Hypothesis Testing
z-test & t-test


INF-604: Data Analysis

Lecturer: Dr. Sothea HAS

Outline

  • Motivation Example

  • Probability Review

  • z-Test

  • t-test

  • Examples

Motivation Example

Motivation Example

Newborn baby weights

  • 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?

Motivation Example

Newborn baby weights

  • Yes! We should first estimate the average weight of newborn babies.
  • A collected sample of size \(n=100\) gives \(\overline{W}=3.5\) kg.
  • Can we trust this estimate?

  • If \(\mu\) is the true mean of baby weight, there’re 2 problems:
    • If \(\mu < 3.5\), healthy babies may be considered unhealthy.
    • If \(\mu > 3.5\), unhealthy babies may be considered healthy.
  • One of these two mistakes is much much more important than another! Which one (call it Type I Mistake)?

:::

Motivation Example

Hypothesis setting and goal

  • In this case, we want to build a decision rule that
    • Judges babies based on the estimate \(\overline{W}=3.5\) and
    • Controls the risk of committing Type I Mistake to be low.
  • Here is the corresponding hypothesis test: \[\begin{cases}\color{red}{H_0}:\mu\geq 3.5\\ \color{green}{H_1}: \mu<3.5,\end{cases}\] where
    • \(\color{red}{H_0}:\) Null hypothesis to be rejected with low risk of comitting Type I Mistake.
    • \(\color{green}{H_1}:\) Alternative hypothesis to be supported.

Motivation Example

Hypothesis setting and goal (summary)

  • 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 \(\alpha\in(0,1)\) (significance level, a common choice is \(0.05\)), we aim to design a rule to reject \(\color{red}{H_0}\) and guarantees that \(\mathbb{P}(\color{red}{\text{Reject }H_0}|H_0\text{ is True})\leq \alpha\).

  • To this goal, we need some tools from Probability!

  • Q1: I would open a ☕ coffee shop at a location if the sell count is at least \(500\) cups per day. Please design the null and alternative hypotheses.

Probability Review

Probability

Sample space & Events

  • Sample space: The set of all the outcomes of a random experiment.
  • Events: Subsets of the sample space \(\Omega\).
  • Example: If I toss a coin (H and T) three times, then
    • Sample space: \(\Omega=\{\) HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \(\}\).
    • Events:
      • A: “Head came out twice”?
      • B: “No head before tails”?
      • C: “Number of heads and tails are equal”?

Probability

Sample space & Events

  • Sample space: The set of all the outcomes of a random experiment.
  • Events: Subsets of the sample space \(\Omega\).
  • Example: If I toss a coin (H and T) three times, then
    • Sample space: \(\Omega=\{\) HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \(\}\).
    • Events:
      • A: “Head came out twice”, {HHT, HTH, THH}.
      • B: “No head before tails”, {THH, TTH, TTT}.
      • C: “Number of heads and tails are equal”, { } = ∅.

Probability

Probability on a fintie set

  • Probabilty on a finite set: if \(A\subset \Omega\) then \(\color{blue}{\mathbb{P}(A)=\frac{n(A)}{n(\Omega)}}.\)
  • Previous example: Coin toss (H and T) 3 times:
    • Sample space: \(n(\Omega)=8\).
    • Probability:
      • A: “Head came out twice”, \(\mathbb{P}(A)=3/8\).
      • B: “No head before tails”, \(\mathbb{P}(B)=3/8\).
      • C: “Number of heads and tails are equal”, \(\mathbb{P}(C)=0\).

Probability

Axioms of probability

  • Probability satisfies the following 3 axioms:
    • \(\mathbb{P}(A)\geq 0\) for all \(A\subset \Omega\).
    • \(\mathbb{P}(\Omega)=1\).
    • If \(A,B\subset\Omega\) are two disjoint events, then \[\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B).\]
  • Theorem: \(\color{blue}{A,B\subset\Omega: \mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)}.\)

  • Our example: \(\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)=3/8+3/8-1/8=5/8\).

