In-person
session 6

September 26, 2022

PMAP 8521: Program evaluation
Andrew Young School of Policy Studies

1 / 46

Plan for today2 / 46

Plan for today

Exam 1

2 / 46

Plan for today

Exam 1

FAQs

2 / 46

Plan for today

Exam 1

FAQs

Confidence intervals, credible intervals,
and a crash course on Bayesian statistics

2 / 46

Exam 13 / 46

Tell us about Exam 1!

4 / 46

FAQs5 / 46

Are p-values really misinterpreted
in published research?

6 / 46

Power calculations and sample size

Won't we always be able to find
a significant effect if the
sample size is big enough?

Yes!

7 / 46

Math with computers

andhs.co/live

8 / 46

Are the results from
p-hacking actually a
threat to validity?

9 / 46

https://projects.fivethirtyeight.com/p-hacking/

Do people actually post
their preregistrations?

10 / 46

Yes!

OSF

See this and this for examples

As Predicted

See this

11 / 46

Do you have any tips for identifying the
threats to validity in articles since
they're often not super clear?

Especially things like spillovers,
Hawthorne effects, and John Henry effects?

12 / 46

Using a control group of some kind
seems to be the common fix
for all of these issues.

What happens if you can't do that?
Is the study just a lost cause?

13 / 46

That's the point of DAGs and quasi experiments; simulate having treatment and control groups

Confidence intervals,
credible intervals,
and a crash course on Bayesian statistics14 / 46

In the absence of p-values,
I'm confused about how
we report… significance?

15 / 46

Imbens and p-values

Nobody really cares about p-values

16 / 46

Imbens and p-values

Nobody really cares about p-values

Decision makers want to know
a number or a range of numbers—
some sort of effect and uncertainty

16 / 46

Imbens and p-values

Nobody really cares about p-values

Decision makers want to know
a number or a range of numbers—
some sort of effect and uncertainty

Nobody cares how likely a number would be
in an imaginary null world!

16 / 46

Imbens's solution

Report point estimates and some sort of range

"It would be preferable if reporting standards emphasized confidence intervals or standard errors, and, even better, Bayesian posterior intervals."

17 / 46

Imbens's solution

Report point estimates and some sort of range

"It would be preferable if reporting standards emphasized confidence intervals or standard errors, and, even better, Bayesian posterior intervals."

Point estimate

The single number you calculate
(mean, coefficient, etc.)

Uncertainty

A range of possible values

17 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

18 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

Greek

Letters like $\beta_1$ are the truth

Letters with extra markings like $\hat{\beta_1}$ are our estimate of the truth based on our sample

18 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

Greek

Letters like $\beta_1$ are the truth

Letters with extra markings like $\hat{\beta_1}$ are our estimate of the truth based on our sample

Latin

Letters like $X$ are actual data from our sample

Letters with extra markings like $\bar{X}$ are calculations from our sample

18 / 46

Estimating truth

Data → Calculation → Estimate → Truth

19 / 46

Estimating truth

Data → Calculation → Estimate → Truth

Data	$X$
Calculation	$\bar{X} = \frac{\sum{X}}{N}$
Estimate	$\hat{\mu}$
Truth	$\mu$

19 / 46

Estimating truth

Data → Calculation → Estimate → Truth

Data	$X$
Calculation	$\bar{X} = \frac{\sum{X}}{N}$
Estimate	$\hat{\mu}$
Truth	$\mu$

$\bar{X} = \hat{\mu}$

$X \rightarrow \bar{X} \rightarrow \hat{\mu} \xrightarrow{\text{🤞 hopefully 🤞}} \mu$

19 / 46

Population parameter

Truth = Greek letter

An single unknown number that is true for the entire population

20 / 46

Population parameter

Truth = Greek letter

An single unknown number that is true for the entire population

Proportion of left-handed students at GSU

Median rent of apartments in NYC

Proportion of red M&Ms produced in a factory

ATE of your program

20 / 46

Samples and estimates

We take a sample and make a guess

21 / 46

Samples and estimates

We take a sample and make a guess

This single value is a point estimate

(This is the Greek letter with a hat)

21 / 46

Variability

You have an estimate,
but how different might that
estimate be if you take another sample?

