10.2 Means sd Unknown - Outcome 10.2Be able to use the t...

Hypothesis Tests
Statistical Inference
About Means and
Proportions with Two
Populations

BUQU 1230

Esther Tiessen
Outcome 10.2—Be
able to use the t
distribution to
conduct statistical
inferences about the
difference between
two population
means when σ1 and
σ2 are unknown.

Hypothesis Tests
Two Populations
Statistical inference about the difference between
two population:
ü
means (σ1 and σ2 known)

means (σ1 and σ2 unknown)

means (matched samples)
 proportions

Hypothesis Tests
Two Populations
Statistical inference about the difference between
two population:

means (σ1 and σ2 known)

means (σ1 and σ2 unknown)

means (matched samples)
 proportions
t -Test:Two-Sample for
Means

Hypothesis Tests

population parameter: D = μ 1 – μ 2

to make inferences about D,
◦
take independent simple random samples from 2
populations
◦
use difference between sample means as the
random variable
◦
use sampling distribution of

use s1 and s2 to estimate population standard deviations

use t -distribution for making inferences

expected value is

standard error is
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
d = x
1
- x
2
E d = x
1
- x
2
( 29
= μ
1
- μ
2
s
d
=
s
1
2
n
1
+
s
2
2
n
2

Hypothesis Tests

population parameter: D = μ 1 – μ 2

to make inferences about D,
◦
take independent simple random samples from 2
populations
◦
use difference between sample means as the
random variable
◦
use sampling distribution of

use s1 and s2 to estimate population standard deviations

use t -distribution for making inferences

expected value is

standard error is
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
d = x
1
- x
2
E d = x
1
- x
2
( 29
= μ
1
- μ
2
s
d
=
s
1
2
n
1
+
s
2
2
n
2
Use for
both
interval
estimation
and
hypothesis
testing

Hypothesis Tests

population parameter: D = μ 1 – μ 2

sampling distribution of
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
d = x
1
- x
2

Hypothesis Tests
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
Population Mean
parameter
μ
point estimator
standard error
margin of error
interval estimate
degrees of
freedom
x
σ
x
=
s
n
t
α 2
s
n
x ± t
α 2
s
n
n - 1
Recall from
chapter 8

Hypothesis Tests
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
Population Mean
Difference Between 2
Population Means
parameter
μ D = μ 1 – μ 2
point estimator
standard error
margin of error
interval estimate
degrees of
freedom
x
d = x
1
- x
2
σ
x
=
s
n
s
d
=
s
1
2
n
1
+
s
2
2
n
2
t
α 2
s
n
t
α 2
s
1
2
n
1
+
s
2
2
n
2
x ± t
α 2
s
n
d ± t
α 2
s
1
2
n
1
+
s
2
2
n
2
n - 1
df =
s
1
2
n
1
+
s
2
2
n
2
( 29
2
1
n
1
- 1
s
1
2
n
1
(29
2
+ 1
n
2
- 1
s
2
2
n
2
(29
2

Hypothesis Tests
Statistical Inference about the Difference
between Two Population Means (σ1 and σ2
unknown)
Population Mean
Difference Between 2
Population Means
parameter
μ D = μ 1 – μ