Download Chapter 9. Introduction to Procedures Involving Sample Means Part 2

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Ch9pt2 Intro Sample Means
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©Richard Lowry, 1999-2000
All rights reserved.
Chapter 9.
Introduction to Procedures Involving Sample Means
Part 2
Oh, the joys of creation, especially when it can be done fairly simply with a few lines
of programming code. Once again, somewhere deep within the electronic workings of
your computer I have created a vast population of Xi values. This time, however, all I
will tell you about the population is that it is normally distributed. The rest of its main
properties—mean, variance, and standard deviation—you will have to figure out for
yourself, starting from scratch. Note, incidentally, that this scenario is much closer to
reality than the one described in Part 1 of this chapter. There are many real-life
research situations where we can reasonably suppose that a source population is
normally distributed, but rarely if ever do we know in advance the true values of its
central tendency and variability. We will begin with the easiest part of the task, which
is to figure out the mean of the population.
¶Estimating the Mean of a Normally Distributed Source Population
In Part 1 we noted that a random sample will tend to reflect the properties of the
population from which it is drawn. Among these reflected properties is the
population's central tendency. So here is the simple, straightforward way of
estimating the mean of our current source population. Draw from it a random sample
of size N, and then let the mean of that sample serve as your estimate of the mean of
the population.
The principle underlying such a process is that the mean of any particular sample can
be taken as an unbiased estimate of the mean of the population from which the
sample is drawn. In general, a biased estimate is one that will systematically
underestimate the true value, or systematically overestimate it, while an unbiased
estimate is one that avoids this tendency. The unbiased estimate might prove to be
either under or over the true value in particular cases, but it will not move in either of
these directions systematically. It is roughly analogous to shooting arrows at a
target. The archer who tends to hit below the bull's eye is systematically biased in
one direction, while the archer who tends to hit above it is systematically biased in
the other. An archer without such a systematic bias will hit below and above the bull's
eye in equal measure, and occasionally she will even hit it dead center.
But of course, even an unbiased archer is not necessarily a contender for the world
title. The shots of one might tend to cluster within two or three inches of dead center,
while those of another might scatter out over a distance of two or three feet.
Similarly, some sample means can be regarded as very close estimates of the
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population mean, while others can be taken only as loose ballpark estimates.
In the following table is a button labeled "Sample." Each time you click it, you
computer will draw a random sample of whatever size is indicated in the cell labeled
"sample size." The default value of "sample size" is 5, though you can reset it to any
positive integer value you might wish. I suggest you begin by drawing a few samples
of size N=5, then a few of size N=10, and so on until you have worked your way up to
some fairly large sample sizes. By the time you are drawing samples of size N=100,
you will begin to get a pretty close fix on the mean of the source population. With
samples of larger sizes, it will be closer still. (Note, however, that large sample sizes
will take longer to run, especially if your browser is Internet Explorer. ) To help you
avoid getting lost in the fractional numerical details of your samples, each sample
mean is also shown rounded to the nearest integer value.
[Interactive portion omitted from this print file.]
So, click, click, click away. And then, when you have finished, click here to continue
with the text of this chapter.
If you have taken the time to perform the above exercise, you have almost certainly
arrived at the conclusion that the mean of our current source population is
somewhere in the vicinity of 15. Perhaps a shade more than 15, perhaps a shade
less, but in any event somewhere in that vicinity.
Just how narrowly or broadly you can set the boundaries of "vicinity" depends on the
size of the sample. The fact that the mean of our current source population lies
somewhere in the neighborhood of 15 might not have been very obvious when you
were drawing small samples, but it must surely have become so as the sizes of your
samples increased. In general, a larger sample sets the boundaries of "vicinity" more
narrowly, while a smaller sample sets them more broadly. From what we observed in
Part 1 concerning the sampling distribution of sample means, you will know that these
boundaries are also determined by the amount of variability that exists within the
population from which the sample is drawn. A relatively small amount of variability
within the source population would define "vicinity" fairly narrowly, while a relatively
large amount would define it fairly broadly. Either way, the general structure of this
process of estimation is
estimated
source
= MX ±[definition of "vicinity"]
In Chapter 10 we will show precisely how you go about defining the boundaries of
"vicinity." Meanwhile, so as not to keep you in suspense, I will tell you that your
estimate of "somewhere in the vicinity of 15" is right on target. The mean of the
source population from which you were drawing your samples is precisely
source
= 15.0
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¶Estimating the Variability of a Normally Distributed Source Population
The principle here is akin to the one just discussed in connection with the central
tendencies of samples and populations, thought with one very important difference.
