Hypothesis Testing & Confidence Intervals for Two Population Means - Prof. Mark Woychick, Study notes of Introduction to Business Management

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Inference Overview: Chapters 8- 10

Popu- lation

Method Parameter Distribution Chap- ter One Required sample size n/a z 8 Confidence interval Mean z for sigma known t for sigma unknown

8

Proportion z 8 Hypothesis test Mean z for sigma known t for sigma unknown

9

Proportion z 9 Two Confidence interval Mean z for sigma known t for sigma unknown

10

Hypothesis test Mean z for sigma known t for sigma unknown

10

Hypothesis testing – two populations

• http://blog.asmartbear.com/easy-statistics-for-adwords-ab-testing-and-hamsters.html

Hypothesis testing recap

• The Null: − Is the statement about the population parameter that will be tested − Always includes an equality (= <=, >=) − The “benefit of the doubt” goes to the null hypothesis − The status quo goes into the null hypothesis
• The Alternative − Is the opposite of the null hypothesis − Challenges the status quo

− Never contains the “=” , “≤” or “” sign

− Is generally the hypothesis that is believed (or needs to be supported) by the researcher – a research hypothesis

Process of Hypothesis Testing

• 1. Specify population parameter of interest
• 2. Formulate the null and alternative hypotheses
• 3. Specify the desired significance level, α
• 4. Define the rejection region
• 5. Take a random sample and determine whether or not the sample result is in the rejection region
• 6. Reach a decision and draw a conclusion

Working with two populations

• Form interval estimates
• Test hypotheses
• For two independent population means
  • Standard deviations known
  • Standard deviations unknown

Assumptions

• When σ 1 and σ 2 are known

− Samples are independent

− Sample size >= 30

• When σ 1 and σ 2 are unknown:

− Populations are normally distributed

− Populations have equal variances

− Samples are independent

Confidence Interval Estimate

Point Estimate

Lower Confidence Limit

Upper Confidence Limit Width of confidence interval

Point Estimate  (Critical Value)(Standard Error)

Point estimate and standard error

• Point Estimate  (Critical Value)(Standard Error)
• Point estimate for the difference is x 1 – x 2
• Critical value is z for known sigma; t for unknown
• Standard error formulas:

σ 1 and σ 2 are known: σ 1 and σ 2 are

unknown :

x x

n

σ

n

σ

σ

1 2

 



   

n n 2

n 1 s n 1 s s 1 2

2 2 2

2 1 1 p  

   

Confidence intervals: μ 1 – μ

 

n

σ

n

σ

x  x  z   

(^12) p

n

n

x  x  t s 

n

s

x t

n

σ

x  z

For two populations, formulas are similar: σ 1 and σ 2 are known: σ 1 and σ 2 are unknown :

σ known: σ unknown : Single population

Hypothesis Tests for the Difference

Between Two Means

• Testing Hypotheses about μ 1 – μ 2

• Use the same situations discussed already:

−Standard deviations known

−Standard deviations unknown

Hypothesis Tests for Two Populations

Lower tail test: H 0 : μ 1  μ 2 HA: μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 HA: μ 1 – μ 2 < 0

Upper tail test: H 0 : μ 1 ≤ μ 2 HA: μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 HA: μ 1 – μ 2 > 0

Two-tailed test: H 0 : μ 1 = μ 2 HA: μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 HA: μ 1 – μ 2 ≠ 0

Two Population Means, Independent Samples Two Population Means, Independent Samples

Lower tail test: H 0 : μ 1 – μ 2  0 HA: μ 1 – μ 2 < 0

Upper tail test: H 0 : μ 1 – μ 2 ≤ 0 HA: μ 1 – μ 2 > 0

Two-tailed test: H 0 : μ 1 – μ 2 = 0 HA: μ 1 – μ 2 ≠ 0

a a a/2 a/

• za za - za/2 za/ Reject H 0 if z < - za Reject H 0 if z > za Reject H 0 if z < - za/ or z > za/

Hypothesis tests for μ 1 – μ 2

Example: σ 1 and σ 2 known:

Test statistics: μ 1 – μ 2

σ known: σ unknown :

   

n

σ

n

σ

x x μ μ z



   

   

p

n

1

n

1 s

x x μ μ t



   

n

σ

x μ

z





n

s

x μ

t





Single population

For two populations, formulas are similar: σ 1 and σ 2 are known: σ 1 and σ 2 are unknown :