Probability

Conditional Probability & Independency

  • If \(A,B\subset \Omega:\) with \(\mathbb{P}(B)>0\), the conditional probability of \(A\) given \(B\) is defined by: \[\color{blue}{\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}=\frac{n(A\cap B)}{n(B)}}.\]
  • Example: \(\mathbb{P}(A|B)=\frac{1/8}{3/8}=1/3\).
  • An event \(A\) is independent of \(B\), denoted by \(A\perp B\) if and only if: \[\color{blue}{\mathbb{P}(A|B)=\mathbb{P}(A)}.\]

Probability

Discrete Random Variable

  • Any function that maps outcomes to real numbers are called Random Variables (RV), often denoted by \(X,Y,Z,...\)
  • If \(X\) takes only discrete values (for example, in {0,1,2,3,…}), it’s called a Discrete RV.
  • In coin toss example, if \(X=\) number of heads, then \(X\in\{0,1,2,3\}\):
\(\omega\) \(X\)
TTT \(0\)
HTT, THT, TTH \(1\)
HHT, HTH, HHT \(2\)
HHH \(3\)
  • Compute probability for DRV: \[p_k=\mathbb{P}(X=k)=\mathbb{P}(\{\omega:X(\omega)=k\}).\]
  • Our example: \(\mathbb{P}(X=0)=1/8.\)
    • \(\mathbb{P}(X=1)=\mathbb{P}(X=2)=3/8.\)
    • \(\mathbb{P}(X=3)=1/8.\)

Probability

Discrete Random Variable

  • Any function that maps outcomes to real numbers are called Random Variables (RV), often denoted by \(X,Y,Z,...\)
  • If \(X\) takes only discrete values (for example, in {0,1,2,3,…}), it’s called a Discrete RV.
  • In coin toss example, if \(X=\) number of heads, then \(X\in\{0,1,2,3\}\):
\(\omega\) \(X\) \(p_k\)
TTT \(0\) \(1/8\)
HTT, THT, TTH \(1\) \(3/8\)
HHT, HTH, HHT \(2\) \(3/8\)
HHH \(3\) \(1/8\)
  • Compute probability for DRV: \[p_k=\mathbb{P}(X=k)=\mathbb{P}(\{\omega:X(\omega)=k\}).\]
  • Our example: \(\mathbb{P}(X=0)=1/8.\)
    • \(\mathbb{P}(X=1)=\mathbb{P}(X=2)=3/8.\)
    • \(\mathbb{P}(X=3)=1/8.\)

Probability

Graph: barplot

\(\omega\) \(X\) \(p_k\)
TTT \(0\) \(1/8\)
HTT, THT, TTH \(1\) \(3/8\)
HHT, HTH, HHT \(2\) \(3/8\)
HHH \(3\) \(1/8\)
  • 🔑 Remark: Qualitative data are often modeled using discrete random variables.
  • Ex: Gender, Time of day, Traffic Level,…

Probability

Expectation & variance of DRV

  • If \(X\) is a discrete random variable taking values in \(\{x_1,x_2,\dots,x_n\}\)
  • The Expectation of \(X\): \[\mu=\mathbb{E}(X)=\sum_{k=1}^{n}x_kp_k.\]
  • The Variance of \(X\): \[\sigma^2=\mathbb{V}(x)=\sum_{k=1}^np_k(x_k-\mu)^2.\]

  • Our example: \(X\) = Number of heads: \[\begin{align*}\mathbb{E}(X)&=\sum_{k=1}^{n}x_kp_k\\ &=0(1/8)+1(3/8)+2(3/8)+3(1/8)=1.5\\ \mathbb{V}(X)&=\sum_{k=1}^{n}p_k(x_n-1.5)^2\\ &=(0-1.5)^2(1/8)+(1-1.5)^2(3/8)+(2-1.5)^2(3/8)\\ &+(3-1.5)^2(1/8)=0.75.\end{align*}\]

Probability

Some well-known DRV: Bernoulli RV

  • For modeling a coin toss or any experiments with only two outcomes: sucess (1) or failure (0), happening with probability \(\color{green}{p}\) and \(\color{red}{1-p}\) respectively.
  • A Bernoulli RV \(X\) of parameter \(p\) is denoted by \(X\sim{\cal B}(p)\), and has the following Probability Mass Function (PMF):
\(X=k\) \(p_k=\mathbb{P}(X=k)\)
\(\color{green}{1}\) \(\color{green}{p}\)
\(\color{red}{0}\) \(\color{red}{1-p}\)
  • Q2: Check that if \(X\) is a Bernoulli RV of success probability \(p\) denoted by \(X\sim{\cal B}(p)\), then:
    • \(\mathbb{E}(X)=p\)
    • \(\mathbb{V}(X)=p(1-p).\)
  • Ex: Data that can be modeled using Bernoulli RV:
    • Gender: Male (1), female (0)
    • Disease: Positive (1), negative (0)…

