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MASTER OF PUBLIC ADMINISTRATION PROGRAM
PAD 5700 -- Statistics for public management Fall 2013
Sampling and confidence intervals
Statistic of the week
√ ̅
The standard deviation
*
We start with some discussion of data gathering. There is an old cliché in statistics (or any sort
of analytical process) that if you put 'junk' in, you will get 'junk' out.
Think of the recent collapse of the US economy, and as a result global economy: no one
(effectively, click here and here for exceptions) saw it coming. The Bush administration's 2009
budget, for instance, was released early in 2008. In the Budget Message, the President
confidently stated that "As we enter this New Year, our economy retains a solid foundation
despite some challenges, revenues have reached record levels, and we have reduced the Federal
deficit by $250 billion since 2004." In Table S-10 of the Summary Tables, the administration
foresaw economic growth ('Real GDP' in the table) of 2.7% in 2008, followed by 3.0% growth in
2009. It wasn't just the Bush administration that was delusional. As this table also indicated, the
CBO (Congressional Budget Office, widely respected as a competent, and unbiased referee in
budget debates) expected growth of 1.7% and 2.8% for these two years, while the 'Blue Chip
consensus' (the mean prediction of a number of highly respected private economic forecasters)
was about halfway between the two. The actual result for 2008 was 1.0% growth, while 2009
saw about a 3.5% contraction in the economy.
Why? Because the data that they plugged into their economic models was faulty. While
numbers can be a very good way to describe reality (as we saw last week), statistics can be very
bad at prediction, and even worse at predicting unprecedented events. The logic of statistics is
that you plug past data into a dataset, develop a model to analyse it, then punch 'run'. It spits out
a result. However if the future diverges sharply from the past -- if something fundamental
changes -- by definition your past data will not have incorporated this. Indeed, even your model
can be wrong.
These same problems can occur even if the future doesn't reflect a break from the past, but the
analyst fails to sample the data correctly. Imagine conducting a survey about voting preferences
of Americans. If you hung out at Ponte Vedra and asked passersby you'd probably get a different
result than if you hung out on the east side of Jacksonville. For two similar examples: how would
you react if I told you that:
1. I played top 20 college hoops?
2. I placed second in a national championship 10k foot race?
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Ethics
On a similar note, it is widely asserted that there are lies, damned lies, and statistics. Not so.
Numbers don’t lie, but people lie about numbers. People are also often clueless about numbers.
Being clueless and wanting to lie are mutually reinforcing, too, as it is easier for those who want
to use numbers to lie, to do so if they don’t know what they’re doing. So note Berman & Wang’s
short section on ethics (pp. 10-15). For me, if the numbers don’t support what I think is reality,
I’ll generally check the numbers. But if the numbers look legit, I flip-flop, or change my views.
Sampling
Statistics is driven largely by the concept of sampling. At its simplest, sampling refers to a
process of measuring what is going on in a part of a larger population, to determine what is
going on in the entire population. One might fairly ask:
Q. Why not just measure the entire population?
A. Because this can be difficult.
Note the 2000 US federal election in Florida. Who won? Well, Governor Bush did, he got the
electoral votes, and became President. But what I mean is: who got the most votes? We don't
know who got the most votes, we will probably never know, and frankly the point is that given
human (and technical) imperfection, in any election that is that close in an electorate that is that
large, we probably can’t measure the vote exactly.
It can even be difficult to measure smaller populations. Take an issue that is near and dear to
most people: seals in Labrador. Assume you want to monitor their weight. Once you set up the
scales, how do you get all of the little blighters to turn up to be weighed? Instead, you have to go
out and catch them. You may not get them all, if you don't, maybe you missed the leaner,
quicker ones, and so your sample is biased by an over-representation of the seal equivalent of
couch potatoes (or perhaps: cod potatoes?). Again: measuring a population can be difficult.
