4.2 Hypothesis Tests - T-Test

Question 1

What is the t-test used for?

Answer 1

-in situations where the exact variance of observed samples is unknown, we can test the hypothesis Ho:μ=μo against the alternative H1:μ≠μo using the test statistic |t|

Question 2

Two Sided T-Test Test Statistic |t|

Definition

Answer 2

|t| = 1/√n * Σ (xi-μo)/σx^

• where σx^ is the sample standard deviation
• since σx^ is not a constant bu depends on the data, even if we compute the t from samples which follow a normal distribution, the t value itself will not be normally distributed

Question 3

T-Test

Defintion

Answer 3

• the t-test rejects Ho:μ=μo if |t|>c for a critical value c

- before we can use the t-test, we need to choose a value of c that will keep the probability of type I errors below α

Question 4

What do we use to derive the critical values for the t-test?

Answer 4

-the chi-squared probability distribution

Question 5

Chi-Squared Distribution

Definition

Answer 5

• let X1,…,Xv ~ N(0,1) be i.i.d
• then the distribution of ΣXi² is called the χ²-distribution with v degrees of freedom, the sum is taken from i=1 to i=v
• the distribution is denoted by χ²(v)

Question 6

Chi-Squared Distribution

Expectation and Variance Lemma

Answer 6

-let Y~χ²(v), then:
E(Y) = v
and:
Var(Y) = 2v

Question 7

Chi-Squared Distribution

Expectation and Variance Lemma Proof

Answer 7

-write Y as the sum of X1,…,Xn~N(0,1) squared

Question 8

CDF of the Chi-Squared Distribution in R

Answer 8

-the R command:
pchisq(x,v)
-gives the value Ф(x) of the CDF of the χ²(v) distribution

Question 9

α-Quantile of the Chi-Squared Distribution in R

Answer 9

-the R command:
qchisq(α,v)
-can be used to obtain the α-quantile of the χ²(v) distribution

Question 10

Chi-Squared Theorem

Answer 10

-let X1,…,Xn~N(μ,σ² ) i.i.d. and consider:
Y = Σ (Xi,X_)² /σ²
-where the sum is taken from i=1 to i=n and X_ is the sample average of Xi
-then:
a) Y~χ²(n-1)
b) the random variables X_ and Y are independent

Question 11

T-Distribution

Definition

Answer 11

-let Z~N(0,1) and Y~χ²(v) be independent
-then the distribution of:
T = Z / √(Y/v)
-is called the t-distribution with v degrees of freedom
-this distribution is denoted by t(v)

Question 12

T-Test Sample of Xi Random Variables Normally Distributed Lemma

Answer 12

-let X1,…,Xn~N(μ,σ²), and let:
T = 1/√n * Σ (Xi-μo)/σx^
-where σx^ is the sample standard deviation of Xi
-then T~t(n-1)

Question 13

CDF of the T-Distribution in R

Answer 13

-the R command pt(x,v) gives the value Ф(x) of the CDF of the t(v)-distribution

Question 14

α-Quantile of the T-Distribution in R

Answer 14

-the R command qt(x,v) gives the value α-quantile the t(v)-distribution

Question 15

Two-Sided T-Test

Description

Answer 15

-let x1,…,xn be observations of Xi~N(μ,σ²) with unknown variance σ² and assume that we want to test Ho:μ=μo against H1:μ≠μo

Question 16

Two-Sided T-Test

Test Statistic

Answer 16

-as a test statistic we use |t|, where:
t = 1/√n * Σ (xi-μo)/σx^
= √n * (x_-μo)/σx^

Question 17

Two-Sided T-Test

Critical Value

Answer 17

• we know that under Ho, the test statistic follows a t(n-1) distribution
• thus we can use critical value tn-1(α/2), the (1-α/2) quantile of t(n-1)

Question 18

Two-Sided T-Test

Summary

Answer 18

data: x1,…,xn∈R
model: X1,…,Xn~N(μ,σ²) i.i.d
test: Ho:μ=μo vs H1:μ≠μo
test statistic: |t| = 1/√n * Σ |xi-μo|/σx^ = √n * |x_-μo|/σx^
critical value: tn-1(α/2), the (1-α/2) quantile of t(n-1)

Question 19

How does the t-test differ from the z-test?

Answer 19

• the critical value in the t-test depends not only on the significance level α, as in the z-test, but also on the sample size n
• we have to consider quantiles of the t-distribution with v=n-1 degrees of freedom

Question 20

One-Sided T-Test

Description

Answer 20

-assume X1,..,Xn~N(μ,σ²) with unknown mean μ and unknown variance σ²
-we want to test Ho:μ≤μo against H1:μ>μo
-we compute the same test statistic as for the two-sided test but without the modulus:
t = 1/√n * Σ (xi-μo)/σx^
-but now we reject Ho if t>tn-1(α) instead of α/2

Question 21

One-Sided T-Test

Summary

Answer 21

data: x1,…,xn∈R
model: X1,…,Xn~N(μ,σ²) i.i.d
test: Ho:μ≤μo vs H1:μ>μo
test statistic: t = 1/√n * Σ (xi-μo)/σx^ = √n * (x_-μo)/σx^
critical value: tn-1(α), the (1-α) quantile of t(n-1)

Question 22

Large Sample Size T-Test

Description

Answer 22

-assume that Xi are not normally distributed but are still independent with Var(Xi)=σ²

Question 23

Large Sample T-Test

Summary

Answer 23

data: x1,…,xn∈R, n large
model: X1,…,Xn i.i.d with mean μ and variance σ²

Question 24

Two Sample T-Test

Description

Answer 24

-can be used to compare the mean of two independent populations with unknown variances
-assume we have observed x1,…,xn ad y1,…,yn
-the joint variance can be estimated using:
σp^ = 1/(n+m-2) (Σ(xi-x_)² + Σ(yi-y_)² )

Question 25

Two Sample T-Test

Summary

Answer 25

data: x1,..,xn,y1,..,ymR
model: X1,…,Xn~N(μx,σx²), Y1,…,Ym~N(μy,σy²) independent
test: Ho:μx=μy vs H1:μx≠μy
test statistic: |t| = Σ |x_-y_|/√(σx^²(1/n + 1/m))
critical value: tn+m-2(α/2), the (1-α/2) quantile of t(m+m-2)

Question 26

p-values

Answer 26

• if we call a general test statistic s and denote the density of s by φ
• let our observed value of the test statistic be s*
• the p-value represents the area of the region from the top of the distribution down to s* i.e. the probability under Ho that s≥s*
• we see that we reject Ho whenever we have s>q_α