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Hypothesis Testing (Binomial Distribution) (Edexcel A Level Maths: Statistics): Flashcards

Exam code: 9MA0

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FrontHypothesis Testing for the Population Proportion of a Binomial Distribution
In a hypothesis test on a binomial distribution, what is the parameter being tested and what is the test statistic?
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
The parameter is $p$ , the probability of success, so both hypotheses are statements about $p$ .
The test statistic is the number of successes observed in the defined number of trials, and its distribution is $X ~ B (n, p)$ .
Keeping the two apart matters: $p$ is the unknown the test is about, while the number of successes is the piece of evidence you actually have.
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
A definition of $p$ in words, taken from the context of the question, such as "let $p$ be the proportion of people shopping at the supermarket who buy Jacques' brand of bread".
Without it, $H_{0} : p = 0.8$ is a statement about nothing, and the conclusion at the end cannot be written in context either.
If the question has not defined $p$ for you, defining it is the first thing you do.
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
The completed hypotheses are:
$H_{0} : p = 0.8$
$H_{1} : p > 0.8$
The null hypothesis takes the value being questioned, and the alternative carries the direction of the claim. "More than 80%" makes this a one-tailed test looking for an increase.
Notice that the baker's claim is the alternative hypothesis, not the null: the test assumes he is wrong and asks whether the evidence is strong enough to say otherwise.

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Cards in this collection (7)

In a hypothesis test on a binomial distribution, what is the parameter being tested and what is the test statistic?
The parameter is $p$ , the probability of success, so both hypotheses are statements about $p$ .
The test statistic is the number of successes observed in the defined number of trials, and its distribution is $X ~ B (n, p)$ .
Keeping the two apart matters: $p$ is the unknown the test is about, while the number of successes is the piece of evidence you actually have.
What must you write down before stating the hypotheses in a binomial test?
A definition of $p$ in words, taken from the context of the question, such as "let $p$ be the proportion of people shopping at the supermarket who buy Jacques' brand of bread".
Without it, $H_{0} : p = 0.8$ is a statement about nothing, and the conclusion at the end cannot be written in context either.
If the question has not defined $p$ for you, defining it is the first thing you do.
A baker claims that more than 80% of shoppers buy his brand of bread. Complete the hypotheses for a test of that claim:
$H_{0} : p =______$
$H_{1} : p______0.8$
The completed hypotheses are:
$H_{0} : p = 0.8$
$H_{1} : p > 0.8$
The null hypothesis takes the value being questioned, and the alternative carries the direction of the claim. "More than 80%" makes this a one-tailed test looking for an increase.
Notice that the baker's claim is the alternative hypothesis, not the null: the test assumes he is wrong and asks whether the evidence is strong enough to say otherwise.
How do you find the critical value for a one-tailed binomial test?
It is the first value that falls inside the critical region, and which way you work depends on where the alternative hypothesis points.
For $H_{1} : p > \dots$ find the smallest $c$ with $P (X \geq c) \leq α %$ , and for $H_{1} : p < \dots$ the largest $c$ with $P (X \leq c) \leq α %$ .
Use the cumulative binomial function and try successive values, checking that the next one takes the probability back above $α %$ ; because $X$ takes only whole-number values, the critical region's probability is usually less than $α %$ rather than equal to it.
What changes when a binomial test is two-tailed?
There are two critical regions, one in each tail, and each end is located exactly as for a one-tailed test but against half the significance level, $\frac{α}{2} %$ .
The part that surprises people is that the two regions are usually very different sizes.
A binomial distribution is only symmetrical when $p = 0.5$ , so with $p$ well away from 0.5 one tail is much longer than the other, and the two critical regions are not the same distance from the mean.
A baker claims more than 80% of shoppers buy his bread. Of 100 shoppers, 86 do. Testing at the 10% level, which probability do you calculate?
The probability of the observed value or something more extreme, assuming $H_{0}$ is true.
With $H_{0} : p = 0.8$ the test statistic is $X ~ B (100, 0.8)$ , and since the test is for an increase, more extreme means larger:
$P (X \geq 86) = 1 - P (X \leq 85) = 0.0804$
which is less than 0.10, so the result is significant.
Note the value 0.8 comes from $H_{0}$ , not from the sample: the whole test is run inside the assumption that the null hypothesis holds.
The probability of the observed value or more extreme comes to 0.0804, and the test is at the 10% level. What do you conclude?
$0.0804 < 0.10$ , so the result is significant and $H_{0}$ is rejected.
Had it come out above the significance level, the conclusion would be that you do not reject $H_{0}$ ; never write "accept $H_{0}$ ", since a test that fails to find evidence against the null hypothesis has not shown it to be true.
The conclusion then has to be turned back into the language of the question, saying that there is sufficient evidence at the 10% level to support the claim that more than 80% of shoppers buy that brand.

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