CHAPTER 6
Statistical Inference & Hypothesis Testing
• 6.1 - One Sample
 Mean μ, Variance σ 2, Proportion π
• 6.2 - Two Samples
 Means, Variances, Proportions
μ1 vs. μ2 σ12 vs. σ22
π1 vs. π2
• 6.3 - Multiple Samples
 Means, Variances,
μ1, …, μk σ12, …, σk2
Proportions
π1, …, πk
CHAPTER 6
Statistical Inference & Hypothesis Testing
• 6.1 - One Sample
 Mean μ, Variance σ 2, Proportion π
• 6.2 - Two Samples
 Means, Variances, Proportions
μ1 vs. μ2 σ12 vs. σ22
π1 vs. π2
• 6.3 - Multiple Samples
 Means, Variances,
μ1, …, μk σ12, …, σk2
Proportions
π1, …, πk
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
σ2
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1
Random Sample,
size n2
Sampling Distribution =?
X1
X2
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1


X 2 ~ N 2 ,  2
n2

X1  X 2 ~ ????
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2

X1  X 2 ~ N  ????, ????
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2

X1  X 2 ~ N  1  2 , ????
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2

X1  X 2 ~ N  1  2 , ????
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2


 12 
X 1  X 2 ~ N  1  2 ,

n

1 
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2


12  2 2 
X1  X 2 ~ N  1  2 ,


n
n

1
2 
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2


X 1  X 2 ~ N  1  2 ,


 12
n1

 22 

n2 
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0  μ0
σ2
(“No mean difference")
Test at signif level α
1
2
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample,
size n1

Random Sample,
size n2
Sampling Distribution =?
X 1 ~ N 1 ,  1
n1

Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

X 2 ~ N 2 ,  2
n2

= 0 under H0  2  2 
1
X 1  X 2 ~ N  1  2 ,
 2 


n
n
1
2


Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
σ2
(“No mean difference")
Test at signif level α
1
2
Null Distribution

X 1  X 2 ~ N  0,


But what if σ1 and σ2 are unknown?

2 

 Then use sample estimates s12 and s22
n1
n2  with Z- or t-test, if n and n are large.
1
2
2
1
s.e.
X1  X 2
0
2
2
2
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
σ1
POPULATION 2
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
σ2
(“No mean difference")
Test at signif level α
1
2
Null Distribution

X 1  X 2 ~ N  0,


s
s2 


n1 n2 
2
1
2
s.e.
X1  X 2
0
But what if σ12 and σ22 are unknown?
Then use sample estimates s12 and s22
with Z- or t-test, if n1 and n2 are large.
Later…
(But what if n1 and n2 are small?)
Example: X = “$ Cost of a certain medical service”
Assume X is known to be normally distributed at each of k = 2 health care facilities (“groups”).
Hospital: X1 ~ N(μ1, σ1)
Clinic: X2 ~ N(μ2, σ2)
• Null Hypothesis H0: μ1 = μ2,
i.e., μ1 – μ2 = 0
(“No difference exists.")
2-sided test at significance level α = .05
• Data Sample 1: n1 = 137
Sample 2: n2 = 140
x1  630
x2  546
s12  788.5
s22  1663.0
4.2
Null Distribution

X 1  X 2  N  0,



N  0,

2
95% Confidence Interval for μ1 – μ2:
(84 – 8.232, 84 + 8.232) = (75.768, 92.232)
788.5 1663.0 


137
140 
N  0, 4.2
0
95% Margin of Error = (1.96)(4.2) = 8.232
s
s2 


n1 n2 
2
1
x1  x2  84
NOTE:
>0
does not contain 0
84  0
= 20 >> 1.96  p << .05
Z-score =
4.2
Reject H0; extremely strong significant difference
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
1
POPULATION 2
X2 ~ N(μ2, σ2)
2
1
Sample
size n1
2
1
s
(“No mean difference")
Test at signif level α
2

X 1  X 2 ~ N  0,


 12
n1

 unknown 12 and  22
Sample
size n2
Null Distribution
 22 

n2 
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
s22
 large n1 and n2
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
1
POPULATION 2
X2 ~ N(μ2, σ2)
2
1
Sample
size n1
s12
(“No mean difference")
Test at signif level α
2

X 1  X 2 ~ N  0,



2 


n1
n2 
2
1
 unknown 12 and  22
Sample
size n2
Null Distribution
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
 small
large n11 and n22
IF the two populations
are equivariant, i.e.,
2
s22
H0 :   
2
1
2
2
then conduct a t-test on
the “pooled” samples.
H 0 :  12   2 2
H A:    2
2
1
s12
2
s22
H0 :    2
2
1
2
H 0 :  12   2 2
H A:    2
2
1
s12
2
s22
Test Statistic
s12
F 2
s2
Sampling Distribution =?
Working Rule of Thumb
Acceptance Region for H0
¼<F<4
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
POPULATION 2
X2 ~ N(μ2, σ2)
1
2
1
(“No mean difference")
Test at signif level α
 unknown 12 and  22
2
Null

