4.1 Hypothesis Tests - Z-Test

Question 1

What are statistical tests used for?

Answer 1

-statistical test answer the question of whether or not data is compatible with a given statistical model

Question 2

Z-Test

Definition

Answer 2

-let z be an observation of Z~N(μ,1)
-the z-test, with significance level α for the null hypothesis
Ho : μ=0
-with alternative:
H1 : μ≠0
-rejects Ho if and only if:
|z| > q_α/2
-where q_α/2 is the (1-α/2) quantile of N(0,1)
-i.e. the null hypothesis Ho is rejected if the observation z falls in the critical region at either the lower or upper end of the expected values

Question 3

Z-Test

Wrong Rejection Probability Lemma

Answer 3

• assume that Ho is true, i.e. that the observed data is a random sample from N(0,1)
• then the z-test with significance level α wrongly rejects Ho with a probability of α

Question 4

Z-Test

Wrong Rejection Probability Proof

Answer 4

-assume Z~N(0,1)
-then:
P(Ho is rejected) = P(|Z|>q_α/2)
= P(Z < -q_α/2 OR Z > q_α/2)
= P(Z < -q_α/2) + P(Z > q_α/2)
-since N(0,1) is symmetrical, these two probabilities are equal so we can just write:
P(Ho is rejected) = 2P(Z>q_α/2)
= 2(1 - P(Z≤q_α/2))
= 2(1 - (1-α/2))
= α

Question 5

Z-Test

Test Statistic, Critical Value and Critical Region

Answer 5

• for the z-test the modulus of the observation |z| is called the test statistic
• the critical value is q_α/2
• the interval (q_α/2,∞) is called the critical interval
• using these terms, Ho is rejected if the test statistic exceeds the critical value or equivalently, if the test statistic falls into the critical region

Question 6

What do we need to know to apply the z-test?

Answer 6

-the numerical values for the (1-α/2) quantiles of the standard normal distribution

Question 7

Z-Test

Type I Errors

Answer 7

• if Ho is true, the test might wrongly reject Ho
• this outcome is a type I error
• they occur with a probability of α

Question 8

Z-Test

Type II Errors

Answer 8

• an error occurs if H1 is true but the test does not reject Ho (despite Ho being false)
• this outcome is a type II error
• the probability of this error depends on the value of μ
• if μ≈0 these errors occur withe a probability of approximately 1-α
• as μ gets further away from 0, the probability of hitting the interval [-q_α/2,q_α/2] and thus the probability of a type II error decreases to 0

Question 9

Z-Test

Errors and Choosing α

Answer 9

• choosing small values of α reduces the probability of type I errors
• but this also reduces the size of the critical region and increases the chance of type II errors
• the two types of error probabilities must be balanced when choosing α

Question 10

Z-Test

Multiple Observations Description

Answer 10

• data sets being tested in practice will consist of more than one sample
• in this case we can apply the z-test by considering the sample average
• assume that we have observed a sample x1,…,xn which can be described by the statistical model X1,…,Xn~N(μ,σ²) i.i.d for known variance σ²
• we want to test the hypothesis Ho : μ=μo against the alternative H1:μ≠μo

Question 11

Z-Test

Multiple Observations Lemma

Answer 11

-let μ, μo ∈ R and X1,…,Xn~N(μ,σ²) be i.i.d
-define:
Z = 1/√n Σ Xi-μo/σ
-then:
Z~N(√n(μ-μo) , nσ²)

Question 12

Z-Test

Multiple Observations Lemma Proof

Answer 12

-we have X1,…,Xn~N(μ,σ²)
-and we know then that:
Σ(Xi-μo)~N(n(μ-μo),nσ²)
-dividing by σ√n gives the result

Question 13

Z-Test

Multiple Observations Null Hypothesis

Answer 13

• using the multiple observations lemma, we see that the hypothesis Ho:μ=μo is equivalent to the hypothesis that Z has mean 0
• we also know that Z has variance 1 and thus we can apply the z-test on Z to decide whether to reject Ho or not

Question 14

Z-Test

Multiple Observations Applying the Test to Z

Answer 14

-we reject Ho at confidence level α if and only if:
|Z| = |1/√n Σ Xi-μo/σ| > q_α/2
-where q_α/2 is the (1-α/2) quantile of the standard normal distribution
-for this test we need to know the value of the sample variance σ² in order to compute the test statistic |Z|

Question 15

Z-Test

Multiple Observations Manually Computing the Test Statistic

Answer 15

-an alternative representation can make the test statistic easier to compute:
Z = 1/√n Σ Xi-μo/σ
= √n * (x^-μo)/σ
-where x^ is the sample average of xi

Question 16

Z-Test

Single Observation Summary

Answer 16

data: z∈R
model: Z~N(μ,1)
test: Ho:μ=0 H1:μ≠0
test statistic: |z|
critical value: q_α/2, the (1-α/2)-quantile of N(0,1)

Question 17

Z-Test

Multiple Observations Summary

Answer 17

data: x1,…,xn∈R
model: X1,…,Xn~N(μ,σ²) i.i.d with known σ
test: Ho:μ=μo H1:μ≠μo
test statistic: |z| = 1/√n Σ(Xi-μo)/σ = √n * (x^-μo)/σ
critical value: q_α/2, the (1-α/2)-quantile of N(0,1)

Question 18

Z-Test

Comparing the Mean of Two Populations Overview

Answer 18

• assume that we have observed data x1,…,xn∈R and y1,…,yn∈R
• these are non-paired data and we write n and m for the sample sizes to allow for them to be different sizes
• assume that the data can be described by the model X1,…,Xn~N(μx,σx²) and Y1,…,Ym~N(μy,σy²)
• where the random variables X1,…,Xn,Y1,…,Ym are independent of each other and where the variances σx² and σy² are known
• we want to test the hypothesis Ho:μx=μy against the alternative H1:μx≠μy

