Elementary Statistics

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Elementary Statistics

• Chapter 1

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The Scientific Method

• Step 1 Formulate a Theory.
• Step 2 Collect data to test the theory.
• Step 3 Analyze the results.
• Step 4 Interpret the results and make a
  decision.

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Definitions

• Population - the entire group of objects or
  individuals under study.
• Sample - a part of the population that is
  actually used to get information.
• Statistical inference - the process of drawing
  conclusions about the population based on
  information from a sample of that population.

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Statistical Hypothesis

• A statistical hypothesis consists of two parts,
  the null hypothesis and the alternate hypothesis.

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Statistical Hypothesis

• Null Hypothesis - denoted by H0 - the
  conventional belief - the status quo, or
  prevailing viewpoint, about a population.

• Alternate Hypothesis - denoted by H1 - an
  alternate to the null hypothesis - the change in
  the population that the researcher hopes to find
  evidence for.

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Statistically Significant

• The data collected are said to be statistically
  significant if they are very unlikely to be
  observed under the assumption that H0 is true.
  If the data are statistically significant, then
  our decision would be to reject H0.

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What if we make a mistake?

• Type I Error Rejecting the null hypothesis when
  in fact it is true.

• Type II Error Accepting the null hypothesis
  when in fact it is not true.

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Decisions, Decisions
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Definition

• The Significance Level number ? is the chance of
  committing a Type I error -- that is the chance
  of rejecting the null hypothesis when it is in
  fact true.

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Frequency (or Dot) Plot

• A frequency plot displays a set of observations
  by representing each observation value with a
  symbol positioned along a horizontal scale. If
  there are two or more observations with the same
  value, the symbols are stacked vertically.

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Definition

• The number n of observations in a sample is
  called the sample size.

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Decision Rule

• A decision rule is a formal rule that states,
  based on the data obtained, when to reject the
  null hypothesis H0. Generally, it specifies a
  set of values based on the data to be collected,
  which are contradictory to the null hypothesis H0
  and which favor the alternative hypothesis H1.

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Direction of Extreme

• The Direction of extreme corresponds to the
  position of the values that are more likely under
  the alternative hypothesis H1 than under the null
  hypothesis H0. If the larger values are more
  likely under H1 than under H0, then the direction
  of extreme is said to be to the right.

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Most Extreme Value

• The value under the null hypothesis H0 that is
  least likely, but at the same time is very likely
  under the alternative hypothesis H1, is called
  the most extreme value.

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Rejection Region

• A rejection region is the set of values for which
  you would reject the null hypothesis H0. Such
  values are contradictory to the null hypothesis
  and favor the alternative hypothesis.

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Rejection Region

• An acceptance region is the set of values for
  which you would accept the null hypothesis H0.

• A cutoff value, or critical value, is a value
  that marks the starting point of a set of values
  that comprise the rejection region.

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One or Two - Sided?

• A rejection region is called one-sided if its set
  of extreme values are all in one direction,
  either all to the right or all to the left.

• A rejection region is called two-sided if its set
  of extreme values are in two directions, both to
  the right and to the left.

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P-value

• The p-value is the chance, computed under the
  assumption that H0 is true, of getting the
  observed value plus the chance of getting all of
  the more extreme values.

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Two Approaches for Decision Making

• Classical Approach Choose a level of
  significance, ?, and use it to create a decision
  rule.

• P-value Approach Collect the data and calculate
  the likelihood of the data based on the
  assumption that the null hypothesis is true. If
  the p-value is less than ?, reject H0.

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Think About It

• The significance level is ? 0.1, the chance of
  committing a Type I error. The corresponding
  decision rule is Reject H0 if the selected
  voucher is 50 or more. A voucher is selected
  and it turns out to be 60. Your decision is to
  reject the null hypothesis and conclude that the
  data are statistically significant at the 10
  level.

• You reject H0. Could you have made a mistake?

• What type of mistake could you have made?

• What is the chance that you have made a mistake?