Probability

Some well-known DRV: Binomial RV

  • For modeling the number of sucesses that occured in \(\color{red}{n}\) times of independent Bernoulli experiments of some parameter \(p\).
  • If \(X\) is a Binomial RV denoted by \(X\sim{\cal Bin}(n,p)\) then:
    • It takes values \(k\in\{0,1,...,n\}\) with
    • PMF: \(p_k=\mathbb{P}(X=k)=C(n,k)p^k(1-p)^{n-k}=\binom{n}{k}p^k(1-p)^{n-k},\) with
      • \(n!=n\times(n-1)\times\dots \times 1\), called “\(n\) factorial”, and
      • \(C(n,k)=\binom{n}{k}=\frac{n!}{(n-k)!k!}\) the number of ways to choose \(k\) objects at once (no order) from \(n\) objects.
\(X\) \(0\) \(1\) \(2\) \(\dots\) \(n\)
\(p_k\) \((1-p)^n\) \(np(1-p)^{n-1}\) \(\frac{n(n-1)}{2}p^2(1-p)^{n-2}\) \(\dots\) \(p^n\)

Probability

Barplot of binomial distribution

  • Table of distribution:
X \(0\) \(1\) \(2\) \(\dots\) \(n\)
\(p_k\) \((1-p)^n\) \(np(1-p)^{n-1}\) \(\frac{n(n-1)}{2}p^2(1-p)^{n-2}\) \(\dots\) \(p^n\)
  • Q3: Check that if \(X\sim{\cal Bin}(n,p)\), then:
    • \(\mathbb{E}(x)=np\)
    • \(\mathbb{V}(x)=np(1-p).\)
  • Ex: If the prob. of mistakenly transferring money in “$” instead of “៛” in each transaction is \(0.01\), what’s the probability of making \(5\) mistakes in 100 transactions?

Probability

Some well-known DRV: Poisson

  • For modeling the number of rare events such as the number of lost luggages at an airport.
  • It requires a key parameter \(\lambda\) which is called intensity parameter.
  • If \(X\) is a Poisson RV of parameter \(\lambda>0\) denoted by \(X\sim{\cal P}(\lambda)\) if and only if for each \(k=0,1,2,...\), \[p_k=\mathbb{P}(X=k)=e^{-\lambda}\frac{\lambda^{k}}{k!}\quad (e\approx 2.7182...)\]
  • We have the table of its distribution:
X \(0\) \(1\) \(2\) \(3\) \(\dots\)
\(p_k\) \(e^{-\lambda}\) \(e^{-\lambda}\lambda\) \(e^{-\lambda}\frac{\lambda^2}{2!}\) \(e^{-\lambda}\frac{\lambda^3}{3!}\) \(\dots\)

Probability

Barplot of Poisson distribution

X \(0\) \(1\) \(2\) \(3\) \(\dots\)
\(p_k\) \(e^{-\lambda}\) \(e^{-\lambda}\lambda\) \(e^{-\lambda}\frac{\lambda^2}{2!}\) \(e^{-\lambda}\frac{\lambda^3}{3!}\) \(\dots\)
  • Q4: Check that if \(X\sim{\cal P}(\lambda)\), then:
    • \(\mathbb{E}(x)=\lambda\)
    • \(\mathbb{V}(x)=\lambda.\)
  • Poisson variable can be used to model:
    • Number of eggs at a given farm/day
    • Number of cars sold/day
    • Number of mistakes in transections/day…

Probability

Continuous Random Variables

  • A RV \(X\) that takes continuous values are called Continuous RV (CRV).
  • Ex: Many things can be modeled using CRV:
    • Newborn baby weights 👶
    • Height of freshmen at AUPP 🫡
    • Time that the latest student arrives to my DA I class 😏…
  • How would we describe probability of \(\color{red}{X=x\in\mathbb{R}}\)?
  • Probability Mass Function (PMF) \[p_k=\mathbb{P}(X=k),k=0,1,...\]
  • It describes the probability that \(X\) takes value \(k\).
  • Probability Density Function (PDF) \[f_X(x)=\text{A function of variable }x\]
  • Roughly, it describes likelihood of \(X\) taking values around \(x\in\mathbb{R}\).

