Page 56 - 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.
–
x – µ 39.4 – 38
z = σ = 6.8 = 2.058
n
 
100
Step 5 : Make a decision.
To make a decision, we compare the value of the test statistic to the critical
value. This value of z = 2.058 is greater than the critical value of 1.96 and thus
it falls in the critical region. We reject H and conclude that the mean length of
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0
the iron nails produced by the manufacturer does not meet the target length of
38 mm.
Note: Example 6: Large sample, known population variance.

Population mean, variance unknown
2
In practice, the population variance σ is usually not known. However as long as the sample size is large, the
–
normal approximation for the sample mean X remains valid even if s is replaced by its unbiased estimate
–
^ X – µ .
σ. We can then apply the general test procedure using the test statistic Z =
^
σ
n

n
2
Note : σ = n s where s is the sample variance given by s = 1 ∑ (x – x) .
2
2
^ 2
– 2
n – 1 n i = 1
Example 7
A sample of 60 watches of a particular brand is checked for accuracy at 10:00:00 hours. Let μ denote the
true mean watch reading when the actual time is 10:00:00 hours. The resulting sample mean and sample
standard deviation are 10:00:01.2 hours and 1.8 seconds respectively. Use a significance level of 1% to
decide whether the evidence from the sample suggests the watches are fast.
–
Solution: Let x be the mean watch reading for the sample. Given information: μ = 10:00:00
–
hours, s = 1.8 seconds, n = 60, x = 10:00:01.2 hours.
2
^
s is calculated as s = n s
^
n – 1
= 60 × 1.8 2 5
59
= 1.815 seconds
We are going to test whether the mean watch reading is fast. The significance
level a is 0.01.
We carry out a hypothesis test using the following five steps.
Step 1 : State the null hypothesis and the alternative hypothesis.
H : μ = 10:00:00 hours,
0
H : μ . 10:00:00 hours.
1
Step 2 : Specify the significance level.
a = 0.01.
Step 3 : Select an appropriate probability distribution and determine the critical
region.

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