Math 186, Winter 2006, Prof. Tesler – March 10, 2006
“One sample” hypothesis tests for µ and σ
A normal distribution has mean µ and standard deviation σ, but the value of µ (and possibly σ) is not known to us! We must determine µ (and
possibly σ) experimentally.
Let y1 , . . . , yn be a random sample of size n from this normal distribution.
$
"#n
#
n
2 − ( n y )2
n
1 !
i=1 yi
i=1 i
2
2
Let ȳ = (y1 + . . . + yn )/n be the sample mean and s =
(yi − ȳ) =
be the sample variance.
n−1
n(n − 1)
i=1
We want to test ȳ as an estimate of µ (is ȳ close to a constant µ0 ?) or s as an estimate of σ (is s close to a constant σ0 ?) under various circumstances.
z-test: µ unknown, σ known; testing µ. See Chap. 5.3 and 6.2.
Note: zα is defined so that P (Z ≥ zα ) = α in the standard normal distribution. Table A.1 gives Φ(z) = P (Z ≤ z) so solve Φ(zα ) = 1 − α.
Test Statistic:
Test H0 at α sig. level:
100(1 − α)% conf. int. for µ
type
Hypotheses
Z
Reject H0 when
% under H0 &
one-sided (right)
H0 : µ = µ0 vs. H1 : µ > µ0
one-sided (left)
H0 : µ = µ0 vs. H1 : µ < µ0
two-sided
H0 : µ = µ0 vs. H1 : µ &= µ0
z=
ȳ − µ0
√
σ/ n
z ≥ zα
z ≤ −zα
%
z ≤ −zα/2 or z ≥ zα/2
Student’s t-test: µ unknown, σ unknown; testing µ. See Chap. 7.2–7.4
Note: tα,k is defined so that P (Tk ≥ tα,k ) = α in the t distribution with k degrees of freedom. Table A.2.
Test Statistic:
Test H0 at α sig. level:
type
Hypotheses
Tn−1
Reject H0 when
one-sided (right)
H0 : µ = µ0 vs. H1 : µ > µ0
one-sided (left)
H0 : µ = µ0 vs. H1 : µ < µ0
two-sided
H0 : µ = µ0 vs. H1 : µ &= µ0
t=
ȳ − µ0
√
s/ n
t ≥ tα,n−1
t ≤ −tα,n−1
t ≤ −tα/2,n−1 or t ≥ tα/2,n−1
%
−∞ , ȳ + zα √σn
%
&
ȳ − zα √σn , ∞
ȳ − zα/2 √σn , ȳ + zα/2 √σn
&
100(1 − α)% conf. int. for µ
under H0
%
&
−∞ , ȳ + tα,n−1 √sn
%
&
ȳ − tα,n−1 √sn , ∞
ȳ − tα/2,n−1 √sn , ȳ + tα/2,n−1 √sn
χ2 -test (“chi-squared test”): µ unknown, σ unknown; testing σ 2 . See Chap. 7.5.
Note: χ2α,k is defined so that P (χ2k ≤ χ2α,k ) = α in the χ2 distribution with k degrees of freedom. This is backwards from tα,k . Table A.3.
Test Statistic:
Test H0 at α sig. level:
100(1 − α)% conf. int. for σ 2
type
Hypotheses
χ2n−1
Reject H0 when
' under H0 (
2
one-sided (right) H0 : σ 2 = σ0 2 vs. H1 : σ 2 > σ0 2
χ2 ≥ χ21−α,n−1
0 , (n−1)s
χ2α,n−1
(
'
(n − 1)s2
(n−1)s2
2
2
2
2
2
2
2
one-sided (left) H0 : σ = σ0 vs. H1 : σ < σ0
χ =
χ ≤ χα,n−1
, ∞
χ21−α,n−1
σ02
'
(
(n−1)s2
(n−1)s2
2
2
2
2
2
2
2
2
two-sided
H0 : σ = σ0 vs. H1 : σ &= σ0
χ ≤ χα/2,n−1 or χ ≥ χ1−α/2,n−1
, 2
2
χ1−α/2,n−1
χα/2,n−1
&