Page 59 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing

5.3 Testing Population Proportion

Often we have to perform a hypothesis test about a population proportion. Evidence concerning the value
of a population proportion is provided by the sample proportion. We shall discuss hypothesis tests about
the population proportion for small sample, where direct evaluation of binomial probabilities is required.
We shall also discuss hypothesis tests about the population proportion for large samples, using the normal
approximation to the binomial distribution.
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Population proportion, small sample
Let a parameter p represent the unknown proportion of a population that possesses a certain characteristic.
If an independent observation is randomly obtained from the population it can then be taken as having a
probability p of showing that particular characteristic. If a random sample of n observations is taken from
the population, the number of observations exhibiting the character of interest can be realised as a random
variable X obtained from a binomial experiment, that is, X ~ B(n, p).
When the binomial parameter p, success probability in a binomial experiment, is to be tested using hypothesis
testing procedure, we will consider that this parameter equals some specified value. A hypothesis testing
problem would then become testing the null hypothesis H that p = p against the alternative hypothesis
0
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which may be one of the usual one-sided or two-sided alternatives: p , p , p . p , or p ≠ p .
0
0
0
The appropriate random variable on which we base our decision criterion is the binomial random variable
X. Values of X that provide significant evidence indicating the success probabilities are far from p will lead
0
to the rejection of the null hypothesis. Consider the hypotheses
H : p = p ,
0
0
H : p , p ,
1
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we use the binomial distribution with p = p and q = 1 – p to determine P(X < x). The value of x represents
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the number of successes in our sample of size n. If P(X < x) , a, we reject H as the result is significant
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at the significance level a. Likewise, for the hypotheses
H : p = p ,
0
0
H : p . p ,
1
0
we obtain P(X > x). If this probability is less than a, we reject H . Lastly, for the hypotheses
0
5 H : p = p ,
0
0
H : p ≠ p ,
1
0
we calculate P(X < x) if x , np , or P(X > x) if x . np . If the probability is less than a , we reject H .
2
0
0
0
The steps for testing a null hypothesis about a proportion versus various alternatives are:
1 State H : p = p and H : p , p , p . p , or p ≠ p 0
0
0
1
0
0
2 Specify the significance level
3 Determine the critical region
4 Calculate the appropriate binomial probability
5 Make a decision
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05 STPM Math(T) T3.indd 250 28/10/2021 10:24 AM