Testing means, part III The two-sample t-test

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Testing means, part IIIThe two-sample t-test
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One-sample t-test
Null hypothesis The population mean is equal to
?o
Sample
Null distribution t with n-1 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Paired t-test
Null hypothesis The mean difference is equal to ?o
Sample
Null distribution t with n-1 df n is the number
of pairs
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Comparing means

• Tests with one categorical and one numerical
  variable
• Goal to compare the mean of a numerical variable
  for different groups.

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Paired vs. 2 sample comparisons
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2 Sample Design

• Each of the two samples is a random sample from
  its population

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2 Sample Design

• Each of the two samples is a random sample from
  its population
• The data cannot be paired

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2 Sample Design - assumptions

• Each of the two samples is a random sample
• In each population, the numerical variable being
  studied is normally distributed
• The standard deviation of the numerical variable
  in the first population is equal to the standard
  deviation in the second population

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Estimation Difference between two means
Normal distribution Standard deviation s1s2s
Since both Y1 and Y2 are normally distributed,
their difference will also follow a normal
distribution
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Estimation Difference between two means
Confidence interval
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Standard error of difference in means
pooled sample variance size of sample 1
size of sample 2
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Standard error of difference in means
Pooled variance
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Standard error of difference in means
Pooled variance
df1 degrees of freedom for sample 1 n1 -1 df2
degrees of freedom for sample 2 n2-1 s12
sample variance of sample 1 s22 sample variance
of sample 2
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Estimation Difference between two means
Confidence interval
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Estimation Difference between two means
Confidence interval
df df1 df2 n1n2-2
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Costs of resistance to disease
2 genotypes of lettuce Susceptible and
Resistant Do these differ in fitness in the
absence of disease?
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Data, summarized
Both distributions are approximately normal.
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Calculating the standard error
df1 15 -114 df2 16-115
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Calculating the standard error
df1 15 -114 df2 16-115
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Calculating the standard error
df1 15 -114 df2 16-115
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Finding t
df df1 df2 n1n2-2 1516-2 29
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Finding t
df df1 df2 n1n2-2 1516-2 29
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The 95 confidence interval of the difference in
the means
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Testing hypotheses about the difference in two
means
2-sample t-test
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2-sample t-test
Test statistic
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Hypotheses
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Null distribution
df df1 df2 n1n2-2
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Calculating t
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Drawing conclusions...
Critical value
t0.05(2),292.05
t lt2.05, so we cannot reject the null hypothesis.
These data are not sufficient to say that there
is a cost of resistance.
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Assumptions of two-sample t -tests

• Both samples are random samples.
• Both populations have normal distributions
• The variance of both populations is equal.

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Two-sample t-test
Null hypothesis The two populations have the
same mean ?1??2
Sample
Null distribution t with n1n2-2 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Quick reference summary Two-sample t-test

• What is it for? Tests whether two groups have the
  same mean
• What does it assume? Both samples are random
  samples. The numerical variable is normally
  distributed within both populations. The
  variance of the distribution is the same in the
  two populations
• Test statistic t
• Distribution under Ho t-distribution with
  n1n2-2 degrees of freedom.
• Formulae

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Comparing means when variances are not equal
Welchs t test
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Burrowing owls and dung traps
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Dung beetles
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Experimental design

• 20 randomly chosen burrowing owl nests
• Randomly divided into two groups of 10 nests
• One group was given extra dung the other not
• Measured the number of dung beetles on the owls
  diets

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Number of beetles caught

• Dung added
• No dung added

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Hypotheses

• H0 Owls catch the same number of dung beetles
  with or without extra dung (m1 m2)
• HA Owls do not catch the same number of dung
  beetles with or without extra dung (m1 ? m2)

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Welchs t
Round down df to nearest integer
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Owls and dung beetles
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Degrees of freedom
Which we round down to df 10
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Reaching a conclusion
t0.05(2), 10 2.23 t4.01 gt 2.23 So we can
reject the null hypothesis with Plt0.05. Extra
dung near burrowing owl nests increases the
number of dung beetles eaten.
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Quick reference summary Welchs approximate
t-test

• What is it for? Testing the difference between
  means of two groups when the standard deviations
  are unequal
• What does it assume? Both samples are random
  samples. The numerical variable is normally
  distributed within both populations
• Test statistic t
• Distribution under Ho t-distribution with
  adjusted degrees of freedom
• Formulae

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The wrong way to make a comparison of two groups
Group 1 is significantly different from a
constant, but Group 2 is not. Therefore Group 1
and Group 2 are different from each other.
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A more extreme case...