Chapter 10, sections 1 and 4Two-sample Hypothesis Testing

• Test hypotheses for the difference between two independent population means (standard deviations known or unknown)

• Use the F table to find critical F values

• Complete an F test for the difference between two variances

Difference Between Two MeansGoal: Test hypothesis or form a confidence interval for the

difference between two population means, μ1 – μ2.Assumptions:

• Different data sources-- populations are Unrelated and Independent

• two samples are randomly and independently drawn from these populations.

• Sample selected from one population has no effect on the sample selected from the other population

• population distributions are normal or both sample sizes are 30

11

111

,:1Population

n,S , :1 Sample

X

22

2 22

,:2Population

n,S , :2 Sample

X

Possible Hypotheses Are:

Upper-tail test:

H0: μ1 ≤ μ2

H1: μ1 > μ2

i.e.,

H0: μ1 – μ2 ≤ 0H1: μ1 – μ2 > 0

Lower-tail test:

H0: μ1 μ2

H1: μ1 < μ2

i.e.,

H0: μ1 – μ2 0H1: μ1 – μ2 < 0

Two-tail test:

H0: μ1 = μ2

H1: μ1 ≠ μ2

i.e.,H0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0

• Population standard deviations are known, σ1 and σ2 known, use Z test.

• The test statistic for μ1 – μ2 , based on sample

sample means, is:

• The Confidence interval for μ1 – μ2 is:

2

2

2

1

2

1

2121

n

σ

n

σ

μμXXZ

)XX( 21

21 XXσ ,population pooledthe of Error Standard

2

22

1

21

21)21n

σ

n

σXX( Z

σ1 and σ2 UnknownAssumptions: Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least

30 Population variances are unknown but assumed equal, σ1 and σ2

unknown, but known to be equal The population variances are assumed equal, so use the two sample

standard deviations and pool them to estimate σ The test Statistic for μ1 – μ2 has a t distribution with a degree

of freedom of (n1+n2-2):

21

2

p

2121

n

1

n

1S

μμXXt

• Where:

• The confidence interval for μ1 – μ2 is:

• Example:• Last week you were given a sample of 69 beer, 54 are U.S.-made and

15 are foreign-made. Additional information were provided with respect to price, calories, and percent alcohol content.

• Let’s assume the two populations are unrelated, independent, and approximately normally distributed with equal variance

• Assume that the two samples are independently drawn.

1)n()1(n

S1nS1nS

21

2

22

2

112

p

21

2p2-nn2121

n

1

n

1SXX)(

21t

Samples Information: Price($) Calories %Alcohol Content

NON U.S. SampleMean 5.86 138.60 4.13

Std Deviation 0.90 27.61 1.74Sample s ize 15 15 15

Variance $0.82 $762.40 $3.04

U.S. SampleMean 4.71 143.39 4.50

Std Deviation 1.48 30.67 1.53Sample s ize 54 54 54

Variance 2.18 940.73 2.35

• Questions:1. Is there evidence of a difference in mean calories

of us and non-U.S. beers?2. What is the 95% confidence interval for the

difference in mean calories?3. Are conclusions in 1 and 2 consistent?4. U.S. beers have about 10% more alcohol than

non-U.S. beers.5. Is the assumption of equal population variances,

that you used for 1 and 4 a valid assumption? 6. Is there evidence that there is less variation in

price of imported beers than price of domestic beers

Hypothesis Tests for Variances• Test of two population variances

• Hypotheses:H0: σ1

2 = σ22

H1: σ12 ≠ σ2

2 Two-tail test

Lower-tail test

Upper-tail test

H0: σ12 σ2

2

H1: σ12 < σ2

2

H0: σ12 ≤ σ2

2

H1: σ12 > σ2

2

H0: σ12 / σ2

2=1H1: σ1

2 / σ22≠1

H0: σ12 / σ2

2 1H1: σ1

2 / σ22 <1

H0: σ12 / σ2

2 ≤1H1: σ1

2 / σ22 >1

• The test statistic from samples is

• F-Distribution can take values from 0 to infinity

• It is a right-skewed distribution

2

2

2

1),1n(),1n(

S

SF 21

= Variance of Sample 1 n1 - 1 = numerator degrees of freedom

n2 - 1 = denominator degrees of freedom = Variance of Sample 2

21S

22S

F 0 /2Do not

reject H0

FU FL

/2

1. Finding the critical lower and upper tail values

2. Find FU from the F table for n1 – 1 numerator and n2 – 1

denominator degrees of freedom

3. Find FL using the formula:

• Where FU* is from the F table with n2 – 1 numerator and n1 – 1

denominator degrees of freedom (i.e., switch the d.f. from FU)

H0: σ12 = σ2

2

H1: σ12 ≠ σ2

2

*U

L F

1F