Question 19

Z-Test

Comparing the Mean of Two Populations Lemma

Answer 19

-let X1,…,Xn~N(μx,σx²) and Y1,…,Ym~N(μy,σy²) be independent
-define:
Z = (X^-Y^) / √(σx²/n + σy²/m)
-where X^ and Y^ are the sample means, then:
Z~N( (μx-μy)/√(σx²/n + σy²/m),1)

Question 20

Z-Test

Comparing the Mean of Two Populations Lemma Proof

Answer 20

E(X^ - Y^) = E(X^) +  E(Y^)
= μx-μy
-and since X^ and Y^ are independent:
Var(X^-Y^) = Var(X^) + Var(-Y^)
= Var(X^) + (-1)²Var(Y^)
= σx²/n + σy²/m
-since X^-Y^ is a linear combination of independent normally distributed random variables:
X^-Y^ ~ N(μx-μy , σx²/n + σy²/m)
-finally dividing by the standard deviation √(σx²/n + σy²/m) gives the result:
Z = (X^-Y^) / √(σx²/n + σy²/m)
~ N((μx-μy)/√(σx²/n + σy²/m),1)

Question 21

Z-Test

Comparing the Mean of Two Populations Summary

Answer 21

data: x1,…,xn,y1,…,ym∈R
model: X1,…,Xn~N(μx,σx²) , Y1,…,Yn~N(μy,σy²) independent
test: Ho:μx=μy H1:μx≠μy
test statistic: |z| = |x^-y^|/√(σx²/n + σy²/m)
critical value: q_α/2, the (1-α/2)-quantile of N(0,1)

Question 22

Z-Test

Comparing the Mean of Two Populations Paired Data Overview

Answer 22

• so far we have looked at whether samples from two different populations have the same mean
• instead we can also consider the case of paired samples, i.e. where we have observed two variables for each individual
• the idea is to simply for paired data (xi,yi) we can consider zi=xi-yi, and then apply the test we already know
• zi has mean μo=0 if and only if xi and yi have the same mean

Question 23

Z-Test

Comparing the Mean of Two Populations Paired Data Test Statistic

Answer 23

-for paired data (xi,yi) and zi=xi-yi, the test statistic is given by:
1/√n Σ(zi-μo)/σz = √n z^/σz
= (x^-y^) / √(σz²/n)
-the variance σz² can either be computed from the variances of x and y (taking any correlation into account) or estimated from the data

Question 24

One-Sided Z-Test

Definition

Answer 24

• let z be an observation of Z~N(μ,1)
• the one-sided z-test with significance level α for the hypothesis Ho:μ≤0 with alternative H1:μ>0 rejects Ho if and only if z>q_α
• where q_α is the (1-α)-quantile of N(0,1)

Question 25

One-Sided Z-Test

Wrong Rejection of Ho Lemma

Answer 25

• assume that Ho is true , i.e. that Z is a random sample from N(μ,1) with μ<0
• then the one-sided z-test with significance level α wrongly rejects Ho with probability of at most α

Question 26

One-Sided Z-Test

Wrong Rejection of Ho Lemma Proof

Answer 26

-let μ≤0 and Z~N(μ,1)
-define Zo=Z-μ
-then Zo~N(0,1) and since μ≤0 we have Zo≥Z
-thus we find:
P(Z > q_α) ≤ P(Zo > q_α)
P(Zo > q_α) = 1 - P(Zo ≤ q_α)
= 1 - (1-α) = α

Question 27

One-Sided Z-Test

Summary

Answer 27

data: x1,…,xn∈R
model: X1,…,Xn~N(μ,σ²) i.i.d
test: Ho:μ≤μo H1:μ>μo
test statistic: z = 1/√n Σ(xi-μo)/σ = √n (x^-μo)/σ
critical value: q_α, the (1-α)-quantile of N(0,1)

Question 28

One-Sided Z-Test

Alternative Ho

Answer 28

• we can derive the one-sided z-test for testing the hypothesis Ho:μ≥0 with alternative H1:μ<0 by exploiting symmetry
• replacing z with -z, we see that for this case we should reject Ho if and only if z

Question 29

Z-Test for Large Sample Size Overview

Answer 29

• assume that we have observed x1,…,xn from a model where X1,…,Xn are i.i.d but not necessarily normally distributed
• we will see that, if n is sufficiently large, we can still apply the z-test

Question 30

Z-Test for Large Sample Sizes

Central Limit Theorem

Answer 30

-Let X1,…,Xn be i.i.d with E(Xi)=μ and Var(Xi)=σ²
-then:
Z = 1/√n Σ(Xi-μ)/σ -> N(0,1)
-as n ->∞, here -> indicates convergence in the distribution
-this statement of the central limit theorem implies that, for large sample size, the test statistic Z is approximately standard normal distributed even if the individual samples come from different distributions
-so we can still apply the z-test

Question 31

Z-Test for Large Sample Sizes

Summary

Answer 31

data: x1,…,xn∈R, where n is large
model: X1,…,Xn i.i.d with E(Xi)=μ and Var(Xi)=σ²
test: Ho:μ=μo H1:μ≠μo
test statistic: |z| = 1/√n Σ|xi-μo|/σ = √n |x^-μo|/σ
critical value: q_α/2, the (1-α/2)-quantile of N(0,1)

Question 32

Z-Test in R

Answer 32

• to perform a z-test in R, only the test statistic and the critical value must be computed
• this is easily done using standard R commands like mean() and qnorm()