The discussion so far also serves to introduce some technical jargon:
Observation -- the individual unit of a sample, population of 'random variable' (e.g. the
individual MPA program)
Sample -- a random (or otherwise representative) portion of a population or random variable
(e.g. a random selection of every tenth MPA program)
Population -- the entire set of individual observations of interest (MPA programs in the US)
Random variable -- the underlying social/historical/biological/etc process generating the
individual observations in a population (the range of possible MPA program outcomes).
From Berman & Wang:
Hypothesis – what we think might be going on, and so want to test for.
o Theory – one of the least understood words in the (American) English language. See an
online dictionary definition. In the social sciences, #1, 3, 5 are what we refer to as
theory. #2, 4 and 6 are what we refer to as hypotheses.
o Dependent variable – what we are trying to explain
o Independent variable – what we think explains variation in the dependent variable, and so
otherwise referred to as ‘explanatory’ variables.
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o Key independent variable – often there is a single variable whose effect on the dependent
variable particularly interests us.
o Control variables – other variables that are added to the model to hold these constant, so
that the independent effect of the key independent variable can be ascertained.
o Example – we saw this in our discussion of week one of determinants of good socio-
economic outcomes. In Figure 2 and Table 3, economic freedom has a strong, positive
effect on socio-economic outcomes (specifically: ‘human development’). Yet when
public services (an indicator of good government) are introduced to the model, the impact
of economic freedom dissipates. We will try to tease out this relationship as the course
goes on, but for now let’s hypothesize that the relationship is a complex one: economic
freedom and good public services are necessary for improved health, education and
income; and these, in turn, make the provision of public services (and realization of
economic freedom?) possible.
o Correlation or association – this is implied by causation, below. In correlation, A and B co-
vary: as one changes, the other does, as well.
o Causal relationships – a change in A results in a change in B. Berman and Wang’s take on
this: “Causation requires both empirical (that is, statistical) correlation and (2) a plausible
cause-and-effect argument” (p. 25).
o Target population and sampling frame
o Unit of analysis
o Sampling error -- random error, an inevitable part of sampling
o Sampling bias -- systematic error, resulting from a conceptual mistake in sampling
Six steps to research design (Berman & Wang, p. 28).
1. Define the activity and goals that are to be evaluated.
2. Identify which key relationships will be studied.
3. Determine the research design that will be used.
4. Define and measure study concepts.
5. Collect and analyze the data.
6. Present study findings.
Sampling designs:
o Simple randomness -- the functional equivalent of a lottery.
o Stratified random sampling -- cheat a little, by holding mini lotteries
within known important sub-sections of the population, to try to overcome the inevitable
sampling bias resulting from typically unrepresentative rates of non-responses.
Re-weighting – On sampling bias, above, when random sampling yields a sample that does
not share known characteristics of the broader population, under-represented groups are re-
weighted, with their results multiplied by an appropriate figure: if left-handed leftwing
lesbian Luddite Lebanese Lutherans make up 5% of your population, but only 2.5% of the
people who responded to your survey; simply count these responses twice.
o Disproportionate stratified sampling -- drawing more people from especially small, but
especially important groups, to ensure that a large enough sub-sample is drawn from this
group to allow for statistically significant results.
o 'Cluster' sampling -- geographical units essentially become the unit of analysis in the
sampling process, with these randomly selected.
o Convenience sampling -- talk to whoever walks by 1st and Main while the longsuffering
intern is standing there conducting interviews.
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o Purposive sampling -- want to know what teens think? Don't hang out at the vets club.
o Quota sampling -- not unlike stratified random sampling.
o Snowball sampling -- follow referrals.
Sampling distributions
We now come to a key transition in the conceptual understanding of this stuff. The idea here is
to estimate the population mean (N). If we could directly measure N, we wouldn't bother
sampling. After all, no one cares what a random sample of 600 people thinks about anything, no
one even knows who those 600 or so people are. The only reason we ask 600 randomly selected
people what they think about stuff is because we really want to know what society thinks about
these issues, but it is too difficult to discern the opinion of all of society (see the earlier
discussion of the 2000 election, for instance). These 600 or so randomly selected people are only
interesting because they may provide an estimate of what the broader society -- N -- thinks.
