TESTING THE MEANS OF

TWO POPULATIONS

DEPENDENT SAMPLES2

With n-1 degrees of freedom

Where:

Sample problems:

1. A study was conducted to investigate some effects

of physical training. At 0.05 level of significance, test

the claim that the mean pre-training weight equals

the mean post-training weight. The weights are in

kilograms. What do you conclude about the effect

of training weight?

Pre-training weight :

99 57 62 69 74 77 59 92 70 85

Post-training weight :

94 57 62 69 66 76 58 88 70 84

3

2. A dose of the drug Captropil, designed to

lower systolic blood pressure, is administered to

10 randomly selected volunteers, with the

following results. Test the effectiveness of the

drug.

Systolic blood pressure before pill :

120 136 160 98 115 110 180 190 138 128

Systolic blood pressure after pill:

118 122 143 105 98 98 180 175 105 112

4

INDEPENDENT AND LARGE

SAMPLES

5

Sample problems:

1. Two methods of teaching statistics are being tried

by a professor. A class of 40 students is taught by

the method A and a class of 36 is taught by

method B. The two classes are given the same final

examination. The mean scores are 78 and 74,

respectively. Using a 0.05 significance level, can we

conclude that the average final examination

scores produced by the two methods are different

if the population standard deviation for each is 5?

6

2. In a study of abstract reasoning, a sample

group of male and female students scored

as shown below. Assuming that the

samples are random, would further testing

of similar groups of male and female

students give virtually the same result?

Sex Sample size Mean SD

Male 95 29.21 11.56

Female 83 30.92 7.81

7

THE ‘RULE’ OF 2.5

Divide the higher variance by the lower

variance of the two independent

samples.

If the ratio is less than 2.5, use the formula

for equal variance

If the ratio is at least 2.5, use the formula for

unequal variance

8

Independent and small samples with

equal variances

9

Sample problems:1. The Telektronic Company manufactures car

batteries with two different production methods.

The lives in years of the batteries are found for a

sample of from each group. At 0.05 level of

significance, test the claim that the two

production methods yield batteries with the same

mean.

Methods Sample size Mean St. Dev.

Traditional 29 4.31 0.37

Experimental 27 4.07 0.31

10

2. A teacher wishes two methods of teaching

verbs to his Spanish class. He divided the class

randomly into two samples with 14 students

each. Using method 1, the students learn an

average of 43.8 verbs while method 2 yield

an average of 38.6 verbs each with a

standard deviation of 5.2. Theory suggest that

method 1 is superior to method 2. Test the

theory using 0.01 level of significance.

11

Independent and small with

unequal variances

12

Sample problems

1. The Gleanson Supermarket’s manager experiments with

two methods of checking out customers. One method

requires that the cashier manually key each price into

their register, and the other method uses a scanner that

automatically registers prices. The time in minutes are

given below for sample of goods checked out. At 0.01

level, test the claim that the two methods have the same

mean.Methods Sample size Mean Variance

Manual 16 157.6 225

Scanning 10 112.4 56

13

2. The Wisconsin Bottling Company employs two

shifts of workers to fill bottle with iced tea. Test

the claim that there is no significant difference

between the mean fill amount for the day shift

and the night shift at 5% level of significance.

Shift Sample size Mean St. Dev.

Day 25 12.02oz. 0.14oz.

Night 28 11.85oz. 1.22oz.

14

Exercises:Do a complete hypothesis for the following problems.

Problem 1:

At the beginning of the school year, the mean

score of the group of 24 students in an

educational achievement test in reading was

45 with standard deviation of 6. At the end of

the school year, the mean score on an

equivalent for of test was 50 with standard

deviation of 5. Did the class make a significant

progress in reading during this year? Use alpha

0.05.

16

Problem 2: cars radial tires belted tires1 4.2 4.12 4.7 4.93 6.6 6.24 7 6.95 6.7 6.86 4.5 4.47 5.7 5.78 6 5.89 7.4 6.9

10 4.9 4.711 6.1 612 5.2 4.9

A taxi company is trying to

decide whether the use of

radial tires instead of regular

belted tires improves fuel

economy. Twelve cars were

driven twice over a

prescribed test course, each

time using a different type of

tires (radial and belted) in

random order. The mileage, in

kilometers per liter were

recorded.

17

Problem 3:

subject Before After

A 6.6 6.8

B 6.5 2.4

C 9 7.4

D 10.3 8.5

E 11.3 8.1

F 8.1 6.1

G 6.3 3.4

H 11.6 2

Based on “An Analysis of Factors That

Contribute to the Efficacy of Hypnotic

Analgesia”, by Price and Barber,

Journal of Abnormal Psychology, Vol.

96, No. 1. A study was conducted to

investigate the effectiveness of

hypnotism in reducing pain. Results for

randomly selected subjects are given

below. At 0.01 level of significance, test

the claim that the sensory

measurements are lower after

hypnotism. Does hypnotism appear

effective in reducing pain?

18