22 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

3 are left-handed. Is that surprising?

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

3 are left-handed. Is that surprising?

40 are left-handed. Is that surprising?

23 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

24 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

Confidence interval is a net

24 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

Confidence interval is a net

We can be X% confident that our net is
picking up that population parameter

If we took 100 samples, at least 95 of them would have the
true population parameter in their 95% confidence intervals

24 / 46

A city manager wants to know the true average property value of single-value homes in her city. She takes a random sample of 200 houses and builds a 95% confidence interval. The interval is ($180,000, $300,000).

25 / 46

A city manager wants to know the true average property value of single-value homes in her city. She takes a random sample of 200 houses and builds a 95% confidence interval. The interval is ($180,000, $300,000).

We're 95% confident that the
interval ($180,000, $300,000)
captured the true mean value

25 / 46

WARNING26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

People do this all the time! People with PhDs!

26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

People do this all the time! People with PhDs!

YOU will try to do this too

26 / 46

OpenIntro Stats p. 186

First, notice that the statements are always about the population parameter, which considers all American adults for the energy polls or all New York adults for the quarantine poll.

We also avoided another common mistake: incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability. Making a probability interpretation is a common error: while it might be useful to think of it as a probability, the confidence level only quantifies how plausible it is that the parameter is in the given interval.

Another important consideration of confidence intervals is that they are only about the population parameter. A confidence interval says nothing about individual observations or point estimates. Confidence intervals only provide a plausible range for population parameters.

Nets

If you took lots of samples,
95% of their confidence intervals
would have the single true value in them

27 / 46

28 / 46

Frequentism

This kind of statistics is called "frequentism"

29 / 46

Frequentism

This kind of statistics is called "frequentism"

The population parameter θ is fixed and singular
while the data can vary

$P(\text{Data} \mid \theta)$

29 / 46

Frequentism

This kind of statistics is called "frequentism"

The population parameter θ is fixed and singular
while the data can vary

$P(\text{Data} \mid \theta)$

You can do an experiment over and over again;
take more and more samples and polls

29 / 46

Frequentist confidence intervals

"We are 95% confident that this net
captures the true population parameter"

30 / 46

Frequentist confidence intervals

"We are 95% confident that this net
captures the true population parameter"

"There's a 95% chance that the
true value falls in this range"

30 / 46

Weekends and
restaurant scores

31 / 46

Bayesian statistics

$P(\theta \mid \text{Data})$

$\color{orange}{P(\text{H} \mid \text{E})} = \frac{\color{red}{P(\text{H})} \times\color{blue}{P(\text{E} \mid \text{H})}}{\color{black}{P(\text{E})}}$

32 / 46

Bayesianism in WWII

33 / 46

$\color{orange}{P(\text{H} \mid \text{E})} = \frac{\color{red}{P(\text{H})} \times\color{blue}{P(\text{E} \mid \text{H})}}{\color{black}{P(\text{E})}}$

$\color{orange}{P(\text{Hypothesis} \mid \text{Evidence})} =$

$\frac{ \color{red}{P(\text{Hypothesis})} \times \color{blue}{P(\text{Evidence} \mid \text{Hypothesis})} }{ \color{black}{P(\text{Evidence})} }$

34 / 46

35 / 46

36 / 46

Bayesian statistics and
more complex questions

37 / 46

38 / 46

But the math is too hard!

So we simulate!