So again, we estimate these values. We want to know N, as it is an important phenomenon. We
can't measure it directly so we sample. Yet this is inaccurate, as the sampling process, even if we
take great care to do representative sampling, will vary depending on pure chance. This is the
issue here: sample statistics vary, just as observations within a sample vary. This distribution of
sample statistics is known as a sampling distribution.
Imagine trying to measure that very important phenomenon: the birth weight of baby seals in
Labrador. You draw a sample of twenty baby seals, calculate a mean and standard deviation,
and so have a nice little estimate of the population mean and variation of baby seal weights.
However, the sample statistics themselves will vary, so that if you came back the next day, or
even that same day, or even if a second researcher drew a second sample of twenty seals from
the same population at the same time you conducted your sample, you would get different
results. They (hopefully) will not vary too much, but they will be different. This isn't because
you've done anything wrong, it is just a result of randomness.
We have two goals in the next couple of lectures:
Use sampling distributions to evaluate the reliability of sample statistics: or to understand
confidence intervals.
Use sampling distributions to make inferences about populations -- hypothesis tests.
Bad news: more notation, as indicated in the equation below for the standard deviation of the
sampling distribution (what will also be referred to as the standard error):
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Good news: easier calculation, as indicated in
the equation above.
The mean of the sampling distribution
equals the sample mean.
The standard error (referred to as the
standard deviation of the sampling
distribution in some statistics texts, with
the notation: sigma, subscript x; with a
line over it, or 'sigma x-bar') equals the
sample standard deviation divided by the
square root of n (the sample size).
With the sample standard deviation we can
draw a numerical picture of the likely
distribution of observations within a
population. With the standard error -- the
standard deviation of the sampling
distribution, or the distribution of means we
would get through repeated sampling -- we
can do the same thing for the likely
distribution of sample means drawn from a
population.
Before we were looking at where an
individual observation -- x -- lies in the
distribution of observations in our sample,
and drawing inferences from this.
Now we look at where the population
mean likely lies in the distribution of
possible sample means, and draw
inferences from this.
Expressed graphically, it might look like the
drawing at right:
Confidence intervals
Large sample confidence interval for
population mean
Getting back to sampling to estimate population parameters: we take a sample of 2000, with
a sample mean of 100.57 units (pounds, say). What is the population mean? It is not necessarily
100.57 pounds, indeed probably is not. So the sample mean is not necessarily the population
mean. It is an estimate of the population mean. How accurate is it? This, intuitively, would
depend on the size of the sample, and the variability in the sample.
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The effect of sample size on accuracy of a population estimate should be relatively easy to get.
If you catch a fish which weighs five pounds, you have little information with which to conclude
that the mean population weight of this type of fish is five pounds. You might just have gotten
unlucky, picked a dwarf or something, or an Alex Rodriguez fish. But if you catch 10 of these
fish, and have a mean weight of five pounds, you can start to assume that the population mean
weight of these is about five, when caught from the place you caught them, at the time you
caught them, using the gear and bait you used, etc. (Note this latter proviso: in addition to
exercising caution regarding the accuracy of one's estimate of the population mean when one has
little data, one also has to be cautious concerning inferences drawn about populations not
relevant to the sampling operation -- or regarding the 'external validity' of the study.)
Getting back to our population estimate: one can especially be a bit more confident about an
estimate of a population mean given a sample of ten fish if the sample is fairly narrowly grouped
around five pounds. At a sample of 500 fish, you become even more confident that the
population mean weight is five pounds.
