Hypothesis Tests
Null and Alternative Hypothesis
- For a hypothesis test on the population mean μ, the null hypothesis H0 proposes a value μ0 for μ
- H0: μ = μ0
- The alternative hypothesis H1 suggests the way in which μ might differ from μ0. H1 can take three forms:
- H1: μ < μ0, a one-tail test for a decrease
- H1: μ > μ0, a one-tail test for an increase
- H1: μ ≠ μ0, a two-tail test for a difference
- The test statistic is calculated from the sample. It value is used to decide whether the null hypothesis should be rejected
- The rejection or critical region gives the values of the test statistic for which the null hypothesis is rejected
- The acceptance region gives the values of the rejection region
- The critical values are the boundary values of the rejection region
- The significance level of a test gives the probability of the test statistic falling in the rejection region
- To carry out a Hypothesis Test:
- Define the null and alternative hypothesis
- Decide on a significance level
- Determine the critical value(s)
- Calculate the test statistic
- Decide on the outcome of the test depending on whether the value of the test statistic lies in the rejection/acceptance region
- State the conclusion in words
- The test statistic Z can be used to test a hypothesis about a population
- where μ is the population mean specified by H0
- The critical values for some commonly used rejection regions:
Testing Different Distributions
- Test for mean, known variance, normal distribution or large sample
X ~ N (μ, σ²/n)
- Use general procedure as outlined above
- Test for mean, large sample, variance unknown
X ~ N (μ, s²/n)
- Use the same procedure, however, you must use an unbiased estimate of the large population variance, s
- Test for large Poisson mean
X ~ N (λ, λ/n)
- Use general procedure but you must approximate the normal distribution by using the mean given
- You must apply corrections
- Test for proportion, large sample (Binomial distribution)
X ~ N (p, pq/n)
- Similar to Poisson approximation; using probability of success and applying continuity corrections
Type I and Type II Errors
- A Type I Error is made when a true null hypothesis is rejected
- A Type II Error is made when a false null hypothesis is accepted
- P(Type I Error) = significance level
- Calculating P(Type II Error):
- Firstly, calculate the acceptance region by leaving x̄ as a variable and equating the test statistic to the significance level
- Next, calculate the conditional probability that μ is now μ’ and x̄ is still in the acceptance region:
- P(x̄ is in acceptance region | μ = μ’)
- Calculate this by substituting the limit of the acceptance region as x̄ (calculated previously) and the new, given μ’ into the test statistic equation and find the probability