(Monte Carlo Markov Chains, or MCMC)

39 / 46

Weekends and
restaurant scores again

40 / 46

Bayesianism and parameters

In the world of frequentism,
there's a fixed population parameter
and the data can hypothetically vary

$P(\text{Data} \mid \theta)$

41 / 46

Bayesianism and parameters

In the world of frequentism,
there's a fixed population parameter
and the data can hypothetically vary

$P(\text{Data} \mid \theta)$

In the world of Bayesianism,
the data is fixed (you collected it just once!)
and the population parameter can vary

$P(\theta \mid \text{Data})$

41 / 46

In frequentism land, the parameter is fixed and singular and the data can vary - you can do an experiment over and over again, take more and more samples and polls

In Bayes land, the data is fixed (you collected it, that's it), and the parameter can vary

Bayesian credible intervals

(AKA posterior intervals)

"Given the data, there is a 95% probability
that the true population parameter
falls in the credible interval"

42 / 46

a Bayesian statistician would say “given our observed data, there is a 95% probability that the true value of θ falls within the credible region” while a Frequentist statistician would say “there is a 95% probability that when I compute a confidence interval from data of this sort, the true value of θ will fall within it”. (https://freakonometrics.hypotheses.org/18117)

Note how this drastically improve the interpretability of the Bayesian interval compared to the frequentist one. Indeed, the Bayesian framework allows us to say “given the observed data, the effect has 95% probability of falling within this range”, compared to the less straightforward, frequentist alternative (the 95% Confidence* Interval) would be “there is a 95% probability that when computing a confidence interval from data of this sort, the effect falls within this range”. (https://easystats.github.io/bayestestR/articles/credible_interval.html)

Intervals

Frequentism

There's a 95% probability
that the range contains the true value

Probability of the range

Few people naturally
think like this

Bayesianism

There's a 95% probability
that the true value falls in this range

Probability of the actual value

People do naturally
think like this!

43 / 46

There's a 95% probability that the range contains the true value (freq) - We are 95% confident that this net captures the true population parameter vs. There's a 95% probability that the the true value falls in this range (bayes)

This is a minor linguistic difference but it actually matters a lot! With frequentism, you have a range of possible values - you don't really know the true parameter, but it's in that range somewhere. Could be at the very edge, could be in the middle. With Bayesianism, you focus on the parameter itself, which has a distribution around it. It could be on the edge, but is most likely in the middle

Probability of range boundaries vs probability of parameter values

Bayesian p-value = probability that it's greater than 0 - you can say that there's a 100% chance that the coefficient is not zero, no more null worlds!

Thinking Bayesianly

We all think Bayesianly,
even if you've never heard of Bayesian stats

Every time you look at a confidence interval, you inherently think that the parameter is around that value, but that's wrong!

44 / 46

Thinking Bayesianly

We all think Bayesianly,
even if you've never heard of Bayesian stats

Every time you look at a confidence interval, you inherently think that the parameter is around that value, but that's wrong!

BUT Imbens cites research that
that's actually generally okay

Often credible intervals are super similar to confidence intervals

44 / 46

Bayesian inference

What do you do without p-values then?

45 / 46

Bayesian inference

What do you do without p-values then?

Probability
of direction

45 / 46

Bayesian inference

What do you do without p-values then?

Probability
of direction

Region of practical
equivalence (ROPE)

45 / 46

Weekends and
restaurant scores
once more

46 / 46

In-person
session 6

September 26, 2022

PMAP 8521: Program evaluation
Andrew Young School of Policy Studies

1 / 46

Plan for today2 / 46

Plan for today

Exam 1

2 / 46

Plan for today

Exam 1

FAQs

2 / 46

Plan for today

Exam 1

FAQs

Confidence intervals, credible intervals,
and a crash course on Bayesian statistics

2 / 46

Exam 13 / 46

Tell us about Exam 1!

4 / 46

FAQs5 / 46

Are p-values really misinterpreted
in published research?

6 / 46

Power calculations and sample size

Won't we always be able to find
a significant effect if the
sample size is big enough?

Yes!

7 / 46

Math with computers

andhs.co/live

8 / 46

Are the results from
p-hacking actually a
threat to validity?

