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1. Sampling variability and sampling distributions

Warm-up

Three quick questions from earlier lessons.

Discover

Five classmates each survey 20 random students about a new rule. Will they all get the same percentage in favor?

No, the percentages will vary
Yes, if they all sample at random

Key idea

A statistic has a distribution of its own.

Step 110 samples: a few values of p̂, scattered.
Step 2100 samples: a shape appears.
Step 31,000 samples (scaled): a mound centered near p — the sampling distribution.

The sampling distribution of a statistic is its distribution over all possible random samples of the same size. Its center tells us about bias; its spread (standard error) shrinks as the sample size grows.

Worked example

Imagine 1,000 samples of 50 students and the p̂ of each.

A histogram of p̂ values

Plot the 1,000 values.

Centered near the true p

Where it is centered.

Narrower: less variability

With samples of 200 instead.

Your turn, guided

Your turn: by what factor does SD(x̄) change when $n$ goes from 25 to 100?

It is halved: factor

\tfrac{1}{2}

√25/√100.

Practice ★

Many random samples of size 161 are taken from the same population. Which quantity changes from sample to sample?
1. The sample proportion p̂ (a statistic)
2. The population proportion p (a parameter)
3. The sample size

Compared with samples of size 73, samples of size 292 give sample means that…
1. vary less around the population mean
2. vary more
3. are more biased

Watch out!

Practice ★★

What is the sampling distribution of the sample mean for samples of size 25?
1. The distribution of x̄ over all possible samples of size 25
2. The distribution of the 25 values in one sample
3. The distribution of the population

Exit ticket

Three questions to check you have got it.

Many random samples of size 161 are taken from the same population. Which quantity changes from sample to sample?
1. The sample proportion p̂ (a statistic)
2. The population proportion p (a parameter)
3. The sample size
Many random samples of size 224 are taken from the same population. Which quantity changes from sample to sample?
1. The sample proportion p̂ (a statistic)
2. The population proportion p (a parameter)
3. The sample size
Many random samples of size 490 are taken from the same population. Which quantity changes from sample to sample?
1. The sample proportion p̂ (a statistic)
2. The population proportion p (a parameter)
3. The sample size

Summary card

Sampling variability

Statistics vary from sample to sample.
Sampling distribution: a statistic over all samples.
Larger $n$ : less variability.

Watch out: data distribution vs sampling distribution.

2. The sampling distribution of a sample proportion

Warm-up

Three quick questions from earlier lessons.

Discover

In a city 40% of adults cycle. In random samples of 100, around which value will p̂ be centered?

0.40
40

Key idea

Mean, spread and shape of p̂.

For random samples of size

n

from a population with proportion

p

\mu_{\hat{p}} = p

and

\sigma_{\hat{p}} = \sqrt{\dfrac{p(1 - p)}{n}}

(if

n

is at most 10% of the population). The shape is approximately normal when

np \ge 10

and

n(1 - p) \ge 10

Worked example

$p = 0.4$ , $n = 100$ .

\mu_{\hat{p}} = 0.4

Center.

\sigma_{\hat{p}} = \sqrt{0.0024} \approx 0.049

√(0.4 × 0.6/100).

Approximately normal

40 ≥ 10 and 60 ≥ 10.

Your turn, guided

Your turn: $p = 0.5$ , $n = 100$ . Find $\sigma_{\hat{p}}$ .

0.05

√(0.25/100).

Practice ★

The population proportion is p = 8/10. Find the standard deviation of p̂ for random samples of size 83.
The population proportion is p = 2/10. Find the standard deviation of p̂ for random samples of size 227.

The population proportion is p = 5/10. What is the mean of the sampling distribution of p̂ for samples of size 351?

Watch out!

Practice ★★

The population proportion is p = 1/8. What is the smallest sample size n for which np ≥ 10 and n(1 − p) ≥ 10?
The population proportion is p = 1/2. What is the smallest sample size n for which np ≥ 10 and n(1 − p) ≥ 10?

Boss problem. In a population, p = 8/10. Random samples of 100 are taken.

Exit ticket

Three questions to check you have got it.