The variability that is observed to exist within a sample can be taken as an estimate
of the variability of the population, but not as an unbiased estimate. In general, the
variability that appears within samples will tend to be smaller than the variability that
exists within the entire population. In a relatively small percentage of cases it will be
larger, and occasionally it might even hit the variability of the population dead center;
but overall there is a strong bias in favor of the observed variability of a sample
coming out as an underestimate of the variability of the population.
Here as well, the basic concept is that a random sample will tend to reflect the
properties of the population from which it is drawn. Samples drawn from a highly
variable source population will tend to contain relatively large amounts of variability,
while those drawn from a fairly homogeneous population will tend to contain smaller
amounts of variability. Moreover, the larger the size of the sample, the closer the
reflection will tend to be.
You will recall from Chapter 2 that the raw measure of variability within a set of
numerical values—X 1 , X 2 , X 3 , etc.—is the sum of squared deviates, SS, for which the
formulas are
conceptual: SS =
(Xi — MX )2
computational: SS =
Xi2 —
Xi)2
(
From Ch. 2, Pt. 2.
N
The variance of the set of Xi values is then the average of these squared deviates:
variance =
SS
N
and the standard deviation is the square root of that average:
standard deviation = sqrt
[ ]
SS
From Ch. 2, Pt. 2.
N
The following demonstration focuses on the variance. Each time you click one of the
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buttons below, your computer will reach into our current source population and draw
10 random samples of the indicated size, either N=5 or N=20. It will also calculate and
display the variance of each sample along with the average variance of the whole set
of 10 samples and the cumulative average variance of all the samples that you draw
through repeated clicking. Please click each of the buttons for as many times as you
can muster the patience, noting in particular the values that appear toward the
bottom of each text box under the heading of "cumulative average variance." If you
click each button 30 or 40 times, so as to accumulate 300 to 400 samples of each
size, you will almost certainly find that the cumulative average variance for samples of
size N=20 will be larger than for samples of size N=5. (I say "almost certainly"
because, whenever you are drawing samples at random, there is always the slight
chance of ending up with something extraordinary. If this should happen in your case,
click the "Clear" buttons and start over.)
[Interactive portion omitted from this print file.]
If you really want to get a hands-on idea of what is going on here, keep clicking the
two buttons again and again, non-stop over the next several weeks, so as to
accumulate a vast number of samples of each size. I do not imagine anyone will
actually do this. But if you were to do it, here is what you would find. For your zillion
random samples of size N=5, the cumulative average sample variance would very
closely approximate the value of 6.0; and for your zillion random samples of size
N=20, it would very closely approximate the value of 7.125.
There are two reasons why I am able to make this claim. The first is merely
adventitious, occasioned by the fact that I needed to construct a specific source
population in order to illustrate these principles. As the designer of this population I
happen to know, and will now share the fact with you, that its variance is exactly
:
2
source
= 7.5
[hence:
source
= ±2.74]
The second reason is one of principle and obtains irrespective of whether you know
the variability of the source population in advance. When drawing random samples of
size N from a normally distributed source population, the average variance of those
samples will end up in the long haul being equal to a certain proportion of the
variance of the population. That proportion is determined by the size of the samples;
its precise value is given by the ratio (N—1)/N. For samples of size N=5 the proportion
is 4/5=0.80, so here we would expect the average sample variance to end up as
0.80x7.5=6.0. For samples of size N=20 it is 19/20=0.95, so we would expect
0.95x7.5=7.125. And so on. For all cases where samples of size N are randomly
drawn from a normally distributed source population, the form of the relationship is
mean sample variance =:
2
source x
N—1
N
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Conversely, the variance of the source population would be equal to the average
sample variance multiplied by N/(N—1):
2
::
source = mean sample variance x
N
N—1
It is this latter version of the relationship that permits us to estimate the variability of
a source population—in those more realistic cases where we do not know it in
advance—on the basis of a single sample. For even though the variability contained
within individual samples might not fall precisely at the average, it will nonetheless
tend to fall somewhere near the average. In effect, therefore, the observed variance
of a single sample can be taken as an estimate of "mean sample variance"; and that,
in turn, can lead us to an estimate of the variance of the source population. What we
end up with, of course, is still only an estimate. It is not precise, it does not pretend
to be precise, and that fact will eventually have to be taken into account. But more of
this later. Our immediate concern is with how the estimate can be obtained, and what
can be done with it once we have it.