9 / 46

https://projects.fivethirtyeight.com/p-hacking/

Do people actually post
their preregistrations?

10 / 46

Yes!

OSF

See this and this for examples

As Predicted

See this

11 / 46

Do you have any tips for identifying the
threats to validity in articles since
they're often not super clear?

Especially things like spillovers,
Hawthorne effects, and John Henry effects?

12 / 46

Using a control group of some kind
seems to be the common fix
for all of these issues.

What happens if you can't do that?
Is the study just a lost cause?

13 / 46

That's the point of DAGs and quasi experiments; simulate having treatment and control groups

Confidence intervals,
credible intervals,
and a crash course on Bayesian statistics14 / 46

In the absence of p-values,
I'm confused about how
we report… significance?

15 / 46

Imbens and p-values

Nobody really cares about p-values

16 / 46

Imbens and p-values

Nobody really cares about p-values

Decision makers want to know
a number or a range of numbers—
some sort of effect and uncertainty

16 / 46

Imbens and p-values

Nobody really cares about p-values

Decision makers want to know
a number or a range of numbers—
some sort of effect and uncertainty

Nobody cares how likely a number would be
in an imaginary null world!

16 / 46

Imbens's solution

Report point estimates and some sort of range

"It would be preferable if reporting standards emphasized confidence intervals or standard errors, and, even better, Bayesian posterior intervals."

17 / 46

Imbens's solution

Report point estimates and some sort of range

"It would be preferable if reporting standards emphasized confidence intervals or standard errors, and, even better, Bayesian posterior intervals."

Point estimate

The single number you calculate
(mean, coefficient, etc.)

Uncertainty

A range of possible values

17 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

18 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

Greek

Letters like $\beta_1$ are the truth

Letters with extra markings like $\hat{\beta_1}$ are our estimate of the truth based on our sample

18 / 46

Greek, Latin, and extra markings

Statistics: use a sample to make inferences about a population

Greek

Letters like $\beta_1$ are the truth

Letters with extra markings like $\hat{\beta_1}$ are our estimate of the truth based on our sample

Latin

Letters like $X$ are actual data from our sample

Letters with extra markings like $\bar{X}$ are calculations from our sample

18 / 46

Estimating truth

Data → Calculation → Estimate → Truth

19 / 46

Estimating truth

Data → Calculation → Estimate → Truth

Data	$X$
Calculation	$\bar{X} = \frac{\sum{X}}{N}$
Estimate	$\hat{\mu}$
Truth	$\mu$

19 / 46

Estimating truth

Data → Calculation → Estimate → Truth

Data	$X$
Calculation	$\bar{X} = \frac{\sum{X}}{N}$
Estimate	$\hat{\mu}$
Truth	$\mu$

$\bar{X} = \hat{\mu}$

$X \rightarrow \bar{X} \rightarrow \hat{\mu} \xrightarrow{\text{🤞 hopefully 🤞}} \mu$

19 / 46

Population parameter

Truth = Greek letter

An single unknown number that is true for the entire population

20 / 46

Population parameter

Truth = Greek letter

An single unknown number that is true for the entire population

Proportion of left-handed students at GSU

Median rent of apartments in NYC

Proportion of red M&Ms produced in a factory

ATE of your program

20 / 46

Samples and estimates

We take a sample and make a guess

21 / 46

Samples and estimates

We take a sample and make a guess

This single value is a point estimate

(This is the Greek letter with a hat)

21 / 46

Variability

You have an estimate,
but how different might that
estimate be if you take another sample?

22 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

3 are left-handed. Is that surprising?

23 / 46

Left-handedness

You take a random sample of
50 GSU students and 5 are left-handed.

If you take a different random sample of
50 GSU students, how many would you
expect to be left-handed?

3 are left-handed. Is that surprising?

40 are left-handed. Is that surprising?

23 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

24 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

Confidence interval is a net

24 / 46

Nets and confidence intervals

How confident are we that the sample
picked up the population parameter?

