Inferential Stats: Confidence Intervals & Hypothesis Testing for Two Population Means, Study notes of Data Analysis & Statistical Methods

Download Inferential Stats: Confidence Intervals & Hypothesis Testing for Two Population Means and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity!

10/18/

Chapter 9

Inferences Based on Two

Samples: Confidence Intervals

and Tests of Hypothesis

Identifying the Target Parameter

Parameter Key Words or Phrases

Mean difference; difference

1 2 in averages

μ −μ

2

Difference between

proportions, percentage,

fractions or rates

p 1 − p 2

Comparing Two Population Means:

Independent Sampling

Confidence Intervals and hypothesis testing

can be done for both large and small

samples

3

Large sample cases use z-statistic, small

sample cases use t-statistic

When comparing two population means, we

test the difference between the means

Comparing Two Population Means:

Independent Sampling

Large Sample Confidence Interval for μ 1 - μ 2

( ) (^) ( ) ( )

2 2

2 1 (^1 221 ) n n

x x z x x x x z

σ σ

− ± α σ − = − ± α +

4

assuming independent sampling, which provides the

following substitution

( ) n 1 n 2

( )

1 2

n

s

n

s

n n

x x

= + ≈ +

−

σ σ σ

Comparing Two Population Means:

Independent Sampling

Properties of the Sampling Distribution of (x 1 -x 2 )

•Mean of Sampling distribution (x 1 -x 2 ) is (μ 1 - μ 2 )

•Assuming two samples are independent, the standard

deviation of the sampling distribution is

5

deviation of the sampling distribution is

•The sampling distribution of (x 1 -x 2 ) is approximately

normal for large samples by the CLT

( ) 2

2 2

1

2 1 x 1 x (^2) n n

σ σ σ (^) − = +

Comparing Two Population Means:

Independent Sampling

Large Sample Test of Hypothesis for μ 1 - μ 2

One-Tailed Test Two-Tailed Test H 0 :( μ 1 - μ 2 ) = D 0 H 0 :( μ 1 - μ 2 ) = D (^0) Ha :( μ 1 - μ 2 ) <D 0 Ha :( μ 1 - μ 2 ) ≠ D (^0)

6

[or Ha:( μ 1 - μ 2 ) >D 0 ] Where D 0 = hypothesized difference between the means Test Statistic ( )

( 12 )

(^120)

x x

x x D z −

− −

=

σ where^ (^ )^2

2 2 1

2 1 x 1 x (^2) n n

σ σ

σ − = +

Rejection region : z < -z α Rejection region : ⏐ z ⏐ > z α / [or z > z α when H a :( μ 1 - μ 2 ) >D 0 ]

10/18/

Comparing Two Population Means:

Independent Sampling

Required conditions for Valid Large-Sample Inferences about μ 1 - μ 2

1 Random independent sample selection

7

1. Random, independent sample selection
2. Sample sizes are both at least 30 to guarantee

that the CLT applies to the distribution of x 1 -x (^2)

Comparing Two Population Means:

Independent Sampling

Small Sample Confidence Interval for μ 1 - μ 2

( )

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ± +

2 (^122)

1 1

n n

x x t α sp

8

where

and tα/2 is based on (n 1 +n 2 -2) degrees of freedom

⎝ n 1 n 2 ⎠

( ) ( )

+ −

− + −

=

n n

n s n s

s p

Comparing Two Population Means:

Independent Sampling

Small Sample Test of Hypothesis for μ 1 - μ 2

One-Tailed Test Two-Tailed Test H 0 :( μ 1 - μ 2 ) = D 0 H 0 :( μ 1 - μ 2 ) = D 0 H a :( μ 1 - μ 2 ) <D 0 [or Ha :( μ 1 - μ 2 ) >D 0 ]

Ha :( μ 1 - μ 2 ) ≠ D 0

9

[or Ha :( μ 1 μ 2 ) D 0 ] Where D 0 = hypothesized difference between the means Test Statistic

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

− −

=

1 2

2

(^120)

n n

s

x x D

t

p

Rejection region : t < -t α Rejection region : ⏐ t ⏐ > t α / [or t > t α when^ H^ a :(^ μ 1 -^ μ 2 ) >D 0 ] Where t α and t α /2 are based on (n 1 +n 2 -2) degrees of freedom

Comparing Two Population Means:

Independent Sampling

Required conditions for Valid Small-Sample Inferences about μ 1 - μ 2

1 Random independent sample selection

10

1. Random, independent sample selection
2. Approximate normal distribution of both sampled

populations

3. Population variances are equal

2 2 σ 1 =σ 2

Comparing Two Population Means:

Independent Sampling

Small Samples – Assume

Confidence interval:( ) ( ) ( )

x 1 − x 2 ± t s n + s n

σ 1 ≠σ 2

11

Confidence interval:

Test Statistic for H 0:

where t is based on degrees of

freedom

( x 1 − x 2 ) ± t α 2 ( s 1 n 1 ) +( s 2 n 2 )

( ) ( ) ( 2 )

t = x 1 − x 2 s 1 n + s n

( )

( ) ( )

1 2 1

2 2

2 2

1

2 1

2 1

2 2

2 1 2

2 1

−

−

=

n

s n

n

s n

s n s n

ν