Material

🎥 Watch Central Limit Theorem

• Slides

Optional: 📖 Read IMS: Chapter 13 - Inference With Mathematical Models

Today's Goal

• Use Central Limit Theorem to define distribution of sample means
• Calculate probabilities from the normal distribution

Quantifying Variability

We can quantify the variability of sample statistics using different approaches:

• Simulation: via bootstrapping or "resampling" techniques

or

• Theory: via the Central Limit Theorem

Today we will focus on Theory.

Quiz

Q - What is a sampling distribution of the sample mean?

From one random sample of size $n$, calculate the sample mean ${\overline{X}}_1$

From a second random sample of size $n$, calculate the sample mean ${\overline{X}}_2$

$⋮$

Repeat this many times.

We call the distribution of $\overline{X}$ the sampling distribution.

Quiz

Q - Apply the central limit theorem (CLT) on sample means.

Let a random variable $X$ have a mean $\mu$ and standard deviation $\sigma$. Then the sampling distribution of the sample mean $\overline{X}=\frac{X_1+X_2+\cdots +X_n}{n}$

• Has the mean $\mu$

• Has the standard error $\sigma /\sqrt{n}$

• If the sample size $n$ is large enough, the sampling distribution of $\overline{X}$ is approximately normally distributed.

Quiz

Q - Describe density of a normal distribution.

Quiz

Q - The CLT holds only if $X\sim N(\mu ,\sigma )$. (T/F)

F

$X$ can be from any distribution with a mean $\mu$ and standard deviation $\sigma$.

Let's play with weird looking original distributions - Click!

Quiz

Q - What are the two conditions for CLT to hold?

Independence

• $\{X_1,\cdots ,X_n\}$ must be independent to one another
• One observation's value should not "influence" another observation's value.
• Rules of thumb to check independence:
  • Completely random sampling
  • If taken without replacement, $n$ should be less than 10% of the population size

Quiz

Q - What are the two conditions for CLT to hold?

Sample size $n$ / distribution

• If $X$ is numerical, $n>30$
• If $X$ is categorical, at least 10 successes and 10 failures
• If $X\sim N(\mu ,\sigma )$, then the distribution of sample means will also be exactly normal, regardless of the sample size.

Recap

• If certain assumptions are satisfied, regardless of the shape of the population distribution, the sampling distribution of the mean follows an approximately normal distribution.

• The center of the sampling distribution is at the center of the population distribution.

• The sampling distribution is less variable than the population distribution by a factor of $1/\sqrt{n}$.

• As $n$ increases, the standard error (the spread of the sampling distribution) decreases.

Quiz

Q - What is an appropriate code to calculate $P(Z<1.2)$ where $Z\sim N(0,1)$?

pnorm(1.2)

## [1] 0.8849303

Quiz

Q - What is an appropriate code to calculate $P(-2<X<7)$ where $X\sim N(1,3)$?

pnorm(7, mean = 1, sd = 3) - pnorm(-2, mean = 1, sd = 3)

## [1] 0.8185946

Quiz

Q - What is an appropriate code to find q s.t. $P(X>q)=0.05$ where $X\sim N(-1,2)$?

qnorm(0.05, mean = -1, sd = 2, lower.tail = FALSE)

## [1] 2.289707

Let's Practice Together!

Go to AE 17: Central Limit Theorem

Bulletin

• Watch videos for Prepare: June 7

• Project proposal feedback released

• Lab06 due Tuesday, June 7 at 11:59pm

• HW03 due Wednesday, June 8 at 11:59pm

• Submit ae17