What does X\simGeo(p) count in a hypothesis test?

The number of independent trials up to and including the first success.

What are the hypotheses for a geometric test written in terms of?

The parameter p, the probability of success on each trial.

If H1:p<p0, which tail is the critical region?

The upper tail, X\geq c, because a lower p means a longer wait.

If H1:pp0, which tail is the critical region?

The lower tail, X\leq c, because a higher p means a shorter wait.

How do you find a critical region for X\geq c?

Solve (1-p0)^{c-1}< significance level using logarithms, then check neighbouring values.

What is the actual significance level?

The probability of the critical region under H0.

How is the significance level split in a two-tailed test?

Half in each tail, for example 5\% each for a 10\% test.

For X\simGeo(0.3), why is there no lower critical region at 5\%?

The smallest possible value has P(X\leq1)=0.30.05.

Why can reversing the inequality be needed when solving for c?

Dividing by \ln(1-p0), which is negative, reverses the inequality.

Name an assumption of the geometric model that may fail.

The probability of success is constant and trials are independent, which can fail if a player improves with practice.

Hypothesis test for the parameter of a geometric distributionEdexcel A-Level Further Maths: Flashcards

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What does $X\sim\mathrm{Geo}(p)$ count in a hypothesis test?

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What does $X\sim\mathrm{Geo}(p)$ count in a hypothesis test?: The number of independent trials up to and including the first success.
State $\mathrm{P}(X\geq x)$ for $X\sim\mathrm{Geo}(p)$ .: $(1-p)^{x-1}$
State $\mathrm{P}(X\leq x)$ for $X\sim\mathrm{Geo}(p)$ .: $1-(1-p)^x$
What are the hypotheses for a geometric test written in terms of?: The parameter $p$ , the probability of success on each trial.
If $\mathrm{H}_1:p<p_0$ , which tail is the critical region?: The upper tail, $X\geq c$ , because a lower $p$ means a longer wait.
If $\mathrm{H}_1:p>p_0$ , which tail is the critical region?: The lower tail, $X\leq c$ , because a higher $p$ means a shorter wait.
$\mathrm{H}_0:p=0.2$ , $\mathrm{H}_1:p<0.2$ . First success on trial $12$ . Find the probability to compare.: $\mathrm{P}(X\geq12)=0.8^{11}=0.0859$
How do you find a critical region for $X\geq c$ ?: Solve $(1-p_0)^{c-1}<$ significance level using logarithms, then check neighbouring values.
What is the actual significance level?: The probability of the critical region under $\mathrm{H}_0$ .
How is the significance level split in a two-tailed test?: Half in each tail, for example $5\%$ each for a $10\%$ test.
For $X\sim\mathrm{Geo}(0.3)$ , why is there no lower critical region at $5\%$ ?: The smallest possible value has $\mathrm{P}(X\leq1)=0.3>0.05$ .
Why can reversing the inequality be needed when solving for $c$ ?: Dividing by $\ln(1-p_0)$ , which is negative, reverses the inequality.
Name an assumption of the geometric model that may fail.: The probability of success is constant and trials are independent, which can fail if a player improves with practice.

Exam questions on Hypothesis test for the parameter of a geometric distribution

A scratch-card company claims that the probability that a card wins a prize is $0.2$ , independently for each card. A customer buys cards one at a time and has her first win on the $12$ th card. She suspects that the probability of winning is lower than the company claims and carries out a test at the $5\%$ significance level. Let $X$ be the number of cards bought up to and including the first win.
Find the critical region for the test.2 marks
A manufacturer says that the probability that a screen it makes is defective is $0.02$ , independently for each screen. An inspector tests screens one at a time and finds the first defective screen at the $2$ nd screen tested. She suspects that the probability of a defective screen is higher than $0.02$ and carries out a test at the $5\%$ significance level. Let $X$ be the number of screens tested up to and including the first defective one.
Write down the conclusion of the test, in context.2 marks
A website claims that each visitor clicks on an advert with probability $0.1$ , independently of other visitors. An analyst suspects that the probability is lower. She counts the number of visitors $X$ up to and including the first visitor who clicks, and will test the claim at the $5\%$ significance level.
State suitable hypotheses for the test, where $p$ is the probability that a visitor clicks on the advert, and state the distribution of $X$ under $\mathrm{H}_0$ .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).