Probability

Continuous Random Variables

  • A RV \(X\) that takes continuous values are called Continuous RV (CRV).
  • Ex: Many things can be modeled using CRV:
    • Newborn baby weights 👶
    • Height of freshmen at AUPP 🫡
    • Time that the latest student arrives to my DA I class 😏…
  • How would we describe probability of \(\color{red}{X=x\in\mathbb{R}}\)?
  • Probability Mass Function (PMF)
  • Probability Density Function (PDF)

Probability

Properties of PDF

  • If \(f_X\) is the PDF of a CRV \(X\) taking values in \(S\subset\mathbb{R}\), then it satisfies:
    • Positivity: \(f_X(x)\geq 0\) for all \(x\in S\)
    • Total Area Under the Curve is 1: \(\int_{S}f_X(x)dx=1\).
  • Ex: Check that \(f(x)=x^2\) is a density of a RV defined on \(S=[0,\sqrt[3]{3}]\).
  • We have \(f(x)=x^2>0\) for all \(x\in [0,\sqrt[3]{3}]\).
  • Check total area under the curve: \[\begin{align*} \int_{S}f(x)dx&=\int_{0}^{\sqrt[3]{3}}x^2dx\\ &=\frac{1}{3}\left[x^3\right]_0^{\sqrt[3]{3}}\\ &= ((\sqrt[3]{3})^3-0^3)/3=1. \end{align*}\]

Probability

Compute Probability with PDF

  • Formula: if \(f_X\) is the density of a random variable \(X\) taking values in \(S\subset\mathbb{R}\), thus for any \(a,b\in S\) with \(a<b\), one has

\[\color{red}{\mathbb{P}(a<X<b)=\mathbb{P}(a\leq X<b)=\mathbb{P}(a<X\leq b)=\mathbb{P}(a\leq X\leq b)=\int_{a}^bf_X(x)dx.}\]

  • Remark: If \(X\) is a CRV, then \(\color{red}{\mathbb{P}(X=a)=0}\) for all \(a\in S\).
  • Ex: If \(X\) is a RV with PDF \(f(x)=x^2\), taking values in \(S=[0,\sqrt[3]{3}]\). Compute:
    • \(\mathbb{P}(0\leq X<\sqrt[3]{3}/2)\)
    • \(\mathbb{P}(\sqrt[3]{3}/2\leq X\leq \sqrt[3]{3})\)

Probability

Expectation and variance of CRV

  • If \(X\) is a CRV with PDF \(f_X\) then
    • Expectation: \(\mu=\mathbb{E}(X)=\int_{S}xf_X(x)dx.\)
    • Variance: \[\sigma^2=\mathbb{V}(X)=\mathbb{E}[(X-\mu)^2]=\int_{S}(x-\mu)^2f_X(x)dx.\]
  • Example: Check that if \(X\) is a CRV with PDF \(f_X(x)=x^2\) defined on \(S=[0,\sqrt[3]{3}]\), one has
    • \(\mathbb{E}(X)=\int_0^{\sqrt[3]{3}}xf_X(x)dx=\frac{3\sqrt[3]{3}}{4}.\)
    • \(\mathbb{V}(X)=\frac{3^{5/3}}{5}-\frac{3^{8/3}}{16}.\)

Probability

Normal/Gaussian Random Variable

  • For modeling natural measurements such as
    • Heights, weights, sizes…
    • Measurement errors…
  • A RV normal or Gaussian with mean \(\mu\) and variance \(\sigma^2\) is denoted by \(X\sim {\cal N}(\color{green}{\mu}, \color{red}{\sigma^2})\).
  • Its PDF is defined for all \(x\in\mathbb{R}\) by \[f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\color{green}{\mu})^2/(2\color{red}{\sigma^2})}.\]
  • We can check that
    • \(\mathbb{E}(X)=\color{green}{\mu}\) and \(\mathbb{V}(X)=\color{red}{\sigma^2}.\)

Probability

Normal/Gaussian Random Variable

  • Consider our French kid dataset:
Gender Age Height Weight
1 M 74 116 18
2 M 69 120 23
3 M 72 121 25

Probability

Standard Normal & Property

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 \[Z=\frac{\overline{X}_n-\color{green}{\mu}}{\color{red}{\sigma}/\sqrt{n}}\sim {\cal N}(0,1).\]

Probability

Standard Normal & Property

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 \[Z=\frac{\overline{X}_n-\color{green}{\mu}}{\color{red}{\sigma}/\sqrt{n}}\sim {\cal N}(0,1).\]

Probability

Standard Normal & Property

  • If \(Z\sim{\cal N}(0,1)\), then we know its range very well!