Distribution X 1  X 2 ~ N  0,


 12
n1

 small n1 and n2
 22 

n2 
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
IF equal variances
H0 : 12   22
is accepted, then estimate their
common value with a “pooled”
sample variance.
2
pooled
pooled
s
(n1  1) s12  (n2  1) s22

n1  n2  2
The pooled variance is a weighted
average of s12 and s22, using the
degrees of freedom as the weights.
Consider two independent populations…and a random variable X, normally distributed in each.
POPULATION 1
X1 ~ N(μ1, σ1)
POPULATION 2
X2 ~ N(μ2, σ2)
1
2
1
 12


n1
2
pooled2
pooled
1
s
s.e.  s
IF equal variances
(“No mean difference")
Test at signif level α
 unknown 12 and  22
2
Null

Distribution X 1  X 2 ~ N  0,
n
H0 : 12   22

 small n1 and n2
 22 

n2 
 1spooled1 
  
 n1 n2n2 
2
IF equal variances
s
then use Satterwaithe Test, Welch Test, etc.
SEE LECTURE NOTES AND TEXTBOOK.
H0 : 12   22
is accepted, then estimate their
common value with a “pooled”
sample variance.
2
pooled
pooled
is rejected,
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(n1  1) s12  (n2  1) s22

n1  n2  2
The pooled variance is a weighted
average of s12 and s22, using the
degrees of freedom as the weights.
Example: Y = “$ Cost of a certain medical service”
Assume Y is known to be normally distributed at each of k = 2 health care facilities (“groups”).
Hospital: Y1 ~ N(μ1, σ1)
Clinic: Y2 ~ N(μ2, σ2)
• Null Hypothesis H0: μ1 = μ2,
i.e., μ1 – μ2 = 0
(“No difference exists.")
2-sided test at significance level α = .05
• Data: Sample 1 = {667, 653, 614, 612, 604}; n1 = 5
• Analysis via T-test (if equivariance holds):
“Group Means” y1 
“Group
2
Variances” s1 
s2 = SS/df
Pooled
Variance
667  653  614  612  604
5
630))2   (604630)2
(667  630
51
SS1
 630
Point estimates
y2 
 788.5 s 
2
2
df1
593  525  520
3
 546
546))2   (520546
546))2
(593  546
31
y   yi / n
NOTE:
y1  y2  84
>0

1663
 2.11  4
 1663 F  788.5
SS2
( n11)(
1)788.5
s1  ()n
1)s1)(
(3
2
2
2 1663 )
spooled
 (5
 1080
1)
n(5
1 n
2 2 (31)
2
Sample 2 = {593, 525, 520}; n2 = 3
2
df2
The pooled variance is a weighted average of the group
variances, using the degrees of freedom as the weights.
Example: Y = “$ Cost of a certain medical service”
Assume Y is known to be normally distributed at each of k = 2 health care facilities (“groups”).
Hospital: Y1 ~ N(μ1, σ1)
Clinic: Y2 ~ N(μ2, σ2)
• Null Hypothesis H0: μ1 = μ2,
i.e., μ1 – μ2 = 0
(“No difference exists.")
2-sided test at significance level α = .05
• Data: Sample 1 = {667, 653, 614, 612, 604}; n1 = 5
Sample 2 = {593, 525, 520}; n2 = 3
• Analysis via T-test (if equivariance holds):  Point estimates
“Group Means” y1 
“Group
2
Variances” s1 
s2 = SS/df
Pooled
Variance
667  653  614  612  604
5
630))2   (604630)2
(667  630
51
 630
 788.5 s 
2
2
 546
546))2   (520546
546))2
(593  546
31
NOTE:
y1  y2  84
>0
1663
 2.11  4
 1663 F  788.5
SS = 6480
)  (31)(1663)
2
spooled
 (51)( 788.5
 1080
(51)  (31)
df = 6
Standard
Error
y2 
593  525  520
3
y   yi / n
11 1 1
s.e.  1080
  24
s
5n1 3n2
2
pooled
The pooled variance is a weighted average of the group
variances, using the degrees of freedom as the weights.

p-value = 2P(Y1  Y2  84)  2 P T6  84240   2 P T6  3.5
> 2 * (1 - pt(3.5, 6)) Reject H0 at α = .05
stat signif, Hosp > Clinic
[1] 0.01282634

R code:
> y1 = c(667, 653, 614, 612, 604)
> y2 = c(593, 525, 520)
>
> t.test(y1, y2, var.equal = T)
Formal Conclusion
Two Sample t-test
p-value < α = .05
Reject H0 at this level.
data: y1 and y2
t = 3.5, df = 6, p-value = 0.01283
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
25.27412 142.72588
Interpretation
sample estimates:
mean of x mean of y
The samples provide evidence that the
630
546
difference between mean costs is (moderately)
statistically significant, at the 5% level,
with the hospital being higher than the clinic
(by an average of $84).
NEXT UP…
PAIRED MEANS
page 6.2-7, etc.