Still, ten different researchers drawing samples of 500 in an identical manner will get ten
different sample means, or ten different estimates of the population mean. 5 pounds, 5.1, 4.9, 4.9
again, 4.8, 5.2, 5.04, even the odd 4.4 or so. What, then, is the population mean? We don't
know, but can give an estimate within a 'confidence interval', which is the same thing as the
'margin of error' used in polling. We express the population mean in terms of an interval in
which we can have a certain confidence that the true population mean will lie. So we can't say
that the mean is 5.1 pounds, but we can (given details) say something like "we can be 95%
confident that the population mean lies within 0.3 pounds of 5.1; or between 4.8 and 5.4."
An example
Up to now, we have been
doing the following. The
town of Osceola does a study
to see how much residents
spent on energy last winter.
A random sample of 100
residents yields a mean of
$160 for the month of
January, with a standard
deviation of $50. With the
focus on the variation in
individual energy usage
patterns, 95% of individuals
would use between $62 and
$258.
The diagram at right
illustrates the process. A 95% interval includes a probability of .475 (.95 / 2) either side of the
mean. We first need to know the 'z score' for this interval. The z score is just a shorthand way of
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saying how many standard deviations a point is from the mean: a z score of two means
something is two standard deviations from the mean.
You will not have to do this, but the z score can be found in a z table (Appendix A, p. 324 in
Berman & Wang). To find the z score associated with a probability of 0.025 in the 'tails', or
either end of the distribution, we go to a probability of 0.475 (0.5 - 0.025) in the z table. The z
score for this is 1.96, so the interval within which we would expect 95% of the observations to
lie -- or the points at which we will identify the upper and lower tails of p(0.025) -- is 1.96
standard deviations either side of the mean.
This 95% confidence level is a commonly used one, the
margins of error in opinion polling are always expressed
in terms of a 95% confidence level, or 5% margin of
error. Other standard confidence levels (with associated z
score) can be found in Table 1.
Now what we want to do is the following. With that sample of 100 yielding a sample mean of
$160, what can we say about the population mean? The diagram on the right illustrates the
process. To find the z score associated with a 95% confidence interval of the population mean,
we again go to a probability of 0.475 p(0.95/2) in the z table. The z score for this is 1.96, so the
interval within which we can be 95%
confident that the true population mean
lies is 1.96 standard deviations either
side of the mean. Given our standard
deviation of 50, the standard error (the
standard deviation of the sampling
distribution), the distribution of sample
means generated from this population,
from the equation above, will be
50/(square root of 100), or 50/10, or
5. As 5 x 1.96 = 9.8, this interval would
be $9.80 above and below our mean of
$160, or between $150.20 and
$169.80. What this means is that while
we don't know exactly what the
population mean is, we are 95%
confident that it is greater than $150.20
and less than $169.80.
By reconfiguring these equations, you
can work out how large your sample has
to be to get a confidence interval of a given size. In the example above, assume the city is
dissatisfied with this +/- $9.80, 95% confidence interval for the estimate of the population mean
of energy spending. The city would like a more precise estimate, say a 95% confidence interval
of only $5. The sample size needed to obtain an estimate this tight (from the 'sample size
determination' equations below), would be 384 (or 384.16).
Table 1
Confidence levels and z score
Confidence level Z score
99% 2.58
95% 1.96
90% 1.645
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Clear as mud, huh!
SPSS -- tragically, SPSS generally won't calculate this for us. It will give us a mean, and a
standard error, with which we can calculate a confidence interval.
We've seen much of the above in our SPSS exercises so far. Open the income-NSBend file and
we can illustrate a lot of what we've been working on, though:
First, restrict the file only to South Bend. Do this by going to Data, Select Cases, ‘If
condition is satisfied’, click ‘If’, highlight ‘City’ and click the arrow to move it to the right,
then complete the equation: City = 1 (we coded North Bend = 0, South Bend = 1), Continue,
Okay.
