Page 49 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing
The probability of making a Type I errror is called the significance level and is denoted by the Greek letter
a. In our example, a type I error will occur when more than 7 drivers rush through red lights that is actually
an odd sample taken. Hence, if X is the number of drivers who involve running red lights,
20 20
x
P(Type I error) = P(X . 7 when p = 0.2) = ∑ 1 2 (0.2) (1 – 0.2) 20 – x ,
x = 8 x
7 20
x
1 – ∑ 1 2 (0.2) (1 – 0.2) 20 – x = 1 – 0.9679 = 0.0321.
x = 0 x
We say that the null hypothesis is being tested at a significance level of 3.21%.
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The probability of making Type II error is denoted by the Greek letter b. It is impossible to calculate this
probability unless a specific value is stated in the alternative hypothesis. We shall not discuss the determination
of b.
It can be shown that, for a fixed sample size, a decrease in the probability of one error will usually result in
an increase in the probability of the other error. However, we can reduce both types of errors by increasing
the sample size.

Example 3

A food company produces a box of 250 g corn flakes. It periodically conducts a statistical test to decide
whether the mean net mass of all boxes is 250 g. The null and alternative hypotheses are stated below.
H : µ = 250,
0
H : µ ≠ 250.
1
The results of carrying out the hypothesis test lead to no rejection of the null hypothesis. Comment on
the conclusion by error type or as a right decision to make if
(a) µ is in fact 250 g,
(b) µ is in fact not 250 g.

Solution: (a) If in fact µ = 250 g, the null hypothesis is true. Thus, by not rejecting the
null hypothesis, we have made a right decision. This interprets that the
package machine is functioning properly.
(b) If in fact µ ≠ 250 g, the null hypothesis is false. Thus, by not rejecting the
null hypothesis, we have committed a Type II error. This interprets that the
package machine is out of control even though the inspected output sample
indicates a satisfactory position.
5
One-tailed and two-tailed tests

Consider the null hypothesis that the mean weight of new born babies in a certain city is 3 kg. We test
H : µ = 3
0
against H : µ ≠ 3 or H : <3 or H : >3
1
1
1
Only one of these alternative hypotheses can be used at a time. We examine each case in turn.
Two-tailed test (H : µ ≠ 3)
1
A random sample of size n = 64 new born babies is taken. Assume that the standard deviation of the
–
population to be 1.60 kg. From the central limit theorem, we know that the sampling distribution of X is a
approximate normal distribution with standard deviation s – = s n = 1.6 = 0.2.
8
X

240

05 STPM Math(T) T3.indd 240 28/10/2021 10:24 AM