Probability

Standard Normal & Property

  • NO ONE CAN COMPUTE THE EXACT VALUE OF \(\mathbb{P}(a\leq Z\leq b)\) for all \(a<b\)!
  • People use the following standard normal \(z\) table:

Probability

Challenge 👉 Demo

If \(Z\sim{\cal N}(0,1)\), use \(z\)-table to compute:

  • \(\mathbb{P}(Z\leq 0)\)
  • \(\mathbb{P}(Z\leq -1)\)
  • \(\mathbb{P}(Z\leq 1.37)\)
  • \(\mathbb{P}(Z\leq 2.15)\)
  • \(\mathbb{P}(0.25< Z\leq 1.27)\)
  • \(\mathbb{P}(-1.23 \leq Z < 0.14)\)
  • \(\mathbb{P}(Z\geq 2.71)\)
  • \(\mathbb{P}(Z>1.96)\)
  • \(\mathbb{P}(Z<-0.7)\)
  • \(\mathbb{P}(|Z|\geq 1.96)\)
  • \(\mathbb{P}(|Z|> 2.5)\)
  • \(\mathbb{P}(|Z|< 1.37)\)
  • \(\mathbb{P}(|Z|\leq 2.15)\)
  • Find \(z_0\) such that \(\mathbb{P}(Z< z_0)=0.9\)
  • Find \(z_0\) such that \(\mathbb{P}(Z\geq z_0)=0.314\)
  • Find \(z_0\) such that \(\mathbb{P}(|Z|\geq z_0)=0.05\)
  • Find \(z_0\) such that \(\mathbb{P}(|Z|\leq z_0)=0.75\)
  • Find \(z_0\) such that \(\mathbb{P}(|Z|\leq z_0)=0\).



And now, we have enough probability tools!

\(z\)-Test

z-Test

Recall

  • Suppose a sample (weights): \(X_1,X_2,\dots, X_n\in\mathbb{R}\).
  • For any \(\color{blue}{\mu_0}\in\mathbb{R}\) we would like to test the following hypothesis: \[(S_>):\begin{cases}H_0: \text{The true mean }\mu=\color{blue}{\mu_0}\\ H_1: \text{The true mean }\mu>\color{blue}{\mu_0}. \end{cases}\]
  • In other words, we want to reject \(H_0\) with a small risk of committing Type I Error (= reject \(H_0\) while it’s true).
  • This can be done by deploying two important ideas:
    • The original data should be well behaved (Gaussian).
    • Based on the data, build a quantity that we can guess its range.

z-Test

Z-statistics and Rejection Region

Z-test statistics & Rejection Region

  • Key assumption: Assume that \(X_1,\dots,X_n\sim{\cal N}(\mu, \sigma^2)\) are iid with an unknown mean \(\mu\) and a known standard deviation \(\sigma>0\).
  • If \(H_0\) is True, then for any \(\color{red}{\alpha}\in (0,1)\) we have
    • The quantity: \(Z=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\sigma/\sqrt{n}}\sim{\cal N}(0,1)\) “z-value”.
    • If we choose \(\color{red}{z_{\alpha}}\) such that \(\mathbb{P}(Z>\color{red}{z_{\alpha}})=\color{red}{\alpha}\), then “if we reject \(H_0\) whenever \(Z>\color{red}{z_{\alpha}}\), it’s guaranteed that” \[\mathbb{P}(\color{red}{\text{Type I Error}})=\mathbb{P}(\color{red}{\text{Reject }}H_0|H_0\color{blue}{\text{ is True}})=\color{red}{\alpha}.\]

  • The set of \(\color{red}{R(\alpha)=\{z: z>z_{\alpha}\}}\) is called Rejection Region at level \(\alpha\), meaning that if \(Z\in\color{red}{R(\alpha)}\), we can reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).

