When is H0 rejected in a one-tailed test?

When the p-value is less than or equal to the significance level.

What do you compare with in a two-tailed test at 5%?

0.025 (half the significance level) for the relevant tail.

How do you decide which tail in a two-tailed test?

Compare the observation with the expected value np0.

What is the significance level?

The probability of incorrectly rejecting H0 when it is true.

Find the critical region for B(40,0.05), H1:p0.05, 1% level.

X\ge7, as P(X\ge7)=0.0034 and P(X\ge6)=0.0139.

Why is the actual significance level often below the stated one?

The test statistic is discrete, so the critical region cannot always have probability exactly equal to the level.

What does 'do not reject H0' mean?

There is insufficient evidence against H0; it does not prove H0 true.

Write a conclusion after rejecting H0.

'There is evidence at the 5% level that [the claim in context].'

Why can a sample be used to test a claim?

A sample is used to make an inference about the whole population.

Hypothesis tests for a binomial proportionEdexcel A-Level Maths: Flashcards

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What must hypotheses be written in terms of?

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What must hypotheses be written in terms of?: The population parameter $p$ (not the sample proportion).
Under $H_0:p=0.3$ with a sample of 20, what is the distribution of $X$ ?: $X\sim B(20,0.3)$ .
$p$ -value for $H_1:p>p_0$ and observed $x$ ?: $P(X\ge x)$ .
$p$ -value for $H_1:p<p_0$ and observed $x$ ?: $P(X\le x)$ .
When is $H_0$ rejected in a one-tailed test?: When the $p$ -value is less than or equal to the significance level.
What do you compare with in a two-tailed test at 5%?: $0.025$ (half the significance level) for the relevant tail.
How do you decide which tail in a two-tailed test?: Compare the observation with the expected value $np_0$ .
What is the significance level?: The probability of incorrectly rejecting $H_0$ when it is true.
Find the critical region for $B(40,0.05)$ , $H_1:p>0.05$ , 1% level.: $X\ge7$ , as $P(X\ge7)=0.0034$ and $P(X\ge6)=0.0139$ .
Why is the actual significance level often below the stated one?: The test statistic is discrete, so the critical region cannot always have probability exactly equal to the level.
What does 'do not reject $H_0$ ' mean?: There is insufficient evidence against $H_0$ ; it does not prove $H_0$ true.
Write a conclusion after rejecting $H_0$ .: 'There is evidence at the 5% level that [the claim in context].'
Why can a sample be used to test a claim?: A sample is used to make an inference about the whole population.

Exam questions on Hypothesis tests for a binomial proportion

A drug is known to cure 30% of patients with a certain condition. A new drug is tested on 20 randomly chosen patients, of whom 10 are cured. Let $p$ be the probability that a patient given the new drug is cured and let $X$ be the number cured in a sample of 20. The company tests at the 5% significance level whether the new drug is more effective.
State the conclusion of the test, in context.2 marks
A company claims that 85% of its parcels are delivered on time. A customer believes that the proportion is lower. In a random sample of 15 parcels, 10 are delivered on time. Let $p$ be the proportion of all the company's parcels that are delivered on time and let $X$ be the number delivered on time in a sample of 15. The customer tests at the 5% significance level.
State the conclusion of the test, in context.2 marks
A network provider claims that 40% of its customers are on a premium plan. A researcher believes that the proportion has changed. In a random sample of 30 customers, 17 are on a premium plan. Let $X$ be the number of customers on a premium plan in a sample of 30.
State the hypotheses for a test at the 5% significance level, and say what $p$ represents.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).