Statistics - statistical hypothesis testing

Hypothesis testing

The use of statistical techniques to test a particular claim (the hypothesis). A sample from the population is used to see if the result from the sample is consistent with the claim.

Process of a hypothesis test - Set up/ state the hypotheses

the null hypothesis (H₀) - where p takes a particular value/ the value you would expect

alternative hypothesis (H₁) - the probability you’re testing (one-tailed or two-tailed?)

Process of a hypothesis test - Setting the significance level

This is the probability of rejecting the null hypothesis if in fact it is true

Process of a hypothesis test - Carrying out the test (P-values)

Use a sample to obtain a test statistic (the value of X in X ∼ B(n,p))

probability of obtaining a value at least as extreme as the test statistic, if the null hypothesis is true
For X has a value, use X ∼ B(n,p) to calculate…

• P(X > x) if H₁: p > x

• P(X < x) if H₁: p < x

• 2 x P(X > x) if H₁: p ≠ x

Whether the value of X given is in the critical region will determine whether to reject the null hypothesis.

Process of a hypothesis test - Carrying out the test (Critical regions)

Use a sample to obtain a test statistic (the value of X in X ∼ B(n,p))

set of values for the test statistic X for which you would reject the null hypothesis. The critical value is the value for X for which you change from not rejecting the null hypothesis to rejecting it.

• For H₁: p > x, use X ∼ B(n,p) to find the lowest value for r for which P(X > r) is less than the significance level.

• For H₁: p < x, use X ∼ B(n,p) to find the highest value for r for which P(X < r) is less than the significance level.

• For H₁: p ≠ x, the critical region has 2 parts to it, Split the significance level into 2, one for the lower tail and one for the upper tail. Then use X ∼ B(n,p) to find the lowest value for r for which P(X > r) is less than half the significance level and to find the highest value for r for which P(X < r) is less than half the significance level.

Whether the value of X given is in the critical region will determine whether to reject the null hypothesis.

Process of a hypothesis test - The conclusion

Reject H₀ - if p-value is less than sig level/ test statistic lies in critical region → there is sufficient evidence to suggest that H₁ is true

Not reject H₀ - if p-value is more than sig level/ test statistic doesn’t lie in critical region → there is not sufficient evidence to suggest that H₁ is true

Process of a hypothesis test order - 1

Set up/ state the hypotheses

Process of a hypothesis test order - 2

Setting the significance level

Process of a hypothesis test order - 3

Carrying out the test

Process of a hypothesis test order - 4

The conclusion

One-tailed test

When you think a result is more likely H₁: p > x

When you think a result is less likely H₁: p < x

Two-tailed test

When you think there’s a bias in general H₁: p ≠ x