Null and alternative hypotheses

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Lesson 19a
Hypothesis Testing

• Null and alternative hypotheses
• Errors in decision making
• Strategy for statistical hypothesis testing

Main Ideas
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• 1. Choices
• Many decision problems involve making a choice
  between two possibilities
• Process is in control or not in control
• Defendant is guilty or not guilty
• Drug is effective or not effective
• Food product is safe for consumption or not safe

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• 2. Statistical Hypotheses
• A statistical hypothesis is a conjecture about
  the state of nature of something we are studying.
  
• A statistical test of hypothesis is a procedure,
  based on data, for deciding between two
  statistical hypotheses called the null hypothesis
  (denoted H0) and the alternative hypothesis
  (denoted Ha).

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• 3. Illustrations
• Suppose a target value for the amount of liquid
  in a bottle of soft drink is 20 oz. In running a
  control chart for the mean, we decide between
• H0 the process mean is 20 oz. (on target)
• Ha the process mean is not 20 oz. (off target)
• In a jury trial we decide between
• H0 the defendant is not guilty
• Ha the defendant is guilty

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• 4. Two Types of Errors in Hypothesis Testing
• Type I The null hypothesis is true but we say
  the alternative is true.
• Type II The alternative hypothesis is true but
  we say the null hypothesis is true.
• H0 process on target, Ha process out of control
• Type I process on target, but we say it is out
  of control.
• Type II process out of control, but we say it is
  on target.

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• 5. Representation of Errors
• We say this
• H0 is true Ha is true
• H0 is true o.k. Type I
• Reality
• Ha is true Type II o.k.

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• 6. Consequences of Making Errors
• Consequences of making a Type I error are not the
  same as those of making a Type II error.
• H0 defendant not guilty, Ha defendant guilty
• Jury convicts an innocent person (Type I error).
• Jury frees a guilty person (Type II error).
• Ideally we would like both errors to have a small
  chance of occurring. If that is not possible we
  have to decide which error is most important to
  protect against.

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• 7. Strategy in Statistical Hypothesis Testing
• Define the null hypothesis and use the decision
  procedure so that the chance of a Type I error
  small, typically 5.
• For instance, in a jury trial we do not want to
  convict an innocent person, so we require guilt
  beyond a reasonable doubt.
• In statistical terms we let not guilty be the
  null hypothesis, and require a small chance of a
  Type I error.

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• A regulatory agency in charge of drug safety
  would not want an unsafe drug to enter the
  marketplace even if it meant some safe drug is
  rejected (error on the side of caution).
• From a statistical point of view, the agency
  would set the null hypothesis as drug is not
  safe, and require a small chance of a Type I
  error. This has the practical effect of
  requiring strong evidence that the drug is safe
  before it is approved for use.

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There seems to be a lot going on here. How can
I keep it all straight?
First make sure that you understand what the null
and alternative hypotheses are. Then put errors
into the context of the problem. For instance in
the jury example, innocent person sent to jail
or guilty person set free tell us what Type I
and Type II errors mean in this context.
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