Download Hypothesis Testing – One Sample Tests

Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Hypothesis Testing – One Sample Tests
Jiřı́ Neubauer
Department of Econometrics FVL UO Brno
office 69a, tel. 973 442029
email:Jiri.Neubauer@unob.cz
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter µ of a Normal Distribution
Let X1 , X2 , . . . , Xn be a random sample from a normal distribution
N(µ, σ 2 ). A statistic
X − µ√
n
T =
S
has a Student distribution with ν = n − 1 degrees of freedom. We use
this statistic for testing of the parameter µ.
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter µ of a Normal Distribution
Let x1 , x2 , . . . , xn be values of a random sample (measured data),
x denotes an arithmetic mean and s a sample standard deviation.
We test the hypothesis that the parameter µ is equal to a constant µ0 :
H : µ = µ0 ,
the test statistic
x − µ0 √
n,
s
has under the null hypothesis H a Student t-distribution with ν = n − 1
degrees of freedom.
t=
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter µ of a Normal Distribution
According to the alternative hypothesis we construct following regions of
rejection:
alternative hypothesis
rejection region
A : µ > µ0
Wα = {t, t ≥ t1−α (ν)}
A : µ < µ0
Wα = {t, t ≤ −t1−α (ν)}
Wα = t, |t| ≥ t1− α2 (ν)
A : µ 6= µ0
where t1−α (ν), t1− α2 (ν) are quantiles of the Student distribution.
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter σ 2 of a Normal distribution
Let X1 , X2 , . . . , Xn be a random sample from a normal distribution
N(µ, σ 2 ). A statistic
(n − 1)S 2
χ2 =
σ2
has a Pearson distribution with ν = n − 1 degrees of freedom. We use
this statistic for testing of the parameter σ 2 .
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter σ 2 of a Normal distribution
Let x1 , x2 , . . . , xn be values of a random sample (measured data),
s 2 denotes a sample variance.
We test the hypothesis that the parameter σ 2 is equal to a constant σ02 :
H : σ 2 = σ02 ,
the test statistic
χ2 =
(n − 1)s 2
σ02
has under the null hypothesis H a Pearson χ2 -distribution with ν = n − 1
degrees of freedom.
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Tests on Mean of a Normal distribution
Tests on Variance of a Normal distribution
Test of the Parameter σ 2 of a Normal distribution
According to an alternative hypothesis we construct following regions of
rejection:
alternative hypothesis
2
A:σ >
σ02
A : σ 2 < σ02
A : σ 2 6= σ02
rejection region
Wα = χ2 , χ2 ≥ χ21−α (ν)
Wα = χ2 , χ2 ≤ χ2α (ν)
n
o
Wα = χ2 , χ2 ≤ χ2α (ν) or χ2 ≥ χ21− α (ν)
2
2
where χ21−α (ν), χ21− α (ν) are quantiles of the Pearson χ2 -distribution.
2
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Large Sample Test on Mean
Let X1 , X2 , . . . , Xn be a random sample from any distribution with the
mean µ. A statistic
X − µ√
U=
n
S
has for large n approximately a normal distribution N(0, 1) – see the
central limit theorems. We use this statistic for testing of the
parameter µ.
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Large Sample Test on Mean
Let x1 , x2 , . . . , xn be values of a random sample (measured data),
x denotes an arithmetic mean and s a sample standard deviation.
We test the hypothesis that the parameter µ is equal to a constant µ0 :
H : µ = µ0 ,
the test statistic
x − µ0 √
n,
s
has under the null hypothesis H asymptotically a normal distribution
N(0, 1).
u=
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Large Sample Test on Mean
According to an alternative hypothesis we construct following regions of
rejection:
alternative hypothesis
rejection region
A : µ > µ0
Wα = {u, u ≥ u1−α }
A : µ < µ0
Wα = {u, u ≤ −u1−α }
Wα = u, |u| ≥ u1− α2
A : µ 6= µ0
where u1−α , u1− α2 are quantiles of N(0, 1).
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Test on a Population Proportion
Suppose that a random sample of size n has been taken from a large
(possibly infinite) population and that m observations in this sample
belong to a class of interest. Then p = mn is a point estimator of the
proportion of the population π that belongs to this class. A random
variable
π̂ − π
U=p
π(1 − π)/n
has for n → ∞ approximately a normal distribution N(0, 1) – see central
limit theorems. We use this statistic for testing of the population
proportion.
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Test on a Population Proportion
Let π̂ = mn be a point estimator of population proportion.
We test the hypothesis that the parameter π is equal to a constant π0 :
H : π = π0 ,
a test statistic
u=p
π̂ − π0
π0 (1 − π0 )/n
has under the null hypothesis H asymptotically a normal distribution
N(0, 1).
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests
Tests on Mean and Variance of a Normal distribution
Large Sample Test on Mean
Test on a Population Proportion
Test on a Population Proportion
According to an alternative hypothesis we construct following regions of
rejection:
alternative hypothesis
rejection region
A : π > π0
Wα = {u, u ≥ u1−α }
A : π < π0
Wα = {u, u ≤ −u1−α }
Wα = u, |u| ≥ u1− α2
A : π 6= π0
where u1−α , u1− α2 are quantiles of N(0, 1).
Jiřı́ Neubauer
Hypothesis Testing – One Sample Tests