z-Test

Z-test Summary for (\(S_>\), one-tailed)

Z-test Summary (\(S_>\))

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\sigma^2)\) with known \(\sigma^2\) and unknown \(\mu\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu=\color{blue}{\mu_0}\) against \(H_1:\mu>\color{blue}{\mu_0}\) we do:
    • Compute \(Z=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\sigma/\sqrt{n}}\)
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{z_\alpha}\) such that \(\mathbb{P}(Z\geq\color{red}{z_\alpha})=\color{red}{\alpha}\).
    • Decision:
      • If \(Z\geq\color{red}{z_\alpha}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(Z< \color{red}{z_\alpha}\), we CANNOT \(H_0\) due to insufficient evidence.

z-Test

Z-test Summary for (\(S_<\), one-tailed)

Z-test Summary (\(S_<\))

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\sigma^2)\) with known \(\sigma^2\) and unknown \(\mu\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu=\color{blue}{\mu_0}\) against \(H_1:\mu<\color{blue}{\mu_0}\) we do:
    • Compute \(Z=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\sigma/\sqrt{n}}\)
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{z_\alpha}\) such that \(\mathbb{P}(Z\leq\color{red}{z_\alpha})=\color{red}{\alpha}\).
    • Decision:
      • If \(Z\leq\color{red}{z_\alpha}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(Z> \color{red}{z_\alpha}\), we CANNOT reject \(H_0\) due to insufficient evidence.

  • Ex: (Baby weight) Test \(H_0:\mu=3.75\) against \(H_1:\mu<3.75\) at \(\alpha=0.05\), knowing that \(\sigma=0.5\) kg and \(\overline{W}=3.5\) kg (assuming that baby weights are normally distributed). How about when \(\overline{W}=2.9\) kg?

z-Test

Z-test Summary for (\(S_\neq\) Two-tailed)

Z-test Summary (\(S_\neq\), two-sided test)

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\sigma^2)\) with known \(\sigma^2\) and unknown \(\mu\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu=\color{blue}{\mu_0}\) against \(H_1:\mu\neq\color{blue}{\mu_0}\) we do:
    • Compute \(Z=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\sigma/\sqrt{n}}\)
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{z_{\alpha/2}}\) such that \(\mathbb{P}(|Z|\geq\color{red}{z_{\alpha/2}})=\color{red}{\alpha}\).
    • Decision:
      • If \(|Z|\geq\color{red}{z_{\alpha/2}}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(|Z|< \color{red}{z_{\alpha/2}}\), we CANNOT reject \(H_0\) due to insufficient evidence.

z-Test

Example: Newborn weights in France, 2016

  • Test \(\color{red}{\mu=3}\) kg against \(\color{green}{\mu\neq 3}\) kg at significance level \(\alpha=0.1\), given that \(n=745000,\overline{X}_n=3.25\) and \(\sigma^2=1.5\). How about with \(\color{blue}{\mu_0=3.75}\)?

Source: Birth weight of newborns delivered in France 2016, Statistia.

\(t\)-Test

t-Test

\(t\)-distribution

  • A Student or \(t\) RV is a CRV that behaves similarly to a Gaussian RV but with slightly heavier tails.
  • A \(t\) RV \(T\) with degree of freedom (df) \(k\) is often denoted by \(T\sim{\cal T}(k)\).
  • 🔑 As \(k\) increases, \({\cal T}(k)\to Z\sim{\cal N}(0,1)\).

t-Test

\(t\)-Table: tdistributiontable

  • Just like Standard normal, we use \(t\)-table to find \(\mathbb{P}(\leq T(k))\).

t-Test

\(t\)-Test Challenge 👉 Demo

If \(T\sim{\cal T}(\text{df})\), use \(t\)-table to estimate:

  • \(\mathbb{P}(T\leq 0)\) with df \(=15\).
  • \(\mathbb{P}(T\leq -1)\) with df \(=10\).
  • \(\mathbb{P}(T\leq 1.37)\) with df \(=15\).
  • \(\mathbb{P}(T\leq 2.15)\) with df \(=13\).
  • \(\mathbb{P}(0.25< T\leq 1.27)\) with df \(=30\).
  • \(\mathbb{P}(-1.23 \leq T < 0.14)\) with df \(=21\).
  • \(\mathbb{P}(T\geq 2.71)\) with df \(=10\).
  • \(\mathbb{P}(T>1.96)\) with df \(=25\).
  • \(\mathbb{P}(T<-0.7)\) with df \(=17\).

