Chapter 9: Hypothesis Testing

Section 7: Testing Differences of Two

Means or Two Proportions

(Independent Samples)

Large Samples (Independent)

Test Statistic

2

22

1

21

2121

nn

xxz

Test Statistic

, as stated in the null hypothesis = sample mean of data = sample mean of data = standard deviation of data = standard deviation of data = sample size of data = sample size of data

021

1x

2x

1

2

1n

2n

Example A consumer group is testing camp stoves. To test the

heating capacity of a stove, they measure the time required to bring 2 quarts of water from 50°F to boiling. Two competing models are under consideration Thirty-six stoves of each model were tested and the following results are obtained.

Model 1: = 11.4 and = 2.5 Model 2: = 9.9 and = 3

Is there any difference between the performances of these two models? Use a 5% level of significance and find the p-value.

1x

2x1s

1s

Small Samples (Independent)

Test Statistic

df = n1 + n2 – 2

21

21

11

nns

xxt

2

11

21

222

211

nn

snsns

Example

Two competing headache remedies claim to give fast-acting relief. An experiment was performed to compare the mean lengths of times required for bodily absorption of brand A and brand B headache remedies. Groups were randomly selected to use the remedies. Results (in minutes) are as follows:

ExampleBrand A Brand B

n1 = 12 n2 = 12

= 20.1 = 18.9

s1 = 8.7 s2 = 7.5

Use a 5% level of significance to test the claim that there is no difference in the mean times required for bodily absorption.

1x 2x

Proportions Test Statistic

21

21

ˆˆˆˆ

ˆˆ

n

qp

n

qp

ppz

21

21ˆnn

rrp

pq ˆ1ˆ

1

11ˆ n

rp

2

22ˆ n

rp

Example

In order to improve voter registration, reminders are sent in the mail to citizens who are eligible to register. To determine if this method will improve voter registration, two groups are used in a study. In the first group of 625 people, no reminders are sent and 295 registered. In the second group of 625 people, reminders are sent and 350 registered. The county clerk claims that the proportion of people to register was significantly greater in the second group. Use a 5% level of significance to test the claim.