Confidence Intervals and Hypothesis Testing

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Confidence Intervals and Hypothesis Testing

• Data Analysis
• Recap
• Sampling Distributions
• Central Limit Theorem
• Mean
• Standard Deviation
• Confidence Intervals
• Null and Alternate Hypothesis
• H0 and H1
• Errors
• Relationships
• Correlation testing

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Data Analysis

• Preliminary project data sets available now
• Obeying rules for data entry
• Final data set submissions close 5pm Friday
• Final project data sets available after Friday

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Normal distribution

• Continuous probability density distribution
• Any value in a range, probability of a specific
  value is zero(there are an infinite number of
  possible values) , area under the curve
  represents probability
• Curve is bell shaped
• Mean, mode, median are equal
• Population mean and standard deviation determine
  probabilities
• Distribution has infinitely long tails

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Standard Scores-Z Scores

• All continuous scores can be standardised
• Using

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Normal distributions
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Normal distributions
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Normal distributions
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Normal distributions
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Normal distributions
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Activity data Sept5

• Descriptive Statistics

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Sampling distributions

• Consider taking samples from a population all of
  size n eg. n100
• Calculate the sample means eg. 99, 98, 102,
  103 etc.
• The distribution of sample means is approximately
  normally distributed with a mean of and a
  standard deviation of

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Central Limit Theorem

• The sampling distribution of sample means can be
  approximated by a normal probability distribution
  whenever the sample size n is large.
• large usually means over 30
• this leads to hypothesis testing on whether a
  sample represents a population or not

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See similar diagrams p106..Even you
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Confidence Intervals

• The sampling distribution of the mean is used to
  construct confidence intervals for sampling means

General equation
95 confidence interval
99 confidence interval
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Hypotheses

• Conjectures about population parameters
• To be accepted or rejected based on a statistical
  test
• All tests have to state their confidence

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Hypotheses

• Null Hypothesis
• That a population parameter is equal to a certain
  value or that population parameters of two or
  more groups are equal
• Alternative Hypothesis
• population parameter or groups not equal
• this is the hypothesis that you are trying to
  prove

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Hypotheses

• Null Hypothesis
• failure to reject this does not mean it is true
• Alternative Hypothesis
• this is proved by rejecting the null hypothesis

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Hypotheses Areas of Rejection

• Confidence of rejection of 95
• Alpha.05 or 5

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Errors

• Type I error
• Error occurs when the null hypothesis is rejected
  when it is true and should not be rejected
• The risk or probability of this occurring is
  alpha
• Type II error
• Error occurs when the null hypothesis is not
  rejected when in fact it is false and should be
  rejected.
• The risk or probability of this occurring is beta
  
• Bigger sample the lower is beta
• the power of a test is measured by

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Types of Tests

• Test of means
• t tests for small samples
• compare sample mean to a population mean
• compare sample means of two groups independent
  groups(pooled variance t-test)
• paired groups
• t tests are the same as Z tests when sample sizes
  are large

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Assumptions Underlying Tests

• Population data is normal(parametric testing)
• check the histograms or box plots
• non normal testing is possible with
  non-parametric methods
• Population variances are equal
• see if sample variances are similar compare
• No extreme values
• remove outliers

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Assumptions Underlying Tests

• Population data is normal(parametric testing)
• check the histograms or box plots
• non normal testing is possible with
  non-parametric methods
• Population variances are equal
• see if sample variances are similar compare
• No extreme values
• remove outliers

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Correlation tests

• determine if there is a relationship between two
  variables
• the correlation coefficient r
• provide a measure of the strength of that
  relationship
• tells us what percentage of the variation in one
  variable is predicted by another
• the square of the correlation coefficient r2
• relation is not causation

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Correlation tests

• Fits a best fit straight line to the data
• Use Pearsons r when data is continuous
• Use Spearmans rho when data is ordinal(ranking
  data)
• tells us what percentage of variation of one
  variable is predicted by another

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References

• References
• chapter 6 and chapter 7 of Levine D. and Stephan
  D. "Even you can learn Statistics" bundled with
  the ActivStats package for SPSS, Prentice Hall
  2005
• Normal Applet http//psych.colorado.edu/mcclella
  /java/normal/normz.html
• Central Limit and Sampling Charts
• http//math.usask.ca/laverty/S244/Lecture12.pdf