Review of material from previous
week: Variance
•
The variance (s2) is the sum of the squared
deviations from the sample mean, divided by N-1
where N is the number of cases. (Note: in the
formula for the population variance, the denominator
is N rather than N-1)
Formula for sample variance
standard deviation--(s or SD)
•To find the variance and its square root, the standard deviation,
use the command Analyze/ Descriptive Statistics/ Frequencies and
move the variable of interest into the Variables box. Click on
Statistics, check standard deviation and variance and then click OK.
Review from previous week: Standard
deviation and normal distribution
•
•
The standard deviation is the square root of the
variance. It measures how much a typical case
departs from the mean of the distribution. The size of
the SD is generally about one-sixth the size of the
value of the range
The standard deviation becomes meaningful in the
context of the normal distribution. In a normal
distribution
•
•
•
The mean, median and mode of responses for a variable
coincide
The distribution is symmetrical in that it divides into two
equal halves at the mean, so that 50% of the scores fall
below the mean and 50% above it (sum of probabilities
of all categories of outcomes = 1.0 (total “area” = 1)
68.26 % of the scores in a normal distribution fall within
plus or minus one standard deviation of the mean.
95.44% fall within 2 SDs. The curve has a standard
deviation of 1
•
•
•
Thus we are able to use the SD to assess the relative
standing of a score within a distribution, to say that it is 2
SDs above or below the average, for example
The normal distribution has a skewness equal to zero
Some normal curves in nature: heights within gender; LDL
cholesterol
Review from last week: Histogram with
superimposed normal curve
•
•
Histogram of the “vehicle weight”
variable with a superimposed
curve. This is what a normal
distribution of a variable with the
same mean and standard
deviation would look like. This
distribution has a positive skew
and is more platykurtic
Definition: “Kurtosis is a measure
50
40
30
20
10
Std. Dev = 849.83
Mean = 2969.6
0
N = 406.00
.0
00
52
.0
00
48
.0
00
44
.0
00
40
.0
00
36
.0
00
32
.0
00
28
.0
00
24
.0
00
20
.0
00
16
.0
00
12
0
0.
80
of whether the data are peaked or
flat relative to a normal distribution.
That is, data sets with high kurtosis
tend to have a distinct peak near
the mean, decline rather rapidly,
and have heavy tails. Data sets with
low kurtosis tend to have a flat top
near the mean rather than a sharp
peak. A uniform distribution would
be the extreme case.”
60
Vehicle Weight (lbs.)
Descriptive vs. Inferential Statistics
•
•
Descriptive statistics: such measures as the mean,
standard deviation, correlation coefficient when used
to summarize sample data and reduce it to
manageable proportions
Inferential (sampling) statistics: use of sample
characteristics to infer parameters or properties of a
population on the basis of known sample results.
Based on probability theory. Statistical inference is the
process of estimating parameters from statistics.
• Inferential statistics require that certain assumptions
be met about the nature of the population.
Assumptions of Inferential Statistics
•
Let’s consider an example of an application of parametric statistics, the ttest. Suppose we have drawn a sample of Chinese Americans and a sample
of Korean Americans and have a belief that the two populations are likely to
differ in their attitudes toward aging. The purpose of the t-test is to
determine if the means of the populations from which these two samples
were drawn are significantly different. There are certain assumptions that
have to be met before you can perform this test:
•
You must have at least interval level data
•
The data from both samples have to represent scores on the same
variable, that is, you can’t measure attitudes toward aging in different
ways in different populations
•
The populations from which the samples are drawn are normally
distributed with respect to the variable
•
The variances of the two populations are equal
•
The samples have been randomly drawn from comprehensive sampling
frames; that is, each element or unit in the population has an equal
chance of being selected (random sampling permits us to invoke
statistical theories about the relationships of sample and population
characteristics)
Population Parameters vs. Sample
Statistics
•
Purpose of statistical generalizations is to infