Two Samples Tests - Statistics - Lecture Slides, Slides of Statistics

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Testing of Hypotheses

Two-Samples Tests

Central Limit Theorem

standard deviation.

distribution with mean and

thedistribution of x approachesa normal

standard deviationof , then asn increases

n from a population with meanof and

If x is the mean of a randomsampleof size

x

x

n

σ

μ

σ

μ

σ

μ

Hypothesis Testing

• A consumer organization might want to test

two brands of light bulbs to determine

whether one burns longer than the other.

• A company wanting to relocate might want to

know the significant difference between the

average prices of homes in two different cities.

Sampling Distribution of the Difference Between Two Sample Means

x (^) 1 =^ ∑ n 1 x

Population 1

Population 2

x (^) 2 =^ ∑ n 2 x

X 1 − X 2

X 1

X 2

X (^) 1 − X 2

x 1

x 1 (^) x 2 x^1 x^2 x 2

Z Formula for the Difference

in Two Sample Means

n n

z x x

2

2 2 1

2 1

(^1212)

When σ 12 andσ 22 are known and

Independent Samples

The Wage Example

• Suppose we want to conduct a hypothesis test

to determine whether the average annual

wage for an advertising manager is different

from the average annual wage of an auditing

manager. Assume significance level as 0.05.

Hypothesis Testing for Differences Between Means: The Wage Example

μ X (^) 1 − X 2

RejectionRegion

Non Rejection Region Critical Values

Rejection Region

X (^) 1 − X 2

H H

o a

: :

1 2 1 2

0 0

μ μ μ μ

− = − ≠

μ (^) x 1 (^) − x 2 x 1 (^) − x 2

Hypothesis Testing for Differences Between Means: The Wage Example Advertising Managers 74.256 57.791 71. 96.234 89.807 65.14596.767 67.57459. 103.030^ 93.261^ 77.24267.056^ 62.48369. 74.195 75.932 64.27674.194 35.39486. 80.742 39.672 65.36073.904 57. 45.652 93.083 54.27059. 63.384 68.

69.962^ Auditing Managers 77.136 43. 55.052 57.828 66.03554.335 63.36959. 63.362 37.194 42.49483.849 54.44946. 99.198 61.254 67.16037.386 71.80472. 73.065 48.036 59.50572.790 56.47067. 60.053 66.359 71.35158.653 71. 61.261 63.

1. 164

7016 .. (^700253) 32

112

(^11)

==

σ

σ^ x

n

1. 411

6212 .. (^187900) 34 222

(^22)

==

σ^ x

n

Demonstration of the Sketch: One Tailed Test

: 0

: 0 1 2

1 2 − <

− = μ μ

μ μ a

o H

H

Non Rejection Region Critical Value

RejectionRegion α =. zZc c == −− 3 3 08.. (^08) 0

• Example 1: Block Enterprises, a manufacturer of chipsof computers, is in the process deciding whether to replace its current semi automated assembly linewith a fully automated assembly line. Block has gathered some preliminary test data about hourlychip production which is summarized in the following table and it would like to know whether it shouldupgrade its assembly line. State (and test α=0.02) appropriate hypotheses to help Block decide. x bar s n Semiautomatic line Automatic line 198206 3229 150200

t Formula to Test the Difference in Means

1 1 2

( 1 ) ( 1 )

( ) ( ) n n n n

s n s n t x x − ++ − − +

Hernandez Manufacturing

Company

• In Hernandez, new employees are trained by seminarmethod and at the end they are tested to measure the knowledge about the company. Management decided toexperiment with a different training procedure which process new employees by using videocassettes. If thisprocedure works, it could save company thousands dollars over a period of several years. However, there issome concern about the effectiveness of new method, and company managers would like to know whetherthere is any difference in the effectiveness of the two training methods.

Hernandez Manufacturing Company

56^ Training Method A 51 45 (^4742 5253 ) (^5047 4244 )

59^ Training Method B (^5253) 54

(^5756) (^5564)

(^5365) (^5357)

Testing Difference b/w Means with

Dependent Sample

• Before and after measurements on the

same individual

• Studies of twins
• Studies of spouses