The general precept is this: If a random sample of size N is drawn from a normally
distributed source population, a useful estimate of the source population's variance (i
2
source ) can be obtained through multiplying the observed variance of the sample
(s2 ) by the ratio N/(N—1). Thus,
estimated:
2
2 x
source = s
N
N—1
For practical computational purposes, this estimate can be reached in a somewhat
more streamlined fashion. As we reminded ourselves just a moment ago, the variance
of an observed set of Xi values is simply the average of the sum of squared deviates:
s2 =
SS
N
Multiply that average by N/(N—1) and you end up with
SS
N
x
N
N—1
=
SS
N—1
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We will symbolize this modified version of a sample's variance as "{ s 2 }" in order to
distinguish it from the plain-vanilla version that we examined in Chapter 2; similarly,
we will use "{ s }" to denote the modified standard deviation that would result from
taking the square root of { s 2 }. Thus, on the basis of any particular sample randomly
drawn from a population that we can reasonably suppose to be normally distributed,
the variance and standard deviation of the population can be estimated as
{s2 }
SS
=
[= estimate of:
2
N—1
source ]
and
{s} = sqrt
[
SS
N—1
]
[= estimate of:
source ]
¶Estimating the Standard Deviation of the Sampling Distribution of Sample Means
We saw in Part 1 of this chapter that the variance of the sampling distribution of
sample means is equal to the variance of the source population divided by N.
:
2
M
When :
=
:
2
source
From Ch. 9, Pt. 1.
N
2
source
2
is unknown, as it usually is, the value of:
substituting { s }, which is the estimate of:
described above. Thus
estimated:
2
M
=
{s2 }
N
2
source
2
M
can be estimated by
obtained through the procedure
Recall that
{s2 } = SS/(N—1)
This in turn would allow you to estimate the standard deviation of the sampling
distribution of sample means ("standard error of the mean") as
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estimated:
Ch9pt2 Intro Sample Means
M
= sqrt
[
{s2 }
N
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]
To give you an idea where this is headed, suppose you have reason to believe that a
certain source population is normally distributed, but you have no precise knowledge
of its central tendency or variability. On the basis of certain theoretical considerations,
however, you do have reason to suspect that the mean of the population is
source =50.0.
To test this hypothesis, you draw a random sample of size N=25 from
the population. Your reasoning for the test is straightforward: If the mean of the
population is 50.0, then the mean of any particular sample randomly drawn from the
population should fall somewhere in the vicinity of 50.0. So you draw your sample and
find its mean to be MX =53.0, as compared with your hypothetical population mean of
source =50.0.
Well, yes, 53.0 is "somewhere in the vicinity" of 50.0—although, depending on your
scale of measurement, that could be a bit like saying that Glasgow is somewhere in
the vicinity of London. The observed sample mean of 53.0 clearly differs from 50.0.
The question is, does it differ significantly? That is: If the mean of the population truly
were 50.0, how likely would it be, by mere chance coincidence, that the mean of a
sample randomly drawn from the population would fall 3 or more points distant from
50.0?
If only you knew the variability of the source population, you could directly calculate:
2 , the standard deviation of the relevant sampling distribution of sample means.
M
And with that value in hand you could then plug your numbers into the appropriate
version of the sample-mean z-ratios described in Part 1 of this chapter:
z=
MX—
:
M
M
53 —50
=
:
From Ch. 9, Pt. 1.
