Geometric Hypothesis Testing (Edexcel A Level Further Maths: Further Statistics 1): Revision Note

Geometric hypothesis testing

How do I test for the parameter p of a Geometric distribution?

• If $X~\mathrm{Geo}\left(p\right)$, test for the probability of success, $p$, using the following hypotheses

  • ${\mathrm{H}}_0:p=...$

  • ${\mathrm{H}}_1:p\ne ...$ or $p<...$or $p>...$

  • with significance level $\alpha$

    • For example, $\alpha =0.05$ for 5%

• You will be given an observed value, $x$, in the question

  • This is the number of trials it takes to see the first success

    • For example, "They thought the coin was fair ($p=\frac{1}{2}$), but last week it took 5 flips to get the first tail ($x=5$)"

  • It can help to compare $x$ with the expected number of trials to see the first success, $\mathrm{E}\left(X\right)=\frac{1}{p}$

    • For example, "they expected a fair coin ($p=\frac{1}{2}$) to take $\frac{1}{p}=2$ attempts to see the first tail"

• Assuming ${\mathrm{H}}_0:p=...$

  • Find the probability that $X$ is the observed value $x$, or more extreme than that

  • For ${\mathrm{H}}_1:p<...$ the extreme values are $X\ge x$

    • Note the "change in inequality direction"

    • A lower probability of success means a higher number of attempts to first reach that success

  • For ${\mathrm{H}}_1:p>...$ the extreme values are $X\le x$

    • A higher probability of success means a lower number of attempts to first reach that success

  • For ${\mathrm{H}}_0:p\ne ...$ compare $x$ with $\mathrm{E}\left(X\right)=\frac{1}{p}$

    • If $x$ is less than $\frac{1}{p}$, then extreme values are $X\le x$

    • If $x$ is more than $\frac{1}{p}$, then the extreme values are $X\ge x$

  • If the total probability of these values is $<\alpha$ (or $<\frac{\alpha }{2}$ for two-tailed tests)

    • Write that "there is sufficient evidence to reject ${\mathrm{H}}_0$"

  • If not, write that "there is insufficient evidence to reject ${\mathrm{H}}_0$" 

• Write a conclusion in context

  • For example

    • "the probability of success is less than $\frac{1}{2}$"

    • or "the probability of success has not changed from $\frac{1}{2}$"

How do I find the critical region for a Geometric hypothesis test?

• If ${\mathrm{H}}_1:p<...$ 

  • Assume that ${\mathrm{H}}_0:p=...$

  • Then test different integer values, $c$, to get $\mathrm{P}\left(X\ge c\right)$ as close to $\alpha$as possible, without exceeding it

    • Use the formula $\mathrm{P}\left(X\ge c\right)={\left(1-p\right)}^{c-1}$ to help

    • The integer that's the nearest is called the critical value

    • Checking one integer lower should show that $\mathrm{P}\left(X\le c-1\right)$ is $>\alpha$

  • The critical region is $X\ge c$

    • Note that the inequality is the opposite way round to $p<...$

  • Instead of testing integers, you can also use logarithms to solve the critical region inequalities

    • Beware when dividing both sides by $\log \left(p\right)$

    • $\log \left(p\right)<0$ so the inequality must be "flipped"

• If ${\mathrm{H}}_1:p>...$

  • It's the same process, but with $\mathrm{P}\left(X\le c\right)$ as close to $\alpha$ as possible, without exceeding it

    • Use the formula $\mathrm{P}\left(X\le c\right)=1-{\left(1-p\right)}^c$ to help

  • The critical region is $X\le c$

• If ${\mathrm{H}}_1:p\ne ...$

  • The critical region is $X\le c_1$ or $X\ge c_2$

    • $\mathrm{P}\left(X\le c_1\right)$ is as close to $\frac{\alpha }{2}$ as possible, without exceeding it

    • $\mathrm{P}\left(X\ge c_2\right)$ is as close to $\frac{\alpha }{2}$ as possible, without exceeding it

• Your calculator may have an 'Inverse Geometric Distribution' function that can help with finding critical values

  • But always check those values against the requirements of the question

  • The calculator may not always give the exact answer you are looking for

What is the actual significance level?

• As the geometric model is discrete, it's not possible to get a critical region whose probability sums to $\alpha$ exactly

  • That's because $X$ can only take integer values

• Whatever it does sum to is called the actual significance level

  • The actual amount of probability in the tail (or tails)

• For example, if ${\mathrm{H}}_1:p<...$ has the critical region $X\ge c$

  • Then $\mathrm{P}\left(X\ge c\right)$ will be just less than $\alpha$

    • It's value is the actual significance level

    • It represents the probability of rejecting ${\mathrm{H}}_0$ incorrectly (when ${\mathrm{H}}_0$ was actually true)

• Some questions want a critical region that's as close to $\alpha$ as possible, even if that means probabilities that exceed $\alpha$

  • For example, if $\mathrm{P}\left(X\ge 12\right)=0.0511$ and $\mathrm{P}\left(X\ge 13\right)=0.0299$ where $\alpha =0.05$

    • Then $X\ge 12$ is the critical region that's as close to $\alpha$ as possible

    • The actual significance level is 0.0511