Run descriptive statistics on the ten South Bend cases for income in 2000; including a mean,
standard deviation and standard error. These figures are 35.0, 5.9 and 1.9, respectively. The
SPSS output:
Table 2
Descriptive Statistics
N Mean Std. Deviation
Statistic Statistic Std. Error Statistic
2000 Income ($1000s) 10 35.0000 1.86190 5.88784
Valid N (listwise) 10
You can also derive these manually, just so that you can see what SPSS is doing:
o The mean of 35
(25 + 29 + 31 + 33 + 35 + 35 + 37 + 39 + 41 + 45)
10
= 35.0
o The standard deviation of 5.9
[(35-25)
2 + (35-29)
2 + (35-31)
2 + (35-33)
2 + (35-35)
2 + (35-35)
2 +
(35-37)2 + (35-39)
2 + (35-41)
2 + (35-45)
2] / (n-1)
so
[(10)2 + (6)
2 + (4)
2 + (2)
2 + (0)
2 + (0)
2 + (-2)
2 + (-4)
2 + (-6)
2 + (-10)
2] / (10-1)
so
100 + 36 + 16 + 4 + 0 + 0 + 4 + 16 + 36 + 100 / 9
so
312 / 9 = 34.67
square root of 34.67 = 5.9
o The standard error (or standard deviation of the sampling distribution)
5.9 / (square root of 10)
so
5.9 / 3.16 = 1.87
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Given our data, a confidence interval estimate of the mean 2000 income for South Bend would
be:
95% C.I. =
95% C.I. = 30 +/- 1.96 x 1.7
95% C.I. = 30 +/- 3.65
or: you can be 95% confident that the population mean lies between 26.35 and 33.65.
Large sample hypothesis tests The general idea in hypothesis testing is to test the reliability of the status quo. In English (or
perhaps Statslish, as the jargon is inevitable), we draw a sample and obtain a sample mean from
this. It differs from what we expected, by which we mean what we thought the value should
have been, based on past experience, for the phenomenon in question. What can we infer from
this, though? Does it differ enough that it suggests that our status quo assumption is now
defunct? Or could our sample mean just have resulted from the randomness associated with
sampling?
McClave and Sincich present the elements of an hypothesis test, which I'll reproduce, modified
below. This is what Berman & Wang present in pages 174-7, though not quite as completely as
McClave & Sincich. Also, this goes through the process if you were doing it longhand. We will
mostly use SPSS, though, so I present the longhand method just in the hopes that the SPSS
output will make more sense. I also drop the ‘critical value’ part, and instead suggest reporting
the statistical significance of the test, for reasons that will be explained.
Elements of a Test of Hypothesis
1. Null hypothesis (H0): A theory about the values of one or more population parameters. The
theory generally represents the status quo, which we accept until it is proven false.
2. Alternative hypothesis (Ha): A theory that contradicts the null hypothesis. The theory
generally represents that which we will accept only when sufficient evidence exists to
establish its truth.
3. Assumptions: Clear statement(s) of any assumptions made about the population(s) being
sampled.
4. Experiment and calculation of test statistic and probability value: Performance of the
sampling experiment and determination of the numerical value of the test statistic.
5. Conclusion: If the numerical value of the test statistic is enough that you are comfortable
rejecting the null hypothesis, do so.
Source: McClave and Sincich, p. 282.
Remember that when rejecting the null hypothesis (the status quo assumption), you should be
careful about what you say about the population mean. When rejecting the null hypothesis, our
old, status quo assumption has been shown to be incorrect. However, we cannot now assume
that our sample mean is the population mean. We can only go back to our confidence intervals,
and provide an estimate of what our sample mean indicates the population mean is. The logic
here is that the null hypothesis, status quo population mean was tried and true, having been
determined some time ago and continually validated through subsequent observation. On
rejecting this, we need to do much more research before we can offer a new population mean, in
any event a single sample does not provide enough information on this.
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Some examples:
Load the Osceola dataset. This is an imaginary dataset with January energy use costs for a
sample of households in three towns.
