Variance and Hypothesis
Tests
Econ. 201 – Econ. Data Analysis
Difference in Means - Design
• Try to isolate the experimental and control
groups from each other while the
experiment is conducted
• Amounts to ensuring either that both
groups are random samples drawn from
the “universe” of such populations or that
each group has equivalent values of any
variable that might also (in addition to the
experiment) alter its outcome
Example
• “Men have higher cholesterol than women”
• Want to isolate gender as a cause for
cholesterol levels to vary. Yet we know many
other things effect cholesterol
– 2 are diet and genetics
• Could isolate gender in two ways
– 1: Draw random sample from the population
• Ask, is cm>cw?
– 2: Draw samples of men and women from populations
with similar diets and genetics --- say 20 yr. old
college students
• Pose the same question as #1
Example, contd.
• The Data in the example take the latter
approach
• Then pose the question formally as:
– Ho: cm=cw (The null hypothesis)
• that which you hope to disprove
– H1: cm≠cw (The alternative hypothesis)
An alternative statement of the
hypothesis
• Define the difference in cholesterol
between the two groups and ask is it zero
or not.
• Using mean levels of the data collected on
each group, form the difference.
• d=cw-cm.
• Ho: d=0 (The null hypothesis)
• H1: d ≠0 (The alternative hypothesis)
Construction of mean and variance
• Look at raw data on p. 56.
• Construct the following from this and the
sheet “Some Basic Statistical Formulae.”
– The mean value of cholesterol for each group.
– The variance and standard deviation of each
group.
Standard Error of the Mean
• Another measure of a distribution is the
Standard Error of the Mean.
– Formula is on the sheet: look at it.
• An estimate of the variability of dispersion
of the sample mean.
– Assuming it were itself constructed from
repeated samples of size n from a population.
– Is a measure of our uncertainty over the true
or population mean, given that we are
“estimating” it.
The Central Limit Theorem
• If the underlying experimental design that
generated the data is a random one, then the
means of various such experiments will be
drawn from a distribution with a mean = (∑x)/n,
and a standard deviation = s/√n.
• Then the area under the standard normal curve
(p. 57) contains various ranges of the mean. A
general rule of thumb says that we have a
95(.4)% confidence level that the true sample
mean lies within +/- 2(s/√n)
Intuition
• Calculate the interval around each group’s
mean with the standard errors of the
means (see page 57).
• The further apart are the means and the
smaller the dispersions around these
means (stnd. errors), the more likely we
are to determine that the mean levels of
the two groups are different.
Alternative formulation, d
• Look at formation and resulting distribution
of d on p. 57.
• d = 173.5 – 163.3 = 10.2
• Now form the variance of this mean
difference
• Defined as the sum of the variances of the
standard errors of each individual mean
– see p. 57 for formulas, = 6.02
Formation of 95% confidence
interval around mean d
• 2+/-(standard error of the difference), here
2+/-(6.02).
• Can be 95% confident that true mean d
lies in the range from: -1.84 to 22.24.
• Cannot be 95% sure d is not 0.
• This interval includes zero, so at the 95%
confidence level, given the data, we
accept the null hypothesis, H0, reject the
alternative, H1.
Cholesterol Example, contd.
• Look at raw data by frequency (p. 57)
• Understand that the two, equivalent, ways
of framing the hypothesis concern either:
• 1. The degree of overlap between the confidence
interval we construct around the mean of men’s
cholesterol observations, and that we construct
around the mean of woman’s cholesterol
observations, and seeing by how much they
overlap, or,
• 2.Whether the distribution of d contains 0 in the
confidence interval we construct around its mean
Construction of the true confidence
level
• We know we can meet the requirement of 1+/(stnd. error of the mean)
• Would give us a 66% level of confidence because 66% of the
area under the standard normal curve lie in this range
• Here = 10.2+/- 6.02: from 4.18 to 16.22
• But we cannot meet the criteria for a 95% level
of confidence – somewhere between 95% and
66%
• So there is weak support for the contention that
cholesterol varies by gender
Or could consult a t-statistic
• t = mean/(it’s standard error)
• “critical values of t, depend on the size of
the sample, and gives a significance value
at which a particular sample mean can be
assumed to be different than zero
• here t = 10.2/6.02 = 1.69
• for a sample of 30, a t–statistic of 1.69, is
significant at approximately the 90% level
Automate the calculation
• Use Excel
• Convenient for “big” data sets, with many
observations
• Use it to calculate:
– 1. avg. cholesterol,
– 2. differences from avg.,
– 3. differences squared
– 4. squared differences summed
Excel Computations, contd.
• Use a calculator and formula sheet for the
rest
• Calculate the variance and the standard
deviations of the two samples
• Calculate the stnd. error of each mean
• Then calculate the stnd. error of the
difference in means