#### Hypothesis Tests

__Null and Alternative Hypothesis__

__Null and Alternative Hypothesis__

- For a hypothesis test on the population mean μ, the
**null hypothesis****H**proposes a value μ_{0}_{0}for μ- H
_{0}: μ = μ_{0}

- H
- The
**alternative hypothesis H**suggests the way in which μ might differ from μ_{1}_{0}. H_{1}can take three forms:- H
_{1}: μ < μ_{0}, a one-tail test for a decrease - H
_{1}: μ > μ_{0}, a one-tail test for an increase - H
_{1}: μ ≠ μ_{0}, a two-tail test for a difference

- H
- 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 H
_{0}

- where μ is the population mean specified by H
- The critical values for some commonly used rejection regions:

__Testing Different Distributions__

__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__

__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