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