Page 60 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing

Example 8

An airline claims that, on average, 6% of its flights are delayed each day. On a given day, of 20 flights, 2
are delayed. Test the hypothesis that the proportion of delayed flights is 6% at the significance level of 0.05.
Solution: Step 1 : State the null hypothesis and alternative hypothesis.
H : p = 0.06,
0
H : p . 0.06.
1
Step 2 : Specify the significance level.
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a = 0.05.
Step 3 : Select an appropriate probability distribution and determine the critical
region.
We have np = 20 × 0.06 nq = 20 × 0.94
= 1.2 = 18.8
Since np , 5, the sample size is small. We will use binomial distribution to
evaluate directly the probability.
This is a one-tailed test with a critical region falling in right side. The sample
proportion can be considered an outcome of a binomial experiment with
p = 0.06 and n = 20. All x values such that P(X > x) , 0.05.
Step 4 : Calculate the appropriate binomial probability.

We have x = 2 and n = 20,
20 20
x
P(X > 2) = ∑ 1 2 0.06 (1 – 0.06) 20 – x
x = 3 x
2 20
x
= 1 – ∑ 1 2 0.06 (1 – 0.06) 20 – x
x = 0 x
= 1 – 0.885
= 0.115
Step 5 : Make a decision.
To make a decision, we compare the value of the binomial probability to the
significance level. This value of 0.115 is greater than the significance level of 0.05
and thus it falls in the nonrejection region. We do not reject H and conclude
0
that there is insufficient reason to question the airline’s claim. 5

Population proportion, large sample

^
When the sample size n is large, the sample proportion p = x is approximately normally distributed with
n
mean and standard deviation equal to µ ^ = p and s ^ = p(1 – p) respectively.
p p n
Hence, for a large sample, we use normal distribution to perform a hypothesis test about the population
proportion p. The sample size is large if np . 5 and nq . 5. Then the test statistic is given by

p – p
^
Z = ,
pq
n

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05 STPM Math(T) T3.indd 251 28/10/2021 10:24 AM