiased estimator of the mean wait for all customers: Right: its sampling distribution is centered at the parameter.
equal to the true mean wait on each of the 100 days: Single sample means vary from day to day; only their center matches the true mean.
a biased estimator, because the wait times vary from day to day: Variation from sample to sample is variability, not bias.
an estimator with no variability once 100 days are averaged: Averaging the 100 sample means shows where they center; each day's mean still varies.

Part 9 · Summary

Summary

A statistic used to estimate a parameter is an estimator, and its value from one sample is a point estimate. Judge an estimator by its sampling distribution: it is unbiased when that distribution is centered at the parameter, and it has low variability when the distribution is narrow. Random selection removes bias; larger random samples reduce variability. Neither fixes the other.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic