Review of Statistics

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Review of Statistics

• AED 616
• Spring 2007
• Dr. Ed Franklin

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Null Hypothesis

• A random sample engineers and psychologists are
  drawn and administered a self-reporting measure
  of sociability.
• The researcher computes the Mean for each group.
• The computed Mean for the engineers is 65.00.
• The computed Mean for psychologists is 70.00.
• Where did the five point difference come from?

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Three possible explanations

• Population of psychologists is truly more
  sociable than the population of engineers, and
  the researchers samples correctly identified the
  difference.
• There may have been bias in the procedures. By
  random sampling the researcher rules out sampling
  bias. Measurement bias may be the cause. Groups
  may have been contacted at different times of the
  year (Psychologists may have been contacted in
  December, and engineers may have been contacted
  in February).
• The populations of psychologists and engineers
  are the same but the samples are unrepresentative
  of their populations because of random sampling
  errors.

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3rd Explanation Null Hypothesis

• General form in which it is stated varies from
  researcher to researcher.
• Three versions in which the null hypothesis is
  stated by the researcher

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Three Versions for Stating the Null Hypothesis

• The observed difference was created by sampling
  error.
• There is not true difference the two groups.
• The true difference between the two groups is
  zero.

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Significance Tests

• Significance Tests determine the probability that
  the null hypothesis is true.
• The researcher conducts a significance test.
• Test reveals the probability that the null
  hypothesis is a correct hypothesis is less than 5
  in 100.
• This would be stated as p lt .05
• Where p stands for probability.
• If the chances that something is true are less
  than 5 in 100, it is likely that it is not true.
• The researcher would reject the null hypothesis.
• Leaving only the first two explanations they
  started with as viable explanations for the
  difference.

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• Researchers normally state in their reports the
  probability level they used to determine whether
  to reject the null hypothesis.
• Typically, it is .05 or less (such as .01 or
  .001).
• When the researcher fails to reject the null
  hypothesis because the probability is greater
  than .05, they do just that They fail to
  reject the null hypothesis, and it remains on
  their list of possible explanations.
• The researcher cannot accept the null
  hypothesis as the only explanation for a
  difference.

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Statistically Significant

• An alternative way to say they rejected the null
  hypothesis is to state that the difference is
  statistically significant.
• If it is stated that the difference is
  statistically significant at the .05 level
  (meaning .05 or less), it is equivalent to
  stating that the null hypothesis has been
  rejected at that level.
• Researchers report which differences were tested
  for significance, which significance test they
  used, and which differences were found to be
  statistically significant.
• More common to find null hypothesis stated in
  thesis and dissertations because committee
  members want to make sure the students they are
  supervising understand the reason they conducted
  the test.

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Type I Error

• The error of rejecting the null hypothesis when
  it is correct.
• Suppose a weather forecaster reports that the
  probability of rain tomorrow is less than 5 in
  100.
• What should we conclude?
• First, we know there is a chance of rain but it
  is very small.
• Most people may conclude that it probably will
  not rain and not make special preparations.
• By not making any special preparations, they are
  acting as though it will not rain they have
  rejected the hypothesis that it will rain
  tomorrow.
• There is always some probability that the null
  hypothesis is true so if we wait for certainty,
  we will never make a decision.
• If we make nor special preparations for rain 100
  days for which the probability of rain is .05, it
  will probably rain 5 of those 100 days.
• In rejecting the null hypothesis, we are taking a
  calculated risk we might be wrong.
• This is a Type I error.

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Levels of Significance

• .06 level not significant do not reject the
  null hypothesis.
• .05 level significant reject the null
  hypothesis.
• .01 level more significant, eject the null
  hypothesis with more confidence that at the .05
  level.
• .001 level highly significant reject the null
  hypothesis with even more confidence than at the
  .01 or .05 levels.

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Type II Error

• Fail to reject the null hypothesis when, in
  reality, it is false.
• This type of error can have serious consequences.
• Suppose a drug company has developed a new drug
  for a serious disease.
• Suppose that, in reality, the new drug is
  effective.
• If, however, the null hypothesis is not rejected
  because the drug company selected a level of
  significance that is too high, the results of the
  study will have to described as insignificant,
  and the drug may not receive government approval.

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Pearson Correlation Coefficient

• Used when a researcher wants to examine the
  relationship between two quantitative sest of
  scores.
• Most widely used coefficient is Pearson
  Product-Moment Correlation Coefficient.
• Symbol is r (called Pearson r).
• See example

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Example 1
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• Employees with high employment test scores have
  high supervisors ratings, and those with low
  test scores have low supervisors ratings.
• Illustration is example of a direct relationship
  or positive relationship.
• If the relationship were perfect, the value of
  the Pearson r would be 1.00.
• Being less than perfect, its actual value is .89

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Values of r
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Inverse Relationship (negative relationship)

• Those high on one variable are low on the other.
• Individuals with high self-concept scores are low
  on depression, while those who are low on
  self-concept are high on depression.
• Relationship is not perfect.
• Value of the Pearson r for the relationship is
  -.86

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Example of Inverse Relationship
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• Relationships in Tables 1 and 2 are strong
  because they are near 1.00 and -1.00.
• A value of 0.00 indicates the complete absence of
  a relationship.
• Pearson r is not a proportion and cannot be
  multiplied by 100 to get a percentage.
• To think about correlation in terms of
  percentages, must convert Pearson r to another
  statistic called the Coefficient of
  Determination, symbol is r2