What is the test statistic in a binomial hypothesis test?

The test statistic is the number of successes observed in a fixed number of trials, which is a whole number rather than a proportion. So in a sample of 100 shoppers of whom 86 buy a particular brand, the test statistic is 86 , and it is 86 that gets compared with a critical value or used to work out a probability. The evidence arrives as a count even though the claim being tested is about a probability, and keeping those apart is what stops 0.86 being used by mistake.

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Hypothesis Testing (Binomial Distribution) (AQA AS Maths: Statistics): Flashcards

Exam code: 7356

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FrontHypothesis Testing for the Population Proportion of a Binomial Distribution
In a binomial hypothesis test, which quantity do the hypotheses describe?
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
The hypotheses describe the probability of success, $p$ , for the population the sample was drawn from, so they take the form $H_{0} : p = \dots$ with $H_{1}$ in terms of $p$ as well.
Say in words what $p$ stands for before writing them down, because the letter means nothing on its own: in a test about a brand of bread it would be the proportion of all shoppers who buy that brand.
BackHypothesis Testing for the Population Proportion of a Binomial Distribution
The test statistic is the number of successes observed in a fixed number of trials, which is a whole number rather than a proportion.
So in a sample of 100 shoppers of whom 86 buy a particular brand, the test statistic is 86, and it is 86 that gets compared with a critical value or used to work out a probability.
The evidence arrives as a count even though the claim being tested is about a probability, and keeping those apart is what stops 0.86 being used by mistake.

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Cards in this collection (5)

In a binomial hypothesis test, which quantity do the hypotheses describe?
The hypotheses describe the probability of success, $p$ , for the population the sample was drawn from, so they take the form $H_{0} : p = \dots$ with $H_{1}$ in terms of $p$ as well.
Say in words what $p$ stands for before writing them down, because the letter means nothing on its own: in a test about a brand of bread it would be the proportion of all shoppers who buy that brand.
A test is to be carried out on 100 trials, before any value has been assumed for the probability of success. Complete the definition of the test statistic:
$X ~ B (______,______)$
The completed definition is:
$X ~ B (100, p)$
The second entry stays as the letter $p$ , because $p$ is the quantity under test and nothing has fixed it yet; it is the null hypothesis that then assumes a value for it.
Putting a number there instead would assume the answer before the test had been run.
What is the test statistic in a binomial hypothesis test?
The test statistic is the number of successes observed in a fixed number of trials, which is a whole number rather than a proportion.
So in a sample of 100 shoppers of whom 86 buy a particular brand, the test statistic is 86, and it is 86 that gets compared with a critical value or used to work out a probability.
The evidence arrives as a count even though the claim being tested is about a probability, and keeping those apart is what stops 0.86 being used by mistake.
True or False?
In a two-tailed binomial test, the two critical regions always contain the same number of values.
False.
A binomial distribution is not symmetrical unless $p = 0.5$ , so the probability allowed in each tail is used up by different numbers of values at the two ends.
One critical region is often much larger than the other, which is why both have to be found separately rather than one being mirrored across from the other.
A baker claims more than 80% of shoppers buy his bread, and 86 of a sample of 100 do. Is his claim supported at the 10% level?
Yes. Taking $p$ as the proportion of all shoppers who buy his bread, $H_{0} : p = 0.8$ and $H_{1} : p > 0.8$ , and assuming $H_{0}$ the count follows $X ~ B (100, 0.8)$ .
The probability of a result at least as extreme as 86 is $P (X \geq 86) = 0.0804$ , which is smaller than 10%.
So there is sufficient evidence at the 10% level to support the claim that more than 80% of shoppers buy his brand.

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