The population proportion is p = 8/10. Find the standard deviation of p̂ for random samples of size 83.
The population proportion is p = 2/10. Find the standard deviation of p̂ for random samples of size 227.
The population proportion is p = 4/10. Find the standard deviation of p̂ for random samples of size 394.

Summary card

Sampling distribution of p̂

$\mu = p$ , $\sigma = \sqrt{p(1 - p)/n}$ ; normal if $np, n(1 - p) \ge 10$ .

Watch out: proportion, not count.

3. The sampling distribution of a sample mean

Warm-up

Three quick questions from earlier lessons.

Discover

Heights have SD 10 cm. Is the SD of the mean of 25 random heights also 10 cm?

No, it is 2 cm
Yes

Key idea

Mean and spread of x̄.

For random samples of size

n

from a population with mean

\mu

and SD

\sigma

\mu_{\bar{x}} = \mu

and

\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}

. If the population is normal,

\bar{x}

is exactly normal.

Worked example

$\mu = 170$ , $\sigma = 10$ , $n = 25$ . How unusual is $\bar{x} = 174$ ?

\sigma_{\bar{x}} = 2

10/5.

z = 2

(174 − 170)/2.

Unusual (about 2.5% are higher)

Beyond 2 SDs.

Your turn, guided

Your turn: $\sigma = 12$ , $n = 16$ . Find $\sigma_{\bar{x}}$ .

3

12/4.

Practice ★

A population has standard deviation σ = 9. Find the SD of the sample mean for samples of size 49.
A population has standard deviation σ = 29. Find the SD of the sample mean for samples of size 4.
A population has standard deviation σ = 22. Find the SD of the sample mean for samples of size 64.

Watch out!

Practice ★★

Samples of size 26 give a certain SD of x̄. By what factor must the sample size be multiplied to halve it?

A population has mean 76 and SD 5. A random sample of 4 has mean 77. Find the z-score of this sample mean.
A population has mean 62 and SD 14. A random sample of 16 has mean 69. Find the z-score of this sample mean.

Exit ticket

Three questions to check you have got it.

A population has standard deviation σ = 9. Find the SD of the sample mean for samples of size 49.
A population has standard deviation σ = 29. Find the SD of the sample mean for samples of size 4.
A population has standard deviation σ = 22. Find the SD of the sample mean for samples of size 64.

Summary card

Sampling distribution of x̄

$\mu_{\bar{x}} = \mu$ , $\sigma_{\bar{x}} = \sigma/\sqrt{n}$ .

Watch out: divide by $\sqrt{n}$ , not $n$ .

4. The central limit theorem

Warm-up

Three quick questions from earlier lessons.

Discover

Waiting times at a clinic are strongly right-skewed. You average the times of 50 random patients, many times over. What shape do these averages have?

Approximately normal
Right-skewed

Key idea

Averages tend to be normal.

Central limit theorem: for random samples of size

n

from any population with finite SD, the sampling distribution of

\bar{x}

is approximately normal when

n

is large (a common guide:

n \ge 30

, more for strong skew). It is about the distribution of sample means, not of the data.

Worked example

Three situations.

Any

n

\bar{x}

is normal

Normal population.

\bar{x}

approximately normal (CLT)

Skewed population, n = 60.

\bar{x}

still skewed: CLT does not apply yet

Skewed population, n = 4.

Your turn, guided

Your turn: a population with $\sigma = 15$ and $n = 36$ . The SD of the (approximately normal) $\bar{x}$ .

2.5

15/6.

Practice ★

Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 52 households?
1. Approximately normal, by the central limit theorem
2. Right-skewed like the population
3. Uniform

A population is strongly skewed. Is the sampling distribution of x̄ for samples of size 3 approximately normal?
1. Not necessarily: n = 3 is too small for the CLT to apply to a strongly skewed population
2. Yes, sample means are always normal
3. No, sample means are never normal

Watch out!

Practice ★★

Which statement is the central limit theorem, for random samples of size 166?
1. The distribution of sample means is approximately normal
2. The data in a large sample are approximately normal
3. The population becomes normal when n is large

Exit ticket

Three questions to check you have got it.

Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 52 households?
1. Approximately normal, by the central limit theorem
2. Right-skewed like the population
3. Uniform
Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 73 households?
1. Approximately normal, by the central limit theorem
2. Right-skewed like the population
3. Uniform
Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 72 households?
1. Approximately normal, by the central limit theorem
2. Right-skewed like the population
3. Uniform

Summary card

Central limit theorem

Large $n$ : $\bar{x}$ approximately normal, whatever the population.

Watch out: it is about means, not about the data.

5. Sampling distributions of differences

Warm-up

Three quick questions from earlier lessons.

Discover

Two independent sample means have SDs 3 and 4. What is the SD of their difference?

Key idea

Differences between independent samples.

For independent samples:

\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2

\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}

; and

\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}

Worked example

$p_1 = 0.5$ , $p_2 = 0.3$ , $n_1 = n_2 = 100$ .

\mu = 0.2

Center.

\sigma = \sqrt{0.0046} \approx 0.068

0.0025 + 0.0021.

Your turn, guided

Your turn: $\sigma_1^2/n_1 = 9$ and $\sigma_2^2/n_2 = 16$ . Find $\sigma_{\bar{x}_1 - \bar{x}_2}$ .

5

√25.

Practice ★

Independent samples of size 4 come from populations with SDs 36 and 48. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 18 and 24. Find the SD of x̄₁ − x̄₂.

Independent random samples of 100 are taken from two populations with proportions 8/10 and 1/10. Find the SD of p̂₁ − p̂₂.

Watch out!

Practice ★★

When are the variances of x̄₁ and x̄₂ added to get the variance of x̄₁ − x̄₂ (samples of 66)?
1. When the two samples are independent
2. Only when the sample sizes are equal
3. Always, even for paired measurements

Exit ticket

Three questions to check you have got it.

Independent samples of size 4 come from populations with SDs 36 and 48. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 18 and 24. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 54 and 72. Find the SD of x̄₁ − x̄₂.

Summary card

Differences

Independent samples: means subtract, variances add.

Watch out: paired data are not independent.

6. 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?

No, it is biased
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 (

\hat{p}

and

\bar{x}

are unbiased). Its variability shrinks as

n

grows. The SD formulas assume independence; when sampling without replacement, keep

n

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.

250

10% of 2,500.

Practice ★

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

A school has 1200 students. Sampling without replacement, what is the largest sample size that meets the 10% condition?
A school has 5500 students. Sampling without replacement, what is the largest sample size that meets the 10% condition?

Watch out!

Practice ★★

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.

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
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
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 $n$ .
10% condition for independence.

Watch out: large does not mean unbiased.

Answers: Sampling distributions

1. Sampling variability and sampling distributions

The sample proportion p̂ (a statistic)

vary less around the population mean

The distribution of x̄ over all possible samples of size 25

The sample proportion p̂ (a statistic)
The sample proportion p̂ (a statistic)
The sample proportion p̂ (a statistic)

2. The sampling distribution of a sample proportion

$\frac{2 \sqrt{83}}{415}$
$\frac{2 \sqrt{227}}{1135}$

$\frac{1}{2}$

$80$
$20$

$\frac{5}{4}$

$\frac{2 \sqrt{83}}{415}$
$\frac{2 \sqrt{227}}{1135}$
$\frac{\sqrt{591}}{985}$

3. The sampling distribution of a sample mean

$\frac{9}{7}$
$\frac{29}{2}$
$\frac{11}{4}$

$\frac{2}{5}$
$2$

$\frac{9}{7}$
$\frac{29}{2}$
$\frac{11}{4}$

4. The central limit theorem

Approximately normal, by the central limit theorem

Not necessarily: n = 3 is too small for the CLT to apply to a strongly skewed population

The distribution of sample means is approximately normal

Approximately normal, by the central limit theorem
Approximately normal, by the central limit theorem
Approximately normal, by the central limit theorem

5. Sampling distributions of differences

$30$
$15$

$\frac{1}{20}$

When the two samples are independent

$30$
$15$
$45$

6. Bias, variability and the 10% condition

The mean of its sampling distribution equals p

$120$
$550$

Biased, with low variability

The mean of its sampling distribution equals p
The mean of its sampling distribution equals p
The mean of its sampling distribution equals p