Restrict the sample only to the town of Osceola (Data, Select cases, If town = 1)
Assume that the mean January energy costs for a household in northern Indiana is
$150. Conduct a test of the null hypothesis that energy costs in Osceola are $150, in other
words, despite Osceola’s sample mean of $160, is it possible that energy costs in Osceola are
the same as costs elsewhere, and the $160 is just a result of randomness in sample selection?
o Analyze, Compare means, One-sample T test, Test value = 150, Energy costs in test
variable, OK
o You get the following:
Table 3a
One-Sample Statistics
N Mean Std. Deviation
Std. Error
Mean
Energy costs, January 100 160.00 18.052 1.805
Table 3b
One-Sample Test
Test Value = 150
t df Sig. (2-tailed)
Mean
Difference
95% Confidence Interval of
the Difference
Lower Upper
Energy costs, January 5.540 99 .000 10.000 6.42 13.58
The One-Sample Statistics are just descriptive statistics.
The One-Sample Test data shows you that, given the standard error of the sample
(1.805), the likelihood that a sample mean of $160 could result from a population with a
mean of $150, is over 5 standard errors from the mean (t = 5.54).
You can do this math, using a ‘z score’ like we discussed in week one: The test
statistic of 160 is ten from the null hypothesis of 150. Is ten a lot, in the sense of
enough that we can be confident Osceola is different? We don’t know, but can get a
sense of this by using the standard error of 1.805. Ten divided by 1.805 is 5.504 (the
t statistic in Table 3b!). Keep in mind that in a normal distribution around that old
population mean of 150, almost all (99.7%) of the population would be expected to
fall within three standard deviations of the mean. The Osceola figure is over five
standard deviations away, which is like a one in 10,000 probability. So either
The people of Osceola have energy use patterns that are not the same as the rest of
the country (you can be over 99.99% confident of this), or
the people of Osceola are no different, this was just an unlikely, random event
(there is about a 0.01% likelihood of this).
So, you can be over 99.9% confident in rejecting the null hypothesis that Osceola has the
same January energy costs as residents of other towns in northern Indiana.
The dataset also gives you mean energy costs for Wakarusa and Nappanee
o Run descriptive statistics, here you might use the Case summaries function, to separate
the three towns:
Remove the 'Select Osceola' function.
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Analyze, Reports, Case Summaries.
Variables = Energy costs; Group Variable = Town name.
Uncheck the Display Cases
Under Statistics ask for a whatever you want. Continue, Okay.
You should get this:
Table 4a
Case Summaries Energy costs, January
Town name Mean Median Minimum Maximum
Std. Error of
Mean Std. Deviation
Osceola 160.00 161.00 121 201 1.805 18.052
Wakarusa 150.00 153.00 120 178 2.600 16.444
Nappanee 158.90 158.50 120 196 3.031 19.171
Total 157.53 159.00 120 201 1.366 18.323
Now do an hypothesis test to see if energy costs differ between Osceola and Wakarusa.
o Analyze, Compare means, Independent-Samples T test, Test variable = Energy costs,
Town = Grouping variable, Define groups, Group 1 = 1 (Osceola), Group 2 = 2
(Wakarusa)
o You should get the following (I’ve reformatted to make it fit, especially getting rid of the
95% Confidence Interval of the Difference):
Table 5
Independent Samples Test
Levene's Test for
Equality of Variances t-test for Equality of Means
F Sig. t df
Sig. (2-
tailed)
Mean
Difference
Std. Error
Difference
Energy
costs,
January
Equal variances assumed .035 .852 3.035 138 .003 10.000 3.295
3.165 Equal variances not assumed
3.159 78.479 .002 10.000
The first table shows the two means: $160 for Osceola, $150 for Wakarusa. I’ll omit it
(we’ve got it above)
The second shows the likelihood that you would get different means for two samples of
these sizes (n = 100 for Osceola; n = 40 for Wakarusa). This is reflected in the Sig. (2-
tailed) figures of .002 or .003.
