Page 62 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing

Step 4 : Calculate the value of the test statistic.
^
p – p
z =
p(1 – p)
n

= 0.7767 – 0.82
0.82(1 – 0.82)
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= –1.952
Step 5 : Make a decision.
We compare the value of the test statistic to the critical value. This value of
z = –1.952 is greater than the critical value of –2.33 and thus it falls in the
nonrejection region. We do not reject H which states that the rate of germination
0
for the hybrid plant is 0.82.

Example 10

A manufacturing company has submitted a claim that 85% of components produced by a certain process
are non-defective. A new process is introduced to lower the proportion of defective components below
the current 15%. In a sample of 100 components produced with the new process, 7 are defective. Is this
evidence sufficient to conclude that the process has been improved? Use the 5% significance level.

Solution: Let p be the proportion of defective components produced by the existing process
and p be the corresponding proportion for the new improved process. Given
^
^
information: p = 0.15, n = 100, p = 7 .
100
We are going to test whether the evidence is sufficient to justify the improvement.
The significance level a is given as 0.05.
The following are the five steps in testing the hypothesis.
Step 1 : Formulate the null hypothesis and the alternative hypothesis.
H : p = 0.15,
0
H : p , 0.15.
1 5
Step 2 : Specify the significance level.
a = 0.05.

Step 3 : Select an appropriate probability distribution and determine the critical
region.

We have
np = 100 × 0.15
= 15
nq = 100 × 0.85
= 85
Since both np and nq are both greater than 5, the sample size is large. We will
use the normal distribution.

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