Statistics and Research Methods

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Statistics and Research Methods
Lecture 4Dr. Kristofer Kinsey
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Text Reading
(1) (2) (3) (4) (5)

• Lecture 1 Getting Started (Chapter 1 Variables
  and research design)
• Lecture 2 Descriptive Statistics (Chapter 2
  Descriptive statistics)
• Lecture 3 Probability (Chapter 3 Probability,
  sampling and distributions)
• Lecture 4 Hypothesis Testing and Statistical
  Significance (Chapter 4 Hypothesis testing and
  statistical significance)
• Lecture 5 Correlations (Chapter 5
  Correlational analysis Pearsons r)
• Lecture 6 t-tests (Chapter 6 Analyses of
  differences between two conditions)
• Lecture 7 Significance Issues Qualitative
  Research (Chapter 7 Issues of significance)
• Lecture 8 One Factor ANOVA (one DV) (Chapter 9
  Analysis of differences between three or more
  conditions one-factor ANOVA)
• Lecture 9 Research Skills (Chapter 10 Analysis
  of variance with more than one IV)

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Text Reading
(1) (2) (3) (4) (5)

• Statistics Without Maths for Psychology
• By Dancey and Reidy
• At end of each chapter work through the even
  number of questions (exam questions)
• 3rd Edition
• 2 copies on 1 hr loan
• 4 copies on 1 day loan
• 23 copies on 7 day loan
• 2nd Edition
• 7 copies on 7 day loan

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Research Testing
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Steps to Research Testing
• Specify the null hypothesis
• 2) Select a significance level
• 3) Calculate a statistic
• 4) Calculate a probability value
• If probability is than or equal to the
  significance level, then the null hypothesis is
  rejected (statistically significant) the
  alternative hypothesis is accepted
• 5) Describe the results and the statistical
  conclusion

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Hypotheses
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• The Null Hypothesis H0
• This hypothesis is set to be nullified or refuted
• Assumed to be true until statistical evidence
  indicates otherwiseusually 95 sure
• Example
• Drinking alcohol does not cause intoxication
• The Alternative Hypothesis H1
• This the hypothesis that supports your prediction
• Example
• Drinking alcohol does cause intoxication

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Significance
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• General convention in research is that a
  probability of 5 is small enough to be a useful
  cut-off point
• SoGiven a true null hypothesis, the probability
  of getting an effect by chance is less than 5 or
  less than 1 in 20
• So if you did your experiment 20 times you would
  find by chance that you get significant results
  once
• This cut-off is often called alpha (a)

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Significance
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Example We wish to test the hypothesis that our
  friend is telepathic. We toss a coin and ask him
  to guess if it is head or tail
• H0 ? He should perform no better than chance
• H1 ? He should perform better than chance
• We toss a coin and he correctly predicts the
  outcome
• Can we say he is telepathic?
• Only with significance level of 0.5 (alpha)
• If he gets the answer right the 2nd timethe
  3rdthe 4th?
• 0.50.5 0.25
• 0.50.50.5 0.125
• 0.50.50.50.5 0.0625
• 0.50.50.50.50.5 0.03125 lt 0.05 (less than
  alpha)
• If he predicts the outcome 5 times in a row we
  can say performance was significantly better than
  chance alonethis could be explained by a
  telepathic ability

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Significance
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Failing to reject the null hypothesis is
  comparable to a verdict of not guilty
• It does not actually mean that the defendant is
  not guilty
• It means that there is insufficient evidence to
  demonstrate that the defendant is guilty
• A failure to reject the null hypothesis does not
  mean that there are no differences
• It means that we are unable to detect any
  differences

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Type I Type II Error
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• What happens if getting it wrong is a VERY bad
  thing?
• Legal Example
• (a) What if we convicted an innocent person for
  life
• (b) What if we set a homicidal maniac free
• Situation (a) is called a Type I error
• Situation (b) is called a Type II error
• Many people find situation (b) disturbing but not
  as horrifying as situation (a)

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Type I Type II Error
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Type I error, also known as
• an "error of the first kind, an a error or a
  "false positive"
• the error of rejecting a null hypothesis when it
  is actually true
• accepting an alternative hypothesis when the
  results can be attributed to chance
• think there is a difference when in truth there
  is none
• Type II error, also known as
• an "error of the second kind", a ß error, or a
  "false negative"
• the error of failing to reject a null hypothesis
  when the alternative hypothesis is the true state
  of nature
• error of failing to observe a difference when in
  truth there is one

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Type I Type II Error
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• In general we use a probability level of 5
• p lt 0.05
• If changed to 1 you reduce type I error but
  increase possibility of a type II error

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Type I Type II Error
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Overlap of distributions representing different
  populations

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One- and Two-Tailed
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• One-tailed
• A one-tailed hypothesis specifies a directional
  relationship between groups

• Two tailed
• A probability computed considering differences in
  both direction

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One- and Two-Tailed
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• If very confident about the nature of the outcome
  we might predict the direction of the effect
• might predict a positive change (or a negative
  change)
• one-tailed hypothesis because only predicting one
  possible outcome
• If we were less confident about the specific
  outcome
• the change in the dependent variable might be
  positive or negative (both of which might be
  acceptable)
• called a two-tailed hypothesistwo outcomes are
  possible
• not predicting the direction of any specific
  outcome

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Assumptions
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Normal Distribution

• Always check that the data from your sample is
  roughly normally distributed before you use a
  parametric test
• Box plots histograms are helpful
• Data transformations may be necessary

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Assumptions
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Homogeneity of Variances
• Assumption that the variances of the populations
  should be approximately equal
• General rule is that that the largest variance
  you are testing is not more than 3x the smallest
• Violation of this assumption isnt too bad as
  long as you have equal numbers in your sample
  sizes
• Boxplots can be helpful

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Assumptions
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Outliers
• Anomalous values in the data
• Tend to increase the estimate of sample variance
• May be due to recording errors
• Everyone in the sample may not be from the same
  population

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t-distribution
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Up to now weve used examples where we know the
  population mean (µ) and standard deviation (s)
• used the Z-distribution
• What happens when we dont know s, but only have
  the sample mean (X) and standard deviation (s)
• use the t-distribution instead
• Gosset (1908) under the pseudonym Student
• a metric of quality for Guinness

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t-distribution
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• Area under curve must be 1
• Must be symmetrical (bell-shaped)
• more observations, the better representation of
  the population

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t-distribution
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• t distribution has only 1 parameter
• the degrees of freedom (df)
• df the number of independent observations
• If we are measuring 10 items, we have 9
  independent observations
• 9 values mean. Trying to estimate mean, after
  the 9 measures are taken, the 10th is
  predetermined
• for the t distribution, df n -1, n number of
  observations

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One Sample t-test
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions
How do we calculate it?
Notice similarity to z
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One Sample t-test
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions

• What is the probability of 4 randomly selected
  people having an average IQ greater than 120?

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One Sample t-test
(1) Research Question (2) Significance
(3) Type I II Error (4) One- Two-Tailed
(5) Assumptions
Probability of getting t34 or larger lt0.05
Probability of getting t34 or more extreme
lt0.025
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