Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is

Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is approximately normal

Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is approximately normal

✅ The distribution is centered at

Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is approximately normal

✅ The distribution is centered at the population parameter (often interest of inference)

Quiz

Q - State the central limit theorem.

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is approximately normal

✅ The distribution is centered at the population parameter (often interest of inference)

✅ The variability of the distribution is inversely proportional to the square root of

Quiz

Q - Why do we care about the distribution of sample mean?

Quiz

Q - What is the distribution of sample mean by CLT?

When the population mean $\mu$ and the population standard deviation $\sigma$ are known,

Quiz

Q - What is the distribution of sample mean by CLT?

When the population mean $\mu$ and the population standard deviation $\sigma$ are known,

$$\overline{X}\sim N(\mu ,\sigma /\sqrt{n})\iff Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$$

approximately, for a large enough $n$.

Quiz

Q - What if $\sigma$ is unknown?

Quiz

Q - What if $\sigma$ is unknown?

• We approximate $\sigma$ with the sample standard deviation.

Quiz

Q - What if $\sigma$ is unknown?

• We approximate $\sigma$ with the sample standard deviation.

$$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}\to T=\frac{\overline{X}-\mu }{S/\sqrt{n}}$$ where $S^2={\sum }_{i=1}^n(X_i-\overline{X}{)}^2/(n-1)$

Quiz

Q - What if $\sigma$ is unknown?

• We approximate $\sigma$ with the sample standard deviation.

$$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}\to T=\frac{\overline{X}-\mu }{S/\sqrt{n}}$$ where $S^2={\sum }_{i=1}^n(X_i-\overline{X}{)}^2/(n-1)$

• $\sigma$ replaced by $S$! (The realized value of $S$ from a sample is $s$.)

Quiz

Q - List properties of the t-distribution.

• Its shape is

Quiz

Q - List properties of the t-distribution.

• Its shape is unimodal, symmetric, centered at 0 similarly to N(0,1).

Quiz

Q - List properties of the t-distribution.

• Its shape is unimodal, symmetric, centered at 0 similarly to N(0,1).

• Its tails are

Quiz

Q - List properties of the t-distribution.

• Its shape is unimodal, symmetric, centered at 0 similarly to N(0,1).

• Its tails are thicker than N(0,1).

Quiz

Q - List properties of the t-distribution.

• Its shape is unimodal, symmetric, centered at 0 similarly to N(0,1).

• Its tails are thicker than N(0,1).

• It is fully defined by

Quiz

Q - What is an appropriate code to calculate $P(T<1.2)$ where $T\sim t_5$?

Quiz

Q - What is an appropriate code to calculate $P(-2<T<3)$ where $T\sim t_{10}$?

Quiz

Q - What is an appropriate code to find q s.t. $P(T>q)=0.25$ where $X\sim t_7$?

Quiz: HT

Let's conduct a hypothesis test for $H_0:\mu =5$ vs. $H_1:\mu \ne 5$. We don't know the population standard deviation. We have a random sample of size 100. The CLT conditions are checked.

Q - What is the test statistic and its null distribution by CLT?

Quiz: HT

Let's conduct a hypothesis test for $H_0:\mu =5$ vs. $H_1:\mu \ne 5$. We don't know the population standard deviation. We have a random sample of size 100. The CLT conditions are checked.

Q - What is the test statistic and its null distribution by CLT?

• The test statistic is calculated by $t=\frac{\overline{x}-5}{s/10}$.

Quiz: HT

Q - What does it mean that the test statistic is 3.5?

Quiz: CI

Q - What is the formula to obtain a $1-\alpha$ confidence interval for $\mu$ ?

Quiz: CI

Q - What is the formula to obtain a $1-\alpha$ confidence interval for $\mu$ ?

$$\overline{x}\pm t_{n-1}^{\ast }\times \frac{s}{\sqrt{n}}$$

where $t_{n-1}^{\ast }$ is a critical value that satisfies $P(T>t_{n-1}^{\ast })=\alpha /2$ for $T\sim t_{n-1}$.

Quiz: CI

Q - What is the formula to obtain a $1-\alpha$ confidence interval for $\mu$ ?

$$\overline{x}\pm t_{n-1}^{\ast }\times \frac{s}{\sqrt{n}}$$

where $t_{n-1}^{\ast }$ is a critical value that satisfies $P(T>t_{n-1}^{\ast })=\alpha /2$ for $T\sim t_{n-1}$.

Q - What is the R function to calculate $t_{n-1}^{\ast }$?

Quiz

Q - What is the function in the infer package to use for CLT-based inference when $\sigma$ is unknown?