Higher Engineering Mathematics

604 STATISTICS AND PROBABILITY

N 2 =50. Hence, equation (9) can be used to deter-

mine thez-value of the difference of the sample

means. From equation (9),

z=

x 1 −x 2

√

√

√

√

(

σ 12

N 1

+

σ^22

N 2

)=

74 − 78

√(

82

40

+

72

50

)

=

− 4

1. 606

=− 2. 49

(a) For a two-tailed test, the results are probably
significant at a 0.05 level of significance when
z lies between −1.96 and+1.96. Hence the
z-value of the difference of means shows there
is ‘no significance’, i.e. thatproduct 1 is signif-
icantly different from product 2 at a level of
significance of 0.05.

(b) For a two-tailed test, the results are highly signif-
icant at a 0.01 level of significance whenzlies
between−2.58 and+2.58. Hence there isno
significant difference between product 1 and
product 2 at a level of significance of 0.01.

Problem 9. The reaction time in seconds of two

people,AandB, are measured by electrodermal

responses and the results of the tests are as shown

below.

PersonA(s) 0.243 0.243 0.239

PersonB(s) 0.238 0.239 0.225

PersonA(s) 0.232 0.229 0.241

PersonB(s) 0.236 0.235 0.234

Find if there is any significant difference

between the reaction times of the two people

at a level of significance of 0.1.

The mean, x, and standard deviation,s, of the

response times of the two people are determined.

xA=

0. 243 + 0. 243 + 0. 239 + 0. 232

+ 0. 229 + 0. 241

6

= 0 .2378 s

xB=

0. 238 + 0. 239 + 0. 225 + 0. 236

+ 0. 235 + 0. 234

6

= 0 .2345 s

sA=

√ √ √ √ √ √ √

⎡

⎢

⎢

⎣

(0. 243 − 0 .2378)^2 +(0. 243 − 0 .2378)^2

+···+(0. 241 − 0 .2378)^2

6

⎤

⎥

⎥

⎦

= 0 .00543 s

sB=

√ √ √ √ √ √ √

⎡

⎢

⎢

⎣

(0. 238 − 0 .2345)^2 +(0. 239 − 0 .2345)^2

+···+(0. 234 − 0 .2345)^2

6

⎤

⎥

⎥

⎦

= 0 .00457 s

The null hypothesis is that there is no difference

between the reaction times of the two people, i.e.

H 0 :xA−xB=0.

The alternative hypothesis is that the reaction

times are different, i.e.H 1 :xA−xB=0 indicating

a two-tailed test.

The sample numbers (combined) are less than 30

and at-distribution is used. The standard deviation of

all the reaction times of the two people is not known,

so an estimate based on the standard deviations of the

samples is used. Applying Bessel’s correction, the

estimate of the standard deviation of the population,

σ^2 =s^2

(

N

N− 1

)

gives σA=(0.00543)

√(

6

5

)

= 0. 00595

and σB=(0.00457)

√(

6

5

)

= 0. 00501

From equation (10), thet-value of the difference of

the means is given by:

|t|=

xA−xB

√

√

√

√

(

σA^2

NA

+

σB^2

NB

)

=

0. 2378 − 0. 2345

√(

0. 005952

6

+

0. 005012

6

)

=1.039

For a two-tailed test and a level of significance

of 0.1, the column heading in thet-distribution of

Table 61.2 (on page 587) ist 0. 95 (refer to Problem 6).

The degrees of freedom due tokbeing 2 isν=

N 1 +N 2 −2, i.e. 6+ 6 − 2 =10. The corresponding

t-value from Table 61.2 is 1.81. Since thet-value