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
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