Topic 6 hypothesis testing, Cheat Sheet of Commercial Law

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Statistics - DADE

◼ Topic 6

◼ Hypothesis Testing: Two Populations

◼ Professor Dr. Carlos Alberto Lastras Rodríguez

Chapter Goals

After completing this chapter, you should be able to: ◼ Test hypotheses for the difference between two population means ◼ Two means, matched pairs ◼ Independent populations, population variances known ◼ Independent populations, population variances unknown but equal ◼ Complete a hypothesis test for the difference between two proportions (large samples) ◼ Use the F table to find critical F values ◼ Complete an F test for the equality of two variances

Dependent Samples

Tests Means of 2 Related Populations ◼ Paired or matched samples ◼ Repeated measures (before/after) ◼ Use difference between paired values: ◼ Assumptions: ◼ Both Populations Are Normally Distributed Dependent Samples

d

i

= x

i

- y i

Tests of the Difference Between Two Normal Population Means: Dependent Samples

Test Statistic:

Dependent Samples

The test statistic for the mean

difference is a t value, with

n – 1 degrees of freedom:

where n s d t d = sd = sample standard dev. of differences n = the sample size (number of pairs) Population Means, Dependent Samples n d d  i For tests of the = following form: H 0 : μ x

• μ y

H 0 : μx – μy ≤ 0 H 0 : μ x

• μ y

◼ Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:

Matched Pairs Example

Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, d i C.B. 6 4 - 2 T.F. 20 6 - 14 M.H. 3 2 - 1 R.K. 0 0 0 M.O. 4 0 - 4

- 21

d =

 di n

n 1 (d d) S 2 i d = − − = 

◼ Has the training made a difference in the number of complaints (at the a = 0.05 level)? d = - 4.

s / n

d

t

d

H

0 : μ x

• μ y

H

1 : μ x

• μ y

Test Statistic: Critical Value = ± 2. d.f. = n − 1 = 4 Reject a /

- 2.776 2. Decision: Do not reject H 0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints.

Matched Pairs: Solution

Reject a /

- 1. a =.

Difference Between Two Means

Population means, independent samples Test statistic is a z value Test statistic is a a value from the Student’s t distribution σx 2 and σy 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σx 2 and σy 2 assumed unequal (continued)

x

and σ

y

Known

Population means, independent samples

Assumptions:

▪ Samples are randomly and independently drawn ▪ both population distributions are normal ▪ Population variances are known

σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown

Test Statistic,

x

and σ

y

Known

Population means, independent samples

σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown

y 2 y x 2 x

n

n

x y

z

The test statistic for

μ x

• μ y

is:

H :μ μ 0 0 x y − =

Hypothesis Tests for

Two Population Means

Lower-tail test: H 0 : μx  μy H 1 : μx < μy i.e., H 0 : μ x

• μ y

H

1 : μ x

• μ y

Upper-tail test: H 0 : μx ≤ μy H 1 : μx > μy i.e., H 0 : μ x

• μ y

H

1 : μ x

• μ y

Two-tail test: H 0 : μx = μy H 1 : μx ≠ μy i.e., H 0 : μ x

• μ y

H

1 : μ x

• μ y

Two Population Means, Independent Samples

x

and σ

y

Unknown,

Assumed Equal

Population means, independent samples

Assumptions:

▪ Samples are randomly and independently drawn ▪ Populations are normally distributed ▪ Population variances are unknown but assumed equal

σx 2 and σy 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σx 2 and σy 2 assumed unequal

x

and σ

y

Unknown,

Assumed Equal

Population means, independent samples (continued) ▪ The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ ▪ use a t value with (n x

• n y

• 2. degrees of freedom

σx 2 and σy 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σx 2 and σy 2 assumed unequal

Decision Rules

Lower-tail test: H 0 : μx – μy  0 H 1 : μ x

• μ y

Upper-tail test: H 0 : μx – μy ≤ 0 H 1 : μ x

• μ y

Two-tail test: H 0 : μx – μy = 0 H 1 : μ x

• μ y

a a a/2 a/

• t a - t a/

t

a

t

a/ Reject H 0 if t < - t (^) (n1+n2 – 2), a Reject H 0 if t > t (^) (n1+n2 – 2), a Reject H 0 if t < - t (^) (n1+n2 – 2), a/2 or t > t (^) (n1+n2 – 2), a/ Two Population Means, Independent Samples, Variances Unknown

Pooled Variance t Test: Example

You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Sample mean 3.27 2. Sample std dev 1.30 1. Assuming both populations are approximately normal with equal variances, is there a difference in average yield (a = 0.05)?