Hypothesis Testing (AQA A-Level Mathematics): Revision Notes

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5.1.1 Hypothesis Testing

Critical Regions in Hypothesis Tests

The critical region of a hypothesis test (also the rejection region) is the set of x values that would lead to the null hypothesis being rejected.

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Example: A single observation, x, is taken from a binomial distribution B(10, p) and a value of 5 is obtained.

Use this observation to test H₀: p = 0.25 against H₁: p > 0.25 using a 5% significance level.

We will use critical regions to perform this test.

We need the set of x values such that P(X ≥ k) ≤ 0.05, i.e., every single value we can observe that would lead to H₀ being rejected.

We do this by trial and improvement:

kP(X≥k)109.537×10−792.958×10−584.158×10−470.019760.075\begin{array}{c|c} k & P(X \geq k) \\ \hline 10 & 9.537 \times 10^{-7} \\ 9 & 2.958 \times 10^{-5} \\ 8 & 4.158 \times 10^{-4} \\ 7 & 0.0197 \\ 6 & 0.07 \\ 5 & \end{array}P(X≥10)=1−P(X≤9)=9.537×10−7 (C)P(X≥9)=1−P(X≤8)=2.958×10−5 (C)P(X≥8)=1−P(X≤7)=4.158×10−4 (C)P(X≥7)=1−P(X≤6)=0.0197 (A)P(X≥6)=1−P(X≤5)=0.07 (A)\begin{aligned} P(X \geq 10) & = 1 - P(X \leq 9) = 9.537 \times 10^{-7} \ (\text{C}) \\ P(X \geq 9) & = 1 - P(X \leq 8) = 2.958 \times 10^{-5} \ (\text{C}) \\ P(X \geq 8) & = 1 - P(X \leq 7) = 4.158 \times 10^{-4} \ (\text{C}) \\ P(X \geq 7) & = 1 - P(X \leq 6) = 0.0197 \ (\text{A}) \\ P(X \geq 6) & = 1 - P(X \leq 5) = 0.07 \ (\text{A}) \\ \end{aligned}

We have seen crossing one boundary from reject to accept; we can be certain we have found the entire critical region.

∴Critical region is X≥6\therefore \text{Critical region is } X \geq 6

Using "List Mode" in Binomial CD to speed this up

  1. Select 'List' on Binomial CD.
  2. Starting at the biggest value (if we are testing the right tail) or the smallest (if we are testing the left tail), type in lots of numbers.

Left Tail:

  1. Look for P(X ≤ k) crossing over sig level boundary.
  2. Summarise.

Right Tail:

  1. Look for the probability crossing over the (1 - SIG LEVEL) boundary.
  2. Write down the calculations either side of this boundary:
P(X≥c)=1−P(X≤5)=1−0.9823P(X \geq c) = 1 - P(X \leq 5) = 1 - 0.9823P(X≥5)=1−P(X≤4)=1−0.9218P(X \geq 5) = 1 - P(X \leq 4) = 1 - 0.9218
  1. Summarise:
P(X≥7)=0.0197<0.05P(X \geq 7) = 0.0197 < 0.05P(X≥5)=0.0782>0.05P(X \geq 5) = 0.0782 > 0.05∴Critical region is X≥6\therefore \text{Critical region is } X \geq 6

A test statistic has a distribution B(10, p). Given that H₀: p = 0.2, H₁: p > 0.2, find the critical region for the test using a 5% significance level.

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Example: Binomial CD image

P(X≥5)=1−P(X≤4)=1−0.967=:highlight[0.033<0.05]P(X \geq 5) = 1 - P(X \leq 4) = 1 - 0.967 = :highlight[0.033 < 0.05]P(X≥4)=1−P(X≤3)=1−0.879=:highlight[0.121>0.05]P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.879 = :highlight[0.121 > 0.05]∴Critical region is :success[X≥5]\therefore \text{Critical region is } :success[X \geq 5]
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Example: A random variable has a distribution B(20, p). A single observation is used to test H₀: p = 0.15 against H₁: p < 0.15.

Using a 5% level of significance, find the critical region of this test.

Since it's a left tail test, we just need to cross the 0.05 boundary.

P(X≤0)=:highlight[0.0387<0.05]P(X \leq 0) = :highlight[0.0387 < 0.05]P(X≤1)=:highlight[0.1755>0.05]P(X \leq 1) = :highlight[0.1755 > 0.05]

Therefore, the critical region is X ≤ 0 (i.e., X = 0 is acceptable).

The test is performed and X = 3 is observed. Conclude 3 > 0. Therefore, Do not reject H₀.

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Example: A random variable has a distribution B(20, p). A single observation is used to test H₀: p = 0.4 against H₁: p ≠ 0.4.

Questions:

a) Using the 5% level of significance, find the critical region of this test.

b) Write down the actual significance level of the test.

a) For a two-tailed test, the significance level for each tail is 0.025.

Left Tail:

P(X≤3)=:highlight[0.0160<0.025]P(X \leq 3) = :highlight[0.0160 < 0.025]P(X≤4)=:highlight[0.0510>0.025]P(X \leq 4) = :highlight[0.0510 > 0.025]

Thus, the left critical region is X ≤ 3.

Right Tail:

P(X≥13)=1−P(X≤12)=1−0.979=:highlight[0.021<0.025]P(X \geq 13) = 1 - P(X \leq 12) = 1 - 0.979 = :highlight[0.021 < 0.025]P(X≥12)=1−P(X≤11)=1−0.943=:highlight[0.057>0.025]P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.943 = :highlight[0.057 > 0.025]

Thus, the right critical region is X ≥ 13.

Therefore, the critical region is X ≤ 3 or X ≥ 13.

b) The actual significance level is the probability contained within the rejection region:

P(X≤3)+P(X≥13)=0.01596+(1−0.978797)=:highlight[0.0370]P(X \leq 3) + P(X \geq 13) = 0.01596 + (1 - 0.978797) = :highlight[0.0370]

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