[2] Hypothesis Testing on Two Populations

•Independent Samples:

Samples taken from two different

populations, where the selection process for one sample is

independent of the selection process for the other sample.

•Dependent Samples:

Samples taken from two populations where either (1) the element sampled is a member of both

populations or (2) the element sampled in the second population is selected because it is similar on all other characteristics, or

“matched,” to the element selected from the first

•Independent Samples:

◦ Testing a company’s claim that its peanut butter contains less fat than that

produced by a competitor.

•Dependent Samples:

◦ Testing the relative fuel efficiency of 10 trucks that run the same route twice, once with the

current air filter installed and once with the new filter.

•Test Statistic

◦ with s₁2 _and s

22 as estimates for s12 and s22

z = [ x 1 – x 2 ] – [ m 1 – m 2 ] 0 s 1 2 n 1 + s 2 2 n 2

•Test Statistic and df = n₁ + n₂ – 2; 2 – 2 1 2 2 ) 1 – 2 ( 2 1 ) 1 – 1 ( 2 where 2 1 1 1 2 0 ] 2 – 1 [ – ] 2 – 1 [ n n s n s n p s n n p s x x t +  +  = + =              

m

m

•Test Statistic





1

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(

1

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(

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(

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(

where

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(

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(

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+

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+

=

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=

n

n

s

n

n

s

n

s

n

s

df

n

s

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x

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t

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•Pooled-variances

t

•The

F

assumption.

•The

F

distribution of

s

s

22 that would result if two

samples were repeatedly drawn from a single normally distributed population.

•If

s

_s

22 , then

s

s

hypotheses can be worded either way.

Test Statistic: whichever is larger

•The critical value of the F will be F(a/2,

n

n

◦ where a = the specified level of significance n₁ = (n – 1), where n is the size of the

sample with the larger variance

n₂ = (n – 1), where n is the size of the sample with the smaller variance

2 1 2 2 or 2 2 2 1 s s s s F =

•Test Statistic

◦ where d = (x₁ – x₂)

= Sd/n, the average difference

n

observations

s

d

df

n

 A study is conducted to whether different

training methods have an effect on the productivity of employees in a company

manufacturing electronic equipment. Twelve recently hired employees were divided into two groups of 6. The first group received a

computer-assisted, individual-based training program, and the other received a collaborative team-based training program. After the training, the employees were evaluated on the time (in

seconds) it takes to assemble an electronic part. The data from the study are tabulated below.

Team

Assembly Time (in seconds)

Computer-assisted

individual-based

19.4 19.4 20.7 21.8 19.3 18.5

Team-based

program

22.4 15.6 16.0 21.7 30.7 20.8

Is there a sufficient evidence to conclude that employees under computer-assisted

individual-based program have significantly faster assembly time than those employees under team-based program? Use 5% level of significance.

(a) Assume that the variances of the assembly

of training methods are equal.

(b) Assume that the variances of the assembly

•Problem : An educator is considering two different videotapes for use in a half-day session designed to introduce students to the basics of economics.

Students have been randomly assigned to two groups, and they all take the same written examination after viewing the videotape. The scores are summarized below. Assuming normal populations with equal standard deviations, does it appear that the two

videos could be equally effective? What is the most

accurate statement that could be made about the

p-value for the test?

Videotape 1: = 77.1, s₁ = 7.8, n₁ = 25 Videotape 2: = 80.0, s₂ = 8.1, n₂ = 25

x ₁ x

•I. H₀:

µ

µ

equally effective. There is no difference in student performance.

H₁:

µ

µ

•II. Rejection Region

a = 0.05 df = 25 + 25 – 2 = 48 Reject H₀ if t > 2.011 or t < –2.011 0.025 0.95 0.025 t=-2.011 t=2.011 Do Not Reject H ₀ 0 0 Reject H Reject H

•Test Statistic 225 . 63 48 64 . 1564 16 . 1460 2 – 25 25 2 ) 1 . 8 ( 24 2 ) 8 . 7 ( 24 2 = + = +  +  = p s 289 . 1 – 25 1 25 1 225 . 63 0 . 80 – 1 . 77 2 1 1 1 2 2 – 1 = + = + =                           n n p s x x t

•IV. Conclusion:

Since the test statistic of t = – 1.289 falls between the critical bounds of t = ± 2.011, we do not reject the null hypothesis with at least 95% confidence.

•V. Implications:

There is not enough evidence for us to conclude that one videotape training session is more effective than the other.

•

p

Using Microsoft Excel, type in a cell: =TDIST(1.289,48,2)

 A taxi company is trying to decide whether

the use of radial tires instead of regular

belted tires improves fuel economy. Twelve cars were equipped radial tires and driven over a prescribed test course. Without

changing drivers, the same cars were then

equipped with regular belted tires and driven once again over the test course. The gasoline consumption, in kilometers per liter, was

Car

1 2 3 4 5 6 7 8 9 10 11 12

Radial Tires

Belted Tires

4.2 4.7 6.6 7.0 6.7 4.5 5.7 6.0 7.4 4.9 6.1 5.2

4.1 4.9 6.2 6.9 6.8 4.4 5.7 5.8 6.9 4.7 6.0 4.9

At the 0.01 level, can we conclude that cars equipped with

radial tires give better fuel economy than those equipped with

belted tires?

•Test Statistic ◦ where p = n 1 p 1 + n 2 p 2 n 1 + n 2

 Suppose that in a poll survey, 925 out of

2500 respondents would like candidate A to be elected as the president of the country, and 840 out of 2500 would like candidate B to succeed as the president. Do we have

reason to believe that the candidate A would win over candidate B as the president? Use 5% level of significance.

 In a test of the quality of two television

commercials, each commercial was shown in a separate test area six times over a one-week period. The following week a telephone survey was conducted to identify individuals who had seen the commercials. Those individuals were asked to state the primary message in the

commercials. The following results were recorded:

 Commercial A Commercial B

 Number Who Saw Commercial 150 200

 Use 5% level of significance and test the

hypothesis that there is no difference in the recall proportions for two commercials.

 The Bureau of Transportation tracks the flight

arrival performances of the 10 biggest airlines in the United States (Wall Street

Journal, 2003). Flights that arrive within 15 minutes of schedule are considered on time. Using sample data below:

 January 2001: A sample of 924 flights

showed 742 on time

 January 2002: A sample of 841 flights

 State the hypotheses that could be tested to

determine whether the major airlines

improved on-time flight performance during the one-year period. What is your conclusion at 5% level of significance.

 A firm is studying the delivery time of two raw material

suppliers. The firm is basically satisfied with supplier A and is prepared to stay with that supplier if the mean delivery time is the same or less than that of supplier B. However, if the firm finds that the mean delivery time of supplier B is less that that of supplier A, it will begin

making raw material purchases from supplier B.

 Supplier A Supplier B

 n₁ = 50 n₂ = 31

 mean= 14 days mean = 12.5 days

 s₁ = 3 days s₂ = 2 days



 What are the null and alternative hypotheses? With 5% level

of significance, what action do you recommend in terms of supplier selection?