Confidence interval is a net

We can be X% confident that our net is
picking up that population parameter

If we took 100 samples, at least 95 of them would have the
true population parameter in their 95% confidence intervals

24 / 46

A city manager wants to know the true average property value of single-value homes in her city. She takes a random sample of 200 houses and builds a 95% confidence interval. The interval is ($180,000, $300,000).

25 / 46

A city manager wants to know the true average property value of single-value homes in her city. She takes a random sample of 200 houses and builds a 95% confidence interval. The interval is ($180,000, $300,000).

We're 95% confident that the
interval ($180,000, $300,000)
captured the true mean value

25 / 46

WARNING26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

People do this all the time! People with PhDs!

26 / 46

WARNING

It is way too tempting to say
“We’re 95% sure that the
population parameter is X”

People do this all the time! People with PhDs!

YOU will try to do this too

26 / 46

OpenIntro Stats p. 186

First, notice that the statements are always about the population parameter, which considers all American adults for the energy polls or all New York adults for the quarantine poll.

Nets

If you took lots of samples,
95% of their confidence intervals
would have the single true value in them

27 / 46

28 / 46

Frequentism

This kind of statistics is called "frequentism"

29 / 46

Frequentism

This kind of statistics is called "frequentism"

The population parameter θ is fixed and singular
while the data can vary

$P(\text{Data} \mid \theta)$

29 / 46

Frequentism

This kind of statistics is called "frequentism"

The population parameter θ is fixed and singular
while the data can vary

$P(\text{Data} \mid \theta)$

You can do an experiment over and over again;
take more and more samples and polls

29 / 46

Frequentist confidence intervals

"We are 95% confident that this net
captures the true population parameter"

30 / 46

Frequentist confidence intervals

"We are 95% confident that this net
captures the true population parameter"

"There's a 95% chance that the
true value falls in this range"

30 / 46

Weekends and
restaurant scores

31 / 46

Bayesian statistics

$P(\theta \mid \text{Data})$

$\color{orange}{P(\text{H} \mid \text{E})} = \frac{\color{red}{P(\text{H})} \times\color{blue}{P(\text{E} \mid \text{H})}}{\color{black}{P(\text{E})}}$

32 / 46

Bayesianism in WWII

33 / 46

$\color{orange}{P(\text{H} \mid \text{E})} = \frac{\color{red}{P(\text{H})} \times\color{blue}{P(\text{E} \mid \text{H})}}{\color{black}{P(\text{E})}}$

$\color{orange}{P(\text{Hypothesis} \mid \text{Evidence})} =$

$\frac{ \color{red}{P(\text{Hypothesis})} \times \color{blue}{P(\text{Evidence} \mid \text{Hypothesis})} }{ \color{black}{P(\text{Evidence})} }$

34 / 46

35 / 46

36 / 46

Bayesian statistics and
more complex questions

37 / 46

38 / 46

But the math is too hard!

So we simulate!

(Monte Carlo Markov Chains, or MCMC)

39 / 46

Weekends and
restaurant scores again

40 / 46

Bayesianism and parameters

In the world of frequentism,
there's a fixed population parameter
and the data can hypothetically vary

$P(\text{Data} \mid \theta)$

41 / 46

Bayesianism and parameters

In the world of frequentism,
there's a fixed population parameter
and the data can hypothetically vary

$P(\text{Data} \mid \theta)$

In the world of Bayesianism,
the data is fixed (you collected it just once!)
and the population parameter can vary

$P(\theta \mid \text{Data})$

41 / 46

In frequentism land, the parameter is fixed and singular and the data can vary - you can do an experiment over and over again, take more and more samples and polls

In Bayes land, the data is fixed (you collected it, that's it), and the parameter can vary

Bayesian credible intervals

(AKA posterior intervals)

"Given the data, there is a 95% probability
that the true population parameter
falls in the credible interval"

