Hypothesis Testing

1
Hypothesis Testing

• Define null hypothesis (H0) (usually no
  effect, or complement of effect you are trying
  to prove)
• Does direction matter?(must decide in advance)
• Choose test statistic.
• Calculate test statistic, t
• Probability of obtaining value at least as
  extreme as t under H0 (P)
• P small reject H0

2
Practice Setting up Hypothesis

• The contaminant levels of two fish populations
  are being compared, one upstream from a nuclear
  waste facility, and one downstream.
• H0 The contaminant levels of the downstream
  population will be less than or equal to () that
  of the upstream population
• H1 The downstream fish are more contaminated
  than the upstream population

3
Types of Errors in Hypothesis Testing

• ? chance of rejecting H0 when H0 is true (higher
  ? means more chance of Type I error)
• ? chance of failing to reject null hypothesis
  when it is false
• (higher ? means more chance of Type II error)

Type I errors (the "false positive") the error
of rejecting something that should have been
accepted e.g., such as finding an innocent
person guilty. Type II errors (the "false
negative") the error of accepting something that
should have been rejected e.g., such as finding
a guilty person innocent.
4
Power of a test

• The power of a statistical test is the
  probability that the test will reject a false
  null hypothesis, or that it will not make a Type
  II error.
• The probability of a Type II error is referred to
  as ß. Therefore power is equal to 1 - ß.
• As power increases, the chances of a Type II
  error decrease

5
Power of a test, cont

• Calculating the power requires first specifying
  the effect size for detection. The greater the
  effect size, the greater the power.
• The most common form of increasing the power is
  increasing the sample size.
• Most researchers who assess the power of their
  tests use 0.80 as a standard for adequacy.

6
Types of Errors in Hypothesis Testing

• H0 The contaminant levels of the downstream
  population will be less than or equal to that of
  the upstream population
• H1 The downstream fish are more contaminated
  than the upstream population
• Type 1 Error Cost?
• Decide that the downstream fish are contaminated
  when they really arent.
• Cost monetary cost for corporations.
• Type 2 Error Cost?
• Decide that the populations have same/lower
  contamination levels when downstream fish are
  actually more contaminated. Cost people may eat
  contaminated fish and die?
• Which should we try to make as low as possible?
• a or ? How?
• We can minimize Type I error by merely decreasing
  alpha our downstream sample population will
  have to have a very abnormal contamination level.
  Easily avoid extraneous costs
• We can minimize Type II error by increasing
  alpha, increasing sample size (number of fish
  sampled), increase the effect size for which we
  will accept a given alpha level