M
As it happens, you do not know the variability of the source population, hence cannot
2
directly calculate either :
M or z. But as Moliere had his character Tartuffe say on
one occasion: Though Heaven forbids certain gratifications, there are nonetheless
"ways and means of accommodation." In the present case, the accommodation comes
about by way of systematic estimates. Suppose that you observe within your sample
a sum of squared deviates of SS =625. On this basis you could estimate the variance
of the population as
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{s2 }
Ch9pt2 Intro Sample Means
SS
=
=
N—1
625
Page: 8
= 26.04
24
And that in turn would allow you to estimate ('est.") the standard deviation of the
sampling distribution as
est.:
M
= sqrt
= sqrt
[
[
{s2 }
N
26.04
25
]
]
= ±1.02
The next step will be obvious. With your estimated value of :
M
in hand, you can
now go on to calculate what is essentially an estimate of the z -ratio examined a
moment ago. To make it clear that what we are now calculating is grounded on
several layers of estimation, the convention is to label it with the letter t rather than
z.
t =
MX —
est.:
=
M
M
53 —50
= +2.94
1.02
We will see a bit later that this distinction between t and z is not simply a matter of
nomenclature.
¶Estimating the Standard Deviation of the Sampling Distribution of Sample-Mean
Differences
There is a certain species of small furry animal known as the golliwump, of which half
are green and the other half are blue. An investigator has framed the hypothesis that
green golliwumps are on average smarter than blue golliwumps. To test this
hypothesis he draws one random sample of N a =20 greens and another of
N b =20 blues. He then administers a standard test of golliwump intelligence to each of
the individual subjects in the two samples, finding that the mean score of the greens
is MXa =105.6, while that of the blues is only MXb =101.3.
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Here again is one of those "if only" scenarios. If only our investigator knew the
variability of test scores within the entire population of golliwumps, he would be able
to calculate directly the standard deviation of the relevant sampling distribution,
2
i
i
M-M
= sqrt
[
source
i
2
+
Na
source
Nb
]
From Ch. 9, Pt. 1.
which would in turn permit the calculation of the two-sample z -ratio described in
Part 1:
z=
MXa —MXb
i
105.6 — 101.3
=
M-M
i
From Ch. 9, Pt. 1.
M-M
The logic of estimation in this case is analogous to what we examined above for the
one-sample situation. Suppose that the values for sum of squared deviates within the
two samples were SS a =4321 and SS b=4563. If you were to estimate the variance of
the source population on the basis of sample A separately, it would be
{s2
a} =
SS a
=
Na —1
4321
= 227.42
19
Estimated on the basis of sample B separately, it would be
2
{s b } =
SS b
Nb —1
=
4563
= 240.16
19
Blending these two separate variance estimates together in just the right way will
give you a composite estimate known as the pooled variance. (Note the subscript "p"
to indicate "pooled.")
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{s2
p}
Ch9pt2 Intro Sample Means
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SS a+SS b
=
(Na —1)+(Nb —1)
4321+4563
=
= 233.79
19+19
Returning now to the formula for the direct calculation of the standard deviation of
the sampling distribution,
2
i
i
M-M = sqrt
[
M-M
+
Na
= sqrt
= sqrt
2
source
Nb
we substitute {s2 p } fori
est.i
i
source
source
[
Na
Nb
[
233.79
233.79
+
{s2 p }
20
From Ch. 9, Pt. 1.
and end up with
{s2 p }
+
]
20
]
]
= ±4.84
The next step is then to calculate a t-ratio on analogy with the two-sample z-ratio
described above:
t =
MXa —MXb
est.i
=
M-M
105.6 — 101.3
= +0.89
4.84
In effect, we are estimating that the true value of the sampling distribution's standard
deviation is somewhere in the vicinity of ±4.84, and accordingly that the true value of
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z is somewhere in the vicinity of +0.89.
So here it stands, estimate piled on estimate, and the whole structure hedged in with
an escape clause that reads "somewhere in the vicinity." At first glance you might
think it hardly possible to squeeze from these rather spongy estimates anything even
remotely resembling a precise probability assessment. Indeed, were it not for the
work of the statistician W.S. Gosset (who wrote under the pseudonym of "Student"),
we might have to conclude at this point that the first-hand impression is accurate. In
the final portion of this chapter you will catch your first glimpse of the extraordinarily
useful inferential tool that Gosset's work created.
End of Chapter 9, Part 2.
Return to Top of Chapter 9, Part 2
Go to Chapter 9, Part 3
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