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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, /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 Error is made when a true null hypothesis is rejected
• 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
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