Brent Griffin

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Unlocking the Mysteries of Hypothesis Testing

• Brent Griffin
• Revised Fall 2006

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Whats this all about?

• Hypothesis
• An educated guess
• A claim or statement about a property of a
  population
• The goal in Hypothesis Testing is to analyze a
  sample in an attempt to distinguish between
  population characteristics that are likely to
  occur and population characteristics that are
  unlikely to occur.

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The Basics

• Null Hypothesis vs. Alternative Hypothesis
• Type I vs. Type II Error
• ? vs. ?

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Null Hypothesis vs. Alternative Hypothesis

• Null Hypothesis
• Statement about the value of a population
  parameter
• Represented by H0
• Always stated as an Equality

• Alternative Hypothesis
• Statement about the value of a population
  parameter that must be true if the null
  hypothesis is false
• Represented by H1
• Stated in on of three forms
• gt
• lt
• ?

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Type I vs. Type II Error
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Alpha vs. Beta

• a is the probability of Type I error
• b is the probability of Type II error
• The experimenters (you and I) have the freedom to
  set the ?-level for a particular hypothesis test.
  That level is called the level of significance
  for the test. Changing a can (and often does)
  affect the results of the testwhether you reject
  or fail to reject H0.

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Alpha vs. Beta, Part II

• It would be wonderful if we could force both ?
  and ? to equal zero. Unfortunately, these
  quantities have an inverse relationship. As ?
  increases, ? decreases and vice versa.
• The only way to decrease both ? and ? is to
  increase the sample size. To make both
  quantities equal zero, the sample size would have
  to be infiniteyou would have to sample the
  entire population.

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Type I and Type II Errors
True State of Nature
The null hypothesis is true
The null hypothesis is false
Type I error (rejecting a true null hypothesis) ?
We decide to reject the null hypothesis
Correct decision
Decision
Type II error (rejecting a false null
hypothesis) ?
We fail to reject the null hypothesis
Correct decision
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Forming Conclusions

• Every hypothesis test ends with the experimenters
  (you and I) either
• Rejecting the Null Hypothesis, or
• Failing to Reject the Null Hypothesis
• As strange as it may seem, you never accept the
  Null Hypothesis. The best you can ever say about
  the Null Hypothesis is that you dont have enough
  evidence, based on a sample, to reject it!

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Seven Steps to Hypothesis Testing Happiness
(Traditional or Classical Method)
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The Seven Steps

1. Describe in words the population characteristic
   about which hypotheses are to be tested
2. State the null hypothesis, Ho
3. State the alternative hypothesis, H1 or Ha
4. Display the test statistic to be used

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The Seven Steps

• Identify the rejection region
• Is it an upper, lower, or two-tailed test?
• Determine the critical value associated with ?,
  the level of significance of the test
• Compute all the quantities in the test statistic,
  and compute the test statistic itself

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The Seven Steps

1. State the conclusion. That is, decide whether
   to reject the null hypothesis, Ho, or fail to
   reject the null hypothesis. The conclusion
   depends on the level of significance of the test.
   Also, remember to state your result in the
   context of the specific problem.

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Types of Hypothesis Tests

• Large Sample Tests, Population Mean (known
  population standard deviation)
• Large Sample Tests, Population Proportion
  (unknown population standard deviation)
• Small Sample Tests, Mean of a Normal Population

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The End

• Actually, its just the beginning...