Higher Engineering Mathematics

SIGNIFICANCE TESTING 605

J

of the difference of the means is within the range
±1.81, there isno significant difference between
the reaction times at a level of significance of 0.1.

Problem 10. An analyst carries out 10 analyses

on equal masses of a substance which is found

to contain a mean of 49.20 g of a metal, with a

standard deviation of 0.41 g. A trainee operator

carries out 12 analyses on equal masses of the

same substance which is found to contain a mean

of 49.30 g, with a standard deviation of 0.32 g.

Is there any significance between the results of

the operators?

Letμ 1 andμ 2 be the mean values of the amounts of
metal found by the two operators.
The null hypothesis is that there is no difference
between the results obtained by the two operators,
i.e.H 0 :μ 1 =μ 2.
The alternative hypothesis is that there is a differ-
ence between the results of the two operators, i.e.
H 1 :μ 1 =μ 2.
Under the hypothesisH 0 the standard deviations
of the amount of metal,σ, will be the same, and from
equation (12)

σ=

√

√

√

√

(

N 1 s^21 +N 2 s^22

N 1 +N 2 − 2

)

=

√(

(10)(0.41)^2 +(12)(0.32)^2

10 + 12 − 2

)

=0.3814

The t-value of the results obtained is given by
equation (11), i.e.,

|t|=

x 1 −x 2

σ

√(

1

N 1

+

1

N 2

)=

49. 20 − 49. 30

(0.3814)

√(

1

10

+

1

12

)

=−0.612

For the results to be probably significant, a two-tailed
test and a level of significance of 0.05 is taken.H 0 is
rejected outside of the ranget− 0. 975 andt 0. 975. The
number of degrees of freedom isN 1 +N 2 −2. For
t 0. 975 ,ν=20, from Table 61.2 on page 587, the range
is from−2.09 to+2.09. Since thet-value based on

the sample data is within this range,there is no sig-

nificant difference between the results of the two

operators at a level of significance of 0.05.

Now try the following exercise.

Exercise 225 Further problems on compar-

ing two sample means

1. A comparison is being made between batter-
   ies used in calculators. Batteries of typeA
   have a mean lifetime of 24 hours with a stan-
   dard deviation of 4 hours, this data being
   calculated from a sample of 100 of the bat-
   teries. A sample of 80 of the typeBbatteries
   has a mean lifetime of 40 hours with a stan-
   dard deviation of 6 hours. Test the hypothesis
   that the typeBbatteries have a mean lifetime
   of at least 15 hours more than those of typeA,
   at a level of significance of 0.05.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
   Ta kexas 24+15,
   i.e. 39 hours,z= 1 .28,z 0. 05 ,
   one-tailed test= 1 .645,
   hence hypothesis is
   accepted

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

2. Two randomly selected groups of 50 opera-
   tives in a factory are timed during an assem-
   bly operation. The first group take a mean
   time of 112 minutes with a standard deviation
   of 12 minutes. The second group take a mean
   time of 117 minutes with a standard devia-
   tion of 9 minutes. Test the hypothesis that the
   mean time for the assembly operation is the
   same for both groups of employees at a level
   of significance of 0.05.
   ⎡

⎢

⎢

⎣

z= 2 .357,z 0. 05 ,

two-tailed test=± 1 .96,

hence hypothesis is

rejected

⎤

⎥

⎥

⎦

3. Capacitors having a nominal capacitance of
   24 μF but produced by two different compa-
   nies are tested. The values of actual capaci-
   tance are:

Company 1 21. 423. 624. 822. 426. 3

Company 2 22. 427. 723. 529. 125. 8

Test the hypothesis that the mean capacitance

of capacitors produced by company 2 are

higher than those produced by company 1 at