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Summary
 Prediction


regression analysis
comparing means
 Statistical inference

significance
Regression
Temp
Gas
15.6
8.8
26.8
8.7
37.8
4.9
Gas
Temp
36.4
5.1.
35.5
5.2
18.6
8.7
Regression
Temp
Gas
15.6
5.2
26.8
6.1
37.8
8.7
Gas
Temp
36.4
8.5.
35.5
8.8
18.6
4.9
Regression equation
Dependent
variable
Slope
Independent
variable
Y intercept
Substitute x value into the equation
and calculate the value of y.
Regression
 Prediction using regression is most secure when the
independent variable x takes a value within the
range of the x values in your data

not about cause and effect
 extrapolation


Using the regression equation for prediction outside the
range of the original data
less secure
R 2: the Coefficient of Determination
 Link between correlation and regression
 tells us the proportion of the variance of one
variable that can be explained by straight line
dependence on the other variable
 How much can we rely on the regression
estimates
R 2: the Coefficient of Determination

.892 = .79



79% of the variance in first year uni marks can be accounted
for by the variance in the sample’s SAT scores
21% of the variance in first year marks is accounted for by
other unknown variables
Eg 2. the correlation between length of car and mpg/l is
-.7


Interpret in terms of r2
percent of variance in the Y scores variable which is
associated with the variance in the X scores.
Regression
 Use when...



1. both the variables are interval
2. for prediction about the scores of individual
cases or groups
3. to measure the amount of impact or change
that one variable produces in another
Comparison of means
 Focus on comparison of data distributions
Mean $
Mean $
N
Income Males
Income Females
Is the difference between the means real or the
result of sampling error???
Comparison of means
 Appropriate when..



Dependent variable is interval
independent variable has few categories (2 or 3)
initial analysis

look for patterns then use tables
Statistical significance
 “Real” or “Chance”?
 Significance


judgements that are made according to agreed on
mathematical rules of probability
used to infer observed differences or
relationships in the sample to the population
studied
Statistical significance
 If we drew 100 samples, how likely is it that we
would get a faulty one
 Probability theory

provides us an estimate of how likely it is that sampling
error is the real explanation for the association that we
are observing
 Tests of significance

a figure from 0.000 to 1.000

the probability of error
P - value
 P = 0.04

in only 4 out of every 100 samples would we
expect to see the association we have noted
purely by chance.

The much stronger likelihood is that the association is
real
Statistical significance
 Every finding derived from a sample is
associated with some probability of error
 How much probability of error should be
tolerated?



Researcher decides
sometimes referred to as tolerance limits
0.05 common
Presenting data
Correlation between wealth (acres
and cows owned) and reproductive
success (number of wives and
surviving offspring
Acres
Cows
Number of wives
.91 ***
.84 **
Number of surviving
offspring
.92 ***
.86 **
N
25
25
**p < 0.01; *** p < 0.001
Another example
Study 1
r(11) = .62
p > .05
Study 2
r(40) = .31
p < 0.05
Y
Y
X
Which study do we trust and why?
X
Means and proportions
 Two means

T-test
 Several means

Analysis of variance (ANOVA)
 Proportions

Chi-square
Conclusion

Univariate analysis

Describing frequency distributions
 shape; central tendency; dispersion

Inferential statistic
 Interval estimates

Bivariate analysis
 cross tabulation; correlation (strength, direction, nature)
 scattergram; regression (prediction)
 statistical significance
 comparison of means (T; ANOVA) and proportions Chi-square