ARCH 21266126

1
ARCH 2126/6126

• Your papers for review please?
• Session 8 Comparing sample means

2
Imagine, then, two batches of numbers...

• ... representing, say, lengths of scrapers
  excavated from two sites
• Are they the same or different?
• Most unlikely to be exactly the same
• Are they similar enough that they could be two
  samples from one population?
• Or are they different enough to make that a very
  unlikely explanation?

3
We have reviewed some theoretical considerations

• Re-sampling central limit theorem
• Statistical significance
• p-values low p high significance
• The problem of many tests
• Type I and type II errors
• Sir R.A. Fisher the magic of 5
• Classical statistics versus exploratory data
  analysis

4
So how do we reach a p-value hence a
conclusion?

• A formal statistical test
• E.g. a t test (Students t)
• Special case of Analysis of Variance
• In general the appropriate test for cases like
  the 2 batches of numbers, scraper lengths or
  human statures
• H0 the 2 batches have equal means
• H1 they have different means

5
t refers to a distribution, like the normal
distribution

• Or rather, a set of distributions
• A different distribution for each possible sample
  size or rather each degree of freedom (d.f.)
• Degrees of freedom n 1
• The larger the sample, the more t resembles the
  normal (z) distribution
• For n ? 30, they are virtually the same
• For n ?, they are the same

6
(No Transcript)
7
The formula looks intimidating

• Its given in both Drennan (p.156) and Madrigal
  (p.98)
• But to calculate t all you need to know is--
  the sample size- the mean - the variance (or
  SD)of each sample
• We have already practised getting these from raw
  data

8
Here is the procedure in words for the (unpaired)
t test

• To test the hypothesis that two samples could
  have been drawn from the same population, or two
  populations with the same mean
• Calculate the difference between the means
• Calculate the joint standard error of the means
• Divide the first by the second

9
Now lets do it with an example Drennans (p.150)

• Calculate pooled standard deviation
• ((n1-1) x s12) ((n2-1) x s22)
• Divide this by (n1 n2 2)
• Take the square root to get Sp
• Multiply by square root of (1/n1 1/n2)
• This is pooled standard error SEp

10
(No Transcript)
11
To complete calculation of t

• We scale the difference between the means by SEp
• I.e. t (X-bar1 X-bar2)/SEp
• When we have this number, we have t
• What does it mean?
• Look up in table (p.125)
• Gives probability of finding such a difference by
  re-sampling one popn

12
(No Transcript)
13
More specifically

• The outcome is a number, a value of the test
  statistic
• E.g. t -3.66 (a different example)
• We also need the degrees of freedom, a number
  usually closely related to the sample size for
  t, the two sample sizes summed e.g. 7 8 2
  13
• We need to know whether to apply a one- or
  two-tailed test (normally two)

14
You can also use

• A statpack e.g. SPSS
• Or website e.g. http//home.clara.net/sisa/index.h
  tm
• Or even just Excel (calculate the long way, then
  check against TTEST which returns p-value)
• Round as late as possible
• Checking is good

15
Interpreting the outcome

• From value of t plus d.f. we can get a p-value
  from a look-up table or an electronic source
  e.g. 0.01gtpgt0.001
• We should normally report- the value of the test
  statistic - the degrees of freedom- the p-value
  or significance level
• Hence in this case t 3.66, d.f. 13,
  0.01gtpgt0.001

16
Assumptions of the t test

• Random sampling
• Independence of observations and samples
• Data are normally distributed
• Variances are homogeneous (or not)
• If these conditions are significantly violated,
  need to discuss this with your statistical adviser

17
This is the unpaired t test

• Other forms of the t test exist
• Comparison of a single observation with a sample
  mean
• And paired samples t test (simpler) for use when
  repeated measures are being taken on the same set
  of cases
• t special case of analysis of variance
• These are all univariate statistics