Which of these you choose to use isn't terribly important (it depends on whether you
think the variances between Osceola and Wakarusa are the same, which the Levene's Test
for Equality of Variances suggests is not the case), but what this tells you is that the
likelihood that you would randomly get means of $160 and $150 for these two samples, if
they came from the same population (and so should have the same mean), is .002, or very
unlikely. Given this, you can reject the null hypothesis that they come from the same
population. Instead, they differ.
Do an hypothesis test to see if energy costs differ between Osceola and Nappanee.
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o Analyze, Compare means, Independent-Samples T test, Test variable = Energy costs,
Town = Grouping variable, Define groups, Group 1 = 1 (Osceola), Group 2 = 3
(Nappanee)
o You should get the following:
Table 6
Independent Samples Test
Levene's Test for
Equality of Variances t-test for Equality of Means
F Sig. t df
Sig. (2-
tailed)
Mean
Difference
Std. Error
Difference
Energy
costs,
January
Equal variances assumed .868 .353 .320 138 .749 1.100 3.438
3.528 Equal variances not assumed
.312 68.191 .756 1.100
The first table shows the two means: $160 for Osceola, $158.90 for Nappanee.
The second shows the likelihood that you would get different means for two samples
of these sizes (n = 100 for Osceola; n = 40 for Nappanee). This is reflected in the Sig.
(2-tailed) figures of .749 or .756.
Again, which of these you choose to use isn't terribly important, but what this tells
you is that the likelihood that you would randomly get means of $160 and $158.90 for
these two samples, if they came from the same population (and so should have
the same mean), is about .750, or very likely. Given this, you can not reject the null
hypothesis that this difference between $160 and $158.90 reflects an underlying
difference between the two communities. Instead, this difference is likely the result
of random error in sample selection.
Some exercises
1. Do an hypothesis test to see if energy costs differ between Osceola, and Wakarusa and
Nappanee combined. The trick here is to use a Cutpoint of 1.5, rather than specify specific
values in defining your test groups. 1.5 will compare Osceola (< 1.5) and the other two (> 1.5).
Load the Global Government dataset. Restrict the sample to countries, so go to Data, Select
Cases, If Type = 1.
2. We will use six variables in these examples, five continuous and one nominal. Run
descriptive statistics of the following variables:
demagg (Democracy combined)
demadmin (Functioning of government)
catoef (Economic freedom)
GDPpci (GDP per capita)
govtsize (Size of Government)
3. Run Frequencies of the sixth (nominal) variable: language (Language spoken in the country)
4. Do a test of the null hypothesis that the average per capita GDP (variable: GDPpci) of the
countries in the sample is $10,000. Use a One-sample T Test.
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5. Many folks fear that the US has slowly become more and more socialistic, like the rest of the
world, over the past couple of decades. The US score on the CATO Institute's Economic
Freedom (catoef) indicator is 8.1. Test to see if this is in line with the global norm. Again, use a
One-sample T Test.
6. Do a test of the null hypothesis that English and Spanish speaking countries do not differ in
terms of income. This would require an Independent-Samples T Test.
7. Do a similar test to see if rich countries have more effective government administrations
(demadmin) than poor ones. Do an Independent-Samples T Test, using $10,000 as a cut point.
8. Do a correlation to see if wealth (GDPpci) and administration (demadmin) are related.
9. Finally, it is widely argued that more government will lead to less democracy. Do a
correlation between Size of Government (govtsize) and Democracy (demagg).
*
References: Gold, Michael and G.G. Candler (2006). "The MPA Program in small markets: an exploratory
analysis." Journal of Public Affairs Education 12(1), pp. 49-62, 2006. Available online.
McClave, James and Terry Sincich (2003). A First Course in Statistics. Upper Saddle River,
NJ: Prentice Hall.