For the followings, df \(=23\):

  • \(\mathbb{P}(|T|\geq 2.1)\)
  • \(\mathbb{P}(|T|> 2.5)\)
  • \(\mathbb{P}(|T|< 1.37)\)
  • \(\mathbb{P}(|T|\leq 2.15)\)
  • Find \(t_0\) such that \(\mathbb{P}(T< t_0)=0.9\)
  • Find \(t_0\) such that \(\mathbb{P}(T\geq t_0)=0.35\)
  • Find \(t_0\) such that \(\mathbb{P}(|T|\geq t_0)=0.05\)
  • Find \(t_0\) such that \(\mathbb{P}(|T|\leq t_0)=0.75\)
  • Find \(t_0\) such that \(\mathbb{P}(|T|\leq t_0)=0\).

t-Test

\(t\)-Table for (\(T_>\) one-tailed)

  • When \(n\) is small (\(<30\)) and \(\sigma>0\) is unknown, we use \(t\)-distribution.

\(t\)-Test Summary (\(T_>\))

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\color{red}{\sigma^2})\) with unknown \(\mu\) and \(\color{red}{\sigma^2}\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu={\mu_0}\) against \(H_1:\mu>\color{blue}{\mu_0}\) we do:
    • Compute \(\color{red}{\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}_n)^2}\) (estimator of \(\color{red}{\sigma^2}\)), then \(T=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\color{red}{\hat{\sigma}}/\sqrt{n}}\)
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{t_\alpha}\) such that \(\mathbb{P}(T\geq\color{red}{t_\alpha})=\color{red}{\alpha}\).
    • Decision:
      • If \(T\geq\color{red}{t_\alpha}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(T< \color{red}{t_\alpha}\), we CANNOT \(H_0\) due to insufficient evidence.



t-Test

\(t\)-Table for (\(T_<\) one-tailed)

\(t\)-Test Summary (\(T_<\))

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\color{red}{\sigma^2})\) with unknown \(\mu\) and \(\color{red}{\sigma^2}\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu={\mu_0}\) against \(H_1:\mu<\color{blue}{\mu_0}\) we do:
    • Compute \(\color{red}{\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}_n)^2}\) (estimator of \(\color{red}{\sigma^2}\)), then \(T=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\color{red}{\hat{\sigma}}/\sqrt{n}}\)
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{t_\alpha}\) such that \(\mathbb{P}(T\leq\color{red}{t_\alpha})=\color{red}{\alpha}\).
    • Decision:
      • If \(T\leq\color{red}{t_\alpha}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(T> \color{red}{t_\alpha}\), we CANNOT \(H_0\) due to insufficient evidence.



t-Test

\(t\)-Table for (\(T_{\neq}\) two-tailed)

\(t\)-Test Summary (\(T_\neq\))

  • Suppose the sample: \(X_1,\dots,X_n\sim{\cal N}(\mu,\color{red}{\sigma^2})\) with unknown \(\mu\) and \(\color{red}{\sigma^2}\).
  • For some \(\color{blue}{\mu_0}\) given, to test \(H_0:\mu={\mu_0}\) against \(H_1:\mu\neq\color{blue}{\mu_0}\) we do:
    • Compute \(\color{red}{\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}_n)^2}\) (estimator of \(\color{red}{\sigma^2}\)), then \(T=\frac{\overline{X}_n-\color{blue}{\mu_0}}{\color{red}{\hat{\sigma}}/\sqrt{n}}\).
    • For a given significance level \(\color{red}{\alpha}\), compute \(\color{red}{t_\alpha}\) such that \(\mathbb{P}(|T|\geq\color{red}{t_\alpha})=\color{red}{\alpha}\).
    • Decision:
      • If \(|T|\geq \color{red}{t_\alpha}\), we reject \(H_0\) with confidence level \(1-\color{red}{\alpha}\).
      • If \(|T|< \color{red}{t_\alpha}\), we CANNOT \(H_0\) due to insufficient evidence.



Summary

🥳 Yeahhhh….









Let’s Party… 🥂