Page 53 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing
Determine the
(a) rejection region,
(b) nonrejection region,
(c) critical value,
(d) significance level.

13. It is given that H : μ = 70 is tested against H : μ > 70. State whether the test is a one-tailed test or
0
1
a two-tailed test. Assuming that the sample mean is normally distributed, sketch a normal curve and
indicate the rejection and nonrejection regions with a critical value arbitrarily chosen.
14. A hypothesis test is stated as H : μ = 15 versus H : μ ≠ 15. State whether the test is a one-tailed test
1
0
or a two-tailed test. Assuming that the sample mean is normally distributed, sketch a normal curve
and indicate the rejection and nonrejection regions with an arbitrarily critical value.
15. The pass rate of driving test in a country is reported to be 0.65. To test whether this claim is true, a
researcher selects a random sample of 20 driving test candidates. If the number of candidates passing
the driving test in the sample is anywhere from 9 to 17, the researcher decides not rejecting the null
hypothesis that p = 0.65; otherwise, he concludes that p ≠ 0.65. Use the binomial distribution to
determine the significance level of the test.
16. A record in a country reveals that the population mean life span of its people last year is 68 years
with a standard deviation of 7.7 years. A random sample of 50 recorded deaths in that country
during this year shows an average life span of 70.1 years. A critical region for the test statistic is such
–
that x > 70.1, find the corresponding value of the test statistic.

= 1 – 5.2 Testing Population Mean
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Evidence concerning the value of a population mean is provided by the sample mean. In the case where the
population variance is known, and a small sample is taken from a normal population or a large sample is
taken from any population, normal distribution is used to test a hypothesis about a population mean. In the
case where the population variance is unknown, a large sample is taken from any population, the hypothesis
test about a population mean can be carried out using a normal distribution as an approximation.

Population mean, variance known

We recall that the sampling distribution of the (sample) mean is normal for samples (any size) drawn from a
5 normal population, and is approximately normal for large sample drawn from any population. The sampling
2
2
distribution has mean µ – = µ and variance σ – = s n 2 , where μ and σ are the mean and variance of the
X
X
population from which we pick random samples of size n.
2
Suppose the population has unknown mean μ and known variance σ . Consider the hypotheses
H : μ = μ ,
0
0
H : μ ≠ μ .
0
1
Under the critical value approach, the significance level a is predetermined. The value of a corresponds to
the total area of the critical region.
–
X – µ
Under the null hypothesis, μ = μ , and the test statistic Z = σ 0 has a standard normal distribution,
0
N(0, 1). 
n
Thus, for a given a, the critical values of the random variable Z are –z a and z a . We have the probability
— 2 — 2
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05 STPM Math(T) T3.indd 244 28/10/2021 10:24 AM