Higher Engineering Mathematics

SIGNIFICANCE TESTING 597

J

Probability of no defective screws in a sample,

qN= 0. 925 =0.0718

Probability of 1 defective screw in a sample,

NqN−^1 P= 25 × 0. 924 × 0. 1 =0.1994

Probability of 2 defective screws in a sample,

N(N−1)

2

qN−^2 p^2

=

25 × 24

2

× 0. 923 × 0. 12 =0.2659

Probability of 0, 1, or 2 defective screws

in a sample =0.5371

That is, the probability of atype II error, i.e. leaving
the machine running even though the defect rate has
risen to 10%,is 53.7%.

Now try the following exercise.

Exercise 223 Further problems on type I

and type II errors

Problems 1 and 2 refer to an automatic machine

producing piston rings for car engines. Random

samples of 1000 rings are drawn from the output

of the machine periodically for inspection pur-

poses. A defect rate of 5% is acceptable to the

manufacturer, but if the defect rate is believed to

have exceeded this value, the machine producing

the rings is stopped and adjusted.

In Problem 1, determine the type I errors which

occur for the decision rules stated.

1. Stop production and adjust the machine if a
   sample contains (a) 54 (b) 62 and (c) 70 or
   more defective rings.
   [
   (a) 28.1% (b) 4.09%
   (c) 0.19%

]

In Problem 2, determine the type II errors which

are made if the decision rule is to stop production

if there are more than 60 defective components

in the sample.

2. When the actual defect rate has risen to (a) 6%
   (b) 7.5% and (c) 9%.
   [(a) 55.2% (b) 4.65% (c) 0.07%]
   3. A random sample of 100 components is
   drawn from the output of a machine whose
   defect rate is 3%. Determine the type I error
   if the decision rule is to stop production when
   the sample contains: (a) 4 or more defec-
   tive components, (b) 5 or more defective
   components, and (c) 6 or more defective
   components.
   [(a) 35.3% (b) 18.5% (c) 8.4%]
   4. If there are 4 or more defective components
   in a sample drawn from the machine given in
   problem 3 above, determine the type II error
   when the actual defect rate is: (a) 5% (b) 6%
   (c) 7%.
   [(a) 26.5% (b) 15.1% (c) 8.18%]

62.3 Significance tests for population

means

When carrying out tests or measurements, it is often

possible to form a hypothesis as a result of these tests.

For example, the boiling point of water is found to

be: 101.7◦C, 99.8◦C, 100.4◦C, 100.3◦C, 99.5◦C and

98.9◦C, as a result of six tests. The mean of these

six results is 100.1◦C. Based on these results, how

confidently can it be predicted, that at this particular

height above sea level and at this particular baromet-

ric pressure, water boils at 100.1◦C? In other words,

are the results based on samplingsignificantly dif-

ferentfrom the true result? There are a variety of

ways of testing significance, but only one or two of

these in common use are introduced in this section.

Usually, in significance tests, some predictions about

population parameters, based on sample data, are

required. In significance tests for population means,

a random sample is drawn from the population and

the mean value of the sample,x, is determined. The

testing procedure depends on whether or not the

standard deviation of the population is known.

(a) When the standard deviation of the

population is known

A null hypothesis is made that there is no differ-

ence between the value of a sample meanxand that

of the population mean,μ, i.e.H 0 :x=μ. If many

samples had been drawn from a population and a

sampling distribution of means had been formed,

then, providedNis large (usually taken asN≥30)

the mean value would form a normal distribution,

having a mean value ofμxand a standard deviation

or standard error of the means (see Section 61.3).