How do you find the critical values for a two-tailed binomial hypothesis test?

Work in each tail separately , splitting the significance level between them, and in each tail find the first value whose probability of being that extreme or more falls below the level for that tail. That gives two critical values, one at each end, and the critical region is everything at or beyond either of them.

A binomial hypothesis test can be done by p-value or by critical region. When is the critical region worth finding?

When more than one observed value has to be judged, or when further testing is going to be done on the same claim. The critical region is worked out from the null hypothesis and the significance level alone, so once it is known any number of observed values can be checked against it without further calculation. For a single observed value the p-value route is usually quicker, since it needs only one probability.

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

Exam code: H240

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FrontHypothesis Testing for the Population Proportion of a Binomial Distribution
In a hypothesis test using a binomial distribution, what is being tested and what is the test statistic?
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
The population proportion $p$ is being tested, so the null hypothesis takes the form $H_{0} : p = \dots$ and the alternative is one of $H_{1} : p > \dots$ or $H_{1} : p < \dots$ or $H_{1} : p \neq \dots$ depending on the claim.
The test statistic is the number of successes observed in a fixed number of $n$ trials, modelled by $X ~ B (n, p)$ with $p$ taking the value the null hypothesis assumes.
Define in words what $p$ stands for before writing the hypotheses, unless the question has already done it.

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In a hypothesis test using a binomial distribution, what is being tested and what is the test statistic?
The population proportion $p$ is being tested, so the null hypothesis takes the form $H_{0} : p = \dots$ and the alternative is one of $H_{1} : p > \dots$ or $H_{1} : p < \dots$ or $H_{1} : p \neq \dots$ depending on the claim.
The test statistic is the number of successes observed in a fixed number of $n$ trials, modelled by $X ~ B (n, p)$ with $p$ taking the value the null hypothesis assumes.
Define in words what $p$ stands for before writing the hypotheses, unless the question has already done it.
A one-tailed binomial test at the $α %$ level has $H_{1} : p > \dots$ and a critical value $c$ for its critical region. Complete the two conditions that fix $c$ itself:
$P (X \geq______) \leq α % and P (X \geq______) > α %$
The completed conditions are:
$P (X \geq c) \leq α % and P (X \geq c - 1) > α %$
Together they say that $c$ is the first value whose tail probability drops below the significance level, so the critical region is $X \geq c$ and everything above it.
For $H_{1} : p < \dots$ the same idea runs the other way, with $P (X \leq c) \leq α %$ and $P (X \leq c + 1) > α %$ instead.
Jacques claims that more than 80% of the customers buying bread at a supermarket buy his brand. In a random sample of 100 bread buyers, 86 bought his brand.
State the hypotheses and the distribution the test uses, at the 10% significance level.
Let $p$ be the proportion of bread buyers at that supermarket who buy his brand, giving $H_{0} : p = 0.8$ and $H_{1} : p > 0.8$ for the claim.
Assuming the null hypothesis is true, the number in the sample buying his brand is modelled by $X ~ B (100, 0.8)$ and the observed value of the test statistic is 86.
The test then compares $P (X \geq 86)$ with the 10% significance level.
True or False?
In a two-tailed binomial hypothesis test, one critical region can be much larger than the other.
True.
A binomial distribution is only symmetrical when $p$ is exactly 0.5, so for any other value of $p$ the two tails fall away at different rates.
Half the significance level then reaches further into one tail than into the other, and the two critical regions end up holding different numbers of values.
How do you find the critical values for a two-tailed binomial hypothesis test?
Work in each tail separately, splitting the significance level between them, and in each tail find the first value whose probability of being that extreme or more falls below the level for that tail.
That gives two critical values, one at each end, and the critical region is everything at or beyond either of them.
A binomial hypothesis test can be done by p-value or by critical region. When is the critical region worth finding?
When more than one observed value has to be judged, or when further testing is going to be done on the same claim.
The critical region is worked out from the null hypothesis and the significance level alone, so once it is known any number of observed values can be checked against it without further calculation.
For a single observed value the p-value route is usually quicker, since it needs only one probability.

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