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

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 $H_0: p = 0.25$ against $H_1: p > 0.25$ using a 5% significance level.

We will use critical regions to perform this test.

We need the set of $x$ values such that $P(X \geq k) \leq 0.05$, i.e., every single value we can observe that would lead to $H_0$ being rejected.

We do this by trial and improvement:

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

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:

3. Look for $P(X \leq k)$ crossing over sig level boundary.
4. Summarize.

Right Tail:

5. Look for the probability crossing over the $(1 - \text{SIG LEVEL})$ boundary.
6. Write down the calculations either side of this boundary:

7. Summarize:

A test statistic has a distribution $B(10, p)$. Given that $H_0: p = 0.2, H_1: p > 0.2$, find the critical region for the test using a $5$% significance level.

Example: Binomial CD

Example: A random variable has a distribution $B(20, p)$. A single observation is used to test $H_0: p = 0.15$ against $H_1: p < 0.15$.

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.

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 $H_0$.

Example: A random variable has a distribution $B(20, p)$. A single observation is used to test $H_0: p = 0.4$ against $H_1$: $p \neq 0.4$.

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:

Thus, the left critical region is X ≤ 3.

Right Tail:

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: