Testing means, part III The twosample ttest

1
Testing means, part IIIThe two-sample t-test
2
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
3
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
4
Comparing means

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

5
Paired vs. 2 sample comparisons
6
2 Sample Design

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

7
2 Sample Design

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

8
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

9
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
10
Estimation Difference between two means
Confidence interval
11
Standard error of difference in means
pooled sample variance size of sample 1
size of sample 2
12
Standard error of difference in means
Pooled variance
13
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
14
Estimation Difference between two means
Confidence interval
15
Estimation Difference between two means
Confidence interval
df df1 df2 n1n2-2
16
Costs of resistance to disease
2 genotypes of lettuce Susceptible and
Resistant Do these differ in fitness in the
absence of disease?
17
Data, summarized
Both distributions are approximately normal.
18
Calculating the standard error
df1 15 -114 df2 16-115
19
Calculating the standard error
df1 15 -114 df2 16-115
20
Calculating the standard error
df1 15 -114 df2 16-115
21
Finding t
df df1 df2 n1n2-2 1516-2 29
22
Finding t
df df1 df2 n1n2-2 1516-2 29
23
The 95 confidence interval of the difference in
the means
24
Testing hypotheses about the difference in two
means
2-sample t-test
25
2-sample t-test
Test statistic
26
Hypotheses
27
Null distribution
df df1 df2 n1n2-2
28
Calculating t
29
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.
30
Assumptions of two-sample t -tests

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

31
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
32
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

33
Comparing means when variances are not equal
Welchs t test
34
Burrowing owls and dung traps
35
Dung beetles
36
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

37
Number of beetles caught

• Dung added
• No dung added

38
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)

39
Welchs t
Round down df to nearest integer
40
Owls and dung beetles
41
Degrees of freedom
Which we round down to df 10
42
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.
43
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

44
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.
45
A more extreme case...