1. 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. σ π1 vs. π2
6.3 - Multiple Samples
Means, Variances, Proportions
μ1, …, μk σ12, …, σk π1, …, πk

2. 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. σ π1 vs. π2
6.3 - Multiple Samples
Means, Variances, Proportions
μ1, …, μk σ12, …, σk π1, …, πk

3. Sampling Distribution =?
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?

4. Sampling Distribution =?
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?

5. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

6. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

7. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

8. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

9. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

10. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
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)

11. Mean(X – Y) = Mean(X) – Mean(Y) Var(X – Y) = Var(X) + Var(Y)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
 μ0
Classic Example: “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control
Random Sample, size n1
Random Sample, size n2
Sampling Distribution =?
Recall from section 4.1 (Discrete Models):
Mean(X – Y) = Mean(X) – Mean(Y)
= 0 under H0
and if X and Y are independent…
Var(X – Y) = Var(X) + Var(Y)

12. But what if σ12 and σ22 are unknown?
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
Null Distribution
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.
s.e.

13. s.e. POPULATION 1 POPULATION 2 X1 ~ N(μ1, σ1) 1 σ1 X2 ~ N(μ2, σ2)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
Null Distribution
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…
s.e.
(But what if n1 and n2 are small?)

14. 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
NOTE:
> 0
4.2
Null Distribution
95% Margin of Error = (1.96)(4.2) = 8.232
95% Confidence Interval for μ1 – μ2:
(84 – 8.232, ) =
(75.768, )
does not contain 0
Z-score =
= 20 >> 1.96 
p << .05
Reject H0; extremely strong significant difference

15. POPULATION 1 POPULATION 2 X1 ~ N(μ1, σ1) X2 ~ N(μ2, σ2)
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1)
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 1 2
Samplesize n1
Sample size n2
 large n1 and n2
Null Distribution

16. IF the two populations are equivariant, i.e.,
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
X1 ~ N(μ1, σ1)
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 1 2
Samplesize n1
Sample size n2
 large n1 and n2
 small n1 and n2
Null Distribution
IF the two populations are equivariant, i.e.,
then conduct a t-test on the “pooled” samples.

18. Sampling Distribution =?
Test Statistic
Sampling Distribution =?
Working Rule of Thumb
Acceptance Region for H0
¼ < F < 4

19. POPULATION 1 POPULATION 2 Null Distribution
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α
X1 ~ N(μ1, σ1)
X2 ~ N(μ2, σ2) 1 2
 small n1 and n2
Null Distribution
IF equal variances
is accepted, then estimate their common value with a “pooled” sample variance.
The pooled variance is a weighted average of s12 and s22, using the degrees of freedom as the weights.

20. POPULATION 1 POPULATION 2 Null Distribution
Consider two independent populations…
and a random variable X, normally distributed in each.
POPULATION 1
POPULATION 2
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α
X1 ~ N(μ1, σ1)
X2 ~ N(μ2, σ2) 1 2
 small n1 and n2
Null Distribution
IF equal variances
is accepted, then estimate their common value with a “pooled” sample variance.
is rejected,
IF equal variances
The pooled variance is a weighted average of s12 and s22, using the degrees of freedom as the weights.
then use Satterwaithe Test, Welch Test, etc. SEE LECTURE NOTES AND TEXTBOOK.

21.  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 = Sample 2 = {593, 525, 520}; n2 = 3
Analysis via T-test (if equivariance holds): Point estimates
NOTE:
> 0
“Group Means” 
“Group Variances”
s2 = SS/df
SS1
SS2
Pooled Variance
The pooled variance is a weighted average of the group variances, using the degrees of freedom as the weights.
df1
df2

22.  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 = Sample 2 = {593, 525, 520}; n2 = 3 
Analysis via T-test (if equivariance holds): Point estimates
NOTE:
> 0
“Group Means”
“Group Variances”
s2 = SS/df
SS = 6480
Pooled Variance
The pooled variance is a weighted average of the group variances, using the degrees of freedom as the weights.
df = 6
p-value =
Standard Error
> 2 * (1 - pt(3.5, 6))
Reject H0 at α = .05
stat signif, Hosp > Clinic
[1]

23. R code: Formal Conclusion Interpretation
> y1 = c(667, 653, 614, 612, 604)
> y2 = c(593, 525, 520)
>
> t.test(y1, y2, var.equal = T)
Two Sample t-test
data: y1 and y2
t = 3.5, df = 6, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y
Formal Conclusion
p-value < α = .05
Reject H0 at this level.
Interpretation
The samples provide evidence that the 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).