42 / 46

a Bayesian statistician would say “given our observed data, there is a 95% probability that the true value of θ falls within the credible region” while a Frequentist statistician would say “there is a 95% probability that when I compute a confidence interval from data of this sort, the true value of θ will fall within it”. (https://freakonometrics.hypotheses.org/18117)

Note how this drastically improve the interpretability of the Bayesian interval compared to the frequentist one. Indeed, the Bayesian framework allows us to say “given the observed data, the effect has 95% probability of falling within this range”, compared to the less straightforward, frequentist alternative (the 95% Confidence* Interval) would be “there is a 95% probability that when computing a confidence interval from data of this sort, the effect falls within this range”. (https://easystats.github.io/bayestestR/articles/credible_interval.html)

Intervals

Frequentism

There's a 95% probability
that the range contains the true value

Probability of the range

Few people naturally
think like this

Bayesianism

There's a 95% probability
that the true value falls in this range

Probability of the actual value

People do naturally
think like this!

43 / 46

Probability of range boundaries vs probability of parameter values

Bayesian p-value = probability that it's greater than 0 - you can say that there's a 100% chance that the coefficient is not zero, no more null worlds!

Thinking Bayesianly

We all think Bayesianly,
even if you've never heard of Bayesian stats

Every time you look at a confidence interval, you inherently think that the parameter is around that value, but that's wrong!

44 / 46

Thinking Bayesianly

We all think Bayesianly,
even if you've never heard of Bayesian stats

Every time you look at a confidence interval, you inherently think that the parameter is around that value, but that's wrong!

BUT Imbens cites research that
that's actually generally okay

Often credible intervals are super similar to confidence intervals

44 / 46

Bayesian inference

What do you do without p-values then?

45 / 46

Bayesian inference

What do you do without p-values then?

Probability
of direction

45 / 46

Bayesian inference

What do you do without p-values then?

Probability
of direction

Region of practical
equivalence (ROPE)

45 / 46

Weekends and
restaurant scores
once more

46 / 46

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In-personsession 6

Plan for today

Plan for today

Plan for today

Plan for today

Exam 1

FAQs

Confidence intervals,credible intervals,and a crash course on Bayesian statistics

Imbens and p-values

Imbens and p-values

Imbens and p-values

Imbens's solution

Imbens's solution

Greek, Latin, and extra markings

Greek, Latin, and extra markings

Greek, Latin, and extra markings

Estimating truth

Estimating truth

Estimating truth

Population parameter

Population parameter

Samples and estimates

Samples and estimates

Variability

Left-handedness

Left-handedness

Left-handedness

Left-handedness

Nets and confidence intervals

Nets and confidence intervals

Nets and confidence intervals

WARNING

WARNING

WARNING

WARNING

Nets

Frequentism

Frequentism

Frequentism

Frequentist confidence intervals

Frequentist confidence intervals

Bayesian statistics

Bayesianism in WWII

Bayesianism and parameters

Bayesianism and parameters

Bayesian credible intervals

Intervals

Thinking Bayesianly

Thinking Bayesianly

Bayesian inference

Bayesian inference

Bayesian inference

Plan for today

Help

Class 6: Validity + confidence intervals

In-personsession 6

Plan for today

Plan for today

Plan for today

Plan for today

Exam 1

FAQs

Confidence intervals,credible intervals,and a crash course on Bayesian statistics

Imbens and p-values

Imbens and p-values

Imbens and p-values

Imbens's solution

Imbens's solution

Greek, Latin, and extra markings

Greek, Latin, and extra markings

Greek, Latin, and extra markings

Estimating truth

Estimating truth

Estimating truth

Population parameter

Population parameter

Samples and estimates

Samples and estimates

Variability

Left-handedness

In-person
session 6

Confidence intervals,
credible intervals,
and a crash course on Bayesian statistics

In-person
session 6

Confidence intervals,
credible intervals,
and a crash course on Bayesian statistics