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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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