Page 57 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing

The population standard deviation is not known and the sample size is large.
We will use the normal distribution to perform the test.
This is a one-tailed test with a critical region at the right tail. To locate the z
value, we look for 0.01 area in the normal distribution table. From the table, the
z value is approximately 2.33.
The critical region: z . 2.33.
Step 4 : Calculate the value of the test statistic.
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x – µ
z = = 1.2 – 0 = 5.122
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σ ^ 1.815
n
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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 = 5.122 is greater than the critical value of 2.33 and thus
it falls in the critical region. We reject H and conclude that the true average
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watch reading is faster than the actual time of 10:00:00 hours.

Relationship between hypothesis tests and confidence intervals

The hypothesis testing is very closely related to the estimation by a confidence interval. For the case of a
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normal population with unknown mean μ and known σ , both hypothesis testing and confidence interval
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estimation are based on the random variable, z = X – µ . It turns out that the testing of H : μ = μ against
σ
0
0
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n
H : μ ≠ μ at a significance level a is similar to calculating a 100(1 – a)% confidence interval for μ. If µ
0
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is outside the 100(1 – a)% confidence interval, then H is rejected at a, and if μ is inside the 100(1 – a)%
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confidence interval, then H is not rejected at a.
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Exercise 5.2
1. Determine the critical region and the critical values for z at a = 0.02 in the hypothesis testing:
H : μ = 50 against H : μ , 50.
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5 2. Find the critical values x given that H : μ = 850, H : μ ≠ 850, σ = 36, n = 80. Use a = 0.05.
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3. Given that H : μ = 250, H : μ ≠ 250 with n = 80, x = 248, σ = 10.7. State whether the null hypothesis
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H should or should not be rejected at a = 0.02.
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4. For the hypothesis test H : μ = 16.8, H : µ . 16.8, what is your conclusion at a = 0.1 if n = 45,
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x = 17.5, s = 2.5?
5. In a random sample of 20 observations from a normal distributed population with standard deviation
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s = 27.6, the sample mean is x = 110. Use a = 0.01 to perform the following hypothesis test:
H : μ = 120, H : μ , 120.
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Next, change the sample size to 200 and perform the test again. Explain why you make different
decisions as the sample size increases.

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05 STPM Math(T) T3.indd 248 28/10/2021 10:24 AM