Exact binomial hypothesis tests | CAIE 9709 Probability & Statistics 2 Notes

Chapters

Probability & Statistics 2

Mathematics9709

Add the whole relevant tail

Lower tail means 0, 1 and 2

P(X ≤ 2) = P(0) + P(1) + P(2) = 0.1028

Compare the whole tail with α = 0.05.

An exact test uses the observed result and every result more extreme in the direction of H₁.

Worked example

Current-paper exact binomial test

A claim says that the probability of a success is $\tfrac16$ . In 30 independent trials there are 2 successes. Test at the 5% level whether the probability is lower.

H_0:p=\frac16,\qquad H_1:p<\frac16

X\sim\mathrm{B}\left(30,\frac16\right)

“Lower” means the observed value 2 and every smaller value:

P(X\le2)=0.103\ \text{(3 s.f.)}

Since $0.103>0.05$ , fail to reject $H_0$ .

Conclusion: There is insufficient evidence that the success probability is lower than $\tfrac16$ .

(9709/62/M/J/25 Q8(a))

Calculator lower tail: use cumulative probability $P(X\le r)$ .
Calculator upper tail: use $1-P(X\le r-1)$ , not $1-P(X\le r)$ .
For a Poisson rate, first scale the null mean to the full observation interval. The same tail rule then applies.

Examiner note

If asked for the critical region, keep the largest lower-tail value

r

for which

P(X\le r)\le\alpha

, or the smallest upper-tail value

r

for which

P(X\ge r)\le\alpha

. For a two-tailed test, replace

\alpha

by the allocated level in that tail. Discrete probabilities jump, so the actual Type I probability may be smaller than the stated level.

Common mistake

P(X=r)

uses only one bar. A hypothesis test needs the observed result and all results more extreme in the direction of H₁.

Poisson bridge: if the claimed rate is 4 per hour and 20 events are observed in 3 hours, then under H₀ $X\sim\mathrm{Po}(12)$ and $P(X\ge20)=0.0213$ . The upper-tail result is significant at 5%.

Your turnExact upper-tail decision [5 marks]

A fair coin is tossed 16 times and gives 12 heads. Test at the 5% level whether it is biased towards heads. You may use $P(X\ge12)=0.0384$ for $X\sim\mathrm{B}(16,0.5)$ .

Show worked answer

H_0:p=0.5,\qquad H_1:p>0.5

0.0384<0.05

Answer

\text{Reject }H_0.

Conclusion: There is sufficient evidence that the coin is biased towards heads.

Add the whole relevant tail

Lower tail means 0, 1 and 2

P(X ≤ 2) = P(0) + P(1) + P(2) = 0.1028

Compare the whole tail with α = 0.05.

An exact test uses the observed result and every result more extreme in the direction of H₁.

Worked example

Current-paper exact binomial test

A claim says that the probability of a success is $\tfrac16$ . In 30 independent trials there are 2 successes. Test at the 5% level whether the probability is lower.

H_0:p=\frac16,\qquad H_1:p<\frac16

X\sim\mathrm{B}\left(30,\frac16\right)

“Lower” means the observed value 2 and every smaller value:

P(X\le2)=0.103\ \text{(3 s.f.)}

Since $0.103>0.05$ , fail to reject $H_0$ .

Conclusion: There is insufficient evidence that the success probability is lower than $\tfrac16$ .

(9709/62/M/J/25 Q8(a))

Calculator lower tail: use cumulative probability $P(X\le r)$ .
Calculator upper tail: use $1-P(X\le r-1)$ , not $1-P(X\le r)$ .
For a Poisson rate, first scale the null mean to the full observation interval. The same tail rule then applies.

Examiner note

If asked for the critical region, keep the largest lower-tail value

r

for which

P(X\le r)\le\alpha

, or the smallest upper-tail value

r

for which

P(X\ge r)\le\alpha

. For a two-tailed test, replace

\alpha

by the allocated level in that tail. Discrete probabilities jump, so the actual Type I probability may be smaller than the stated level.

Common mistake

P(X=r)

uses only one bar. A hypothesis test needs the observed result and all results more extreme in the direction of H₁.

Your turnExact upper-tail decision [5 marks]

A fair coin is tossed 16 times and gives 12 heads. Test at the 5% level whether it is biased towards heads. You may use $P(X\ge12)=0.0384$ for $X\sim\mathrm{B}(16,0.5)$ .

Show worked answer

H_0:p=0.5,\qquad H_1:p>0.5

0.0384<0.05

Answer

\text{Reject }H_0.

Conclusion: There is sufficient evidence that the coin is biased towards heads.