Motivation Example
Probability Review
z-Test
t-test
Examples
Hypothesis Testing
Is all about using
Normal behavior to reject
What’s not so normal!
The weight of new born babies is an important
indicator for the baby’s health.
Assumption: Newborn babies who weigh more
than the average weight of newborns are considered healthy.
Tell me, what should I do to check if my newborn brother is healthy?
:::
In this chapter, we aim to test simple hypothesis testing of the form: \[\begin{cases}\color{red}{H_0}:\mu=\color{blue}{\mu_0}\\{ }\\ \color{green}{H_1}: \mu\underset{<}{\overset{>}{\neq}} \color{blue}{\mu_0},\end{cases}\]
For any significance level \(\color{red}{\alpha}\in(0,1)\), often be \(0.05\), we aim to design a rule to reject \(\color{red}{H_0}\) under \(\color{red}{\text{Type I Error}}=\mathbb{P}(\color{red}{\text{Reject }H_0}|H_0\text{ is True})\leq \color{red}{\alpha}\).
To this goal, we need some tools from Probability!
Practice: I would open a ☕ coffee shop at a location if the sell count is at least \(500\) cups per day. Write the null and alternative hypotheses.
\(\omega\) | \(X\) |
---|---|
TTT | \(0\) |
HTT, THT, TTH | \(1\) |
HHT, HTH, HHT | \(2\) |
HHH | \(3\) |
\(\omega\) | \(X\) | \(p_k\) |
---|---|---|
TTT | \(0\) | \(1/8\) |
HTT, THT, TTH | \(1\) | \(3/8\) |
HHT, HTH, HHT | \(2\) | \(3/8\) |
HHH | \(3\) | \(1/8\) |
\(\omega\) | \(X\) | \(p_k\) |
---|---|---|
TTT | \(0\) | \(1/8\) |
HTT, THT, TTH | \(1\) | \(3/8\) |
HHT, HTH, HHT | \(2\) | \(3/8\) |
HHH | \(3\) | \(1/8\) |
\(X=k\) | \(p_k=\mathbb{P}(X=k)\) |
---|---|
\(\color{green}{1}\) | \(\color{green}{p}\) |
\(\color{red}{0}\) | \(\color{red}{1-p}\) |
\(X\) | \(0\) | \(1\) | \(2\) | \(\dots\) | \(n\) |
---|---|---|---|---|---|
\(p_k\) | \((1-p)^n\) | \(np(1-p)^{n-1}\) | \(\frac{n(n-1)}{2}p^2(1-p)^{n-2}\) | \(\dots\) | \(p^n\) |
X | \(0\) | \(1\) | \(2\) | \(\dots\) | \(n\) |
---|---|---|---|---|---|
\(p_k\) | \((1-p)^n\) | \(np(1-p)^{n-1}\) | \(\frac{n(n-1)}{2}p^2(1-p)^{n-2}\) | \(\dots\) | \(p^n\) |
X | \(0\) | \(1\) | \(2\) | \(3\) | \(\dots\) |
---|---|---|---|---|---|
\(p_k\) | \(e^{-\lambda}\) | \(e^{-\lambda}\lambda\) | \(e^{-\lambda}\frac{\lambda^2}{2!}\) | \(e^{-\lambda}\frac{\lambda^3}{3!}\) | \(\dots\) |
X | \(0\) | \(1\) | \(2\) | \(3\) | \(\dots\) |
---|---|---|---|---|---|
\(p_k\) | \(e^{-\lambda}\) | \(e^{-\lambda}\lambda\) | \(e^{-\lambda}\frac{\lambda^2}{2!}\) | \(e^{-\lambda}\frac{\lambda^3}{3!}\) | \(\dots\) |
\[\color{red}{\mathbb{P}(a<X<b)=\mathbb{P}(a\leq X<b)=\mathbb{P}(a<X\leq b)=\mathbb{P}(a\leq X\leq b)=\int_{a}^bf_X(x)dx.}\]
Gender | Age | Height | Weight | |
---|---|---|---|---|
1 | M | 74 | 116 | 18 |
2 | M | 69 | 120 | 23 |
3 | M | 72 | 121 | 25 |
Standard Normal & The main property
A normal RV with mean \(\mu=0\) and variance \(\sigma^2=1\), denoted by \(X\sim{\cal N}(0,1)\), is called “Standard Normal Random Variable”.
If \(X_1,\dots, X_n\sim{\cal N}(\color{green}{\mu},\color{red}{\sigma^2})\) are Independent and Identically Distributed (iid) and \(\overline{X}_n=\sum_{i=1}^nX_i/n\) then \[Z=\frac{\overline{X}_n-\color{green}{\mu}}{\color{red}{\sigma}/\sqrt{n}}\sim {\cal N}(0,1).\]
Standard Normal & The main property
A normal RV with mean \(\mu=0\) and variance \(\sigma^2=1\), denoted by \(X\sim{\cal N}(0,1)\), is called “Standard Normal Random Variable”.
If \(X_1,\dots, X_n\sim{\cal N}(\color{green}{\mu},\color{red}{\sigma^2})\) are Independent and Identically Distributed (iid) and \(\overline{X}_n=\sum_{i=1}^nX_i/n\) then \[Z=\frac{\overline{X}_n-\color{green}{\mu}}{\color{red}{\sigma}/\sqrt{n}}\sim {\cal N}(0,1).\]
If \(Z\sim{\cal N}(0,1)\), use \(z\)-table to compute:
And now, we have enough probability tools!
Z-test statistics & Rejection Region
Z-test Summary (\(S_>\))
Z-test Summary (\(S_<\))
Z-test Summary (\(S_\neq\), two-sided test)
Source: Birth weight of newborns delivered in France 2016, Statistia.
If \(T\sim{\cal T}(\text{df})\), use \(t\)-table to estimate:
For the followings, df \(=23\):
\(t\)-Test Summary (\(T_>\))
\(t\)-Test Summary (\(T_<\))
\(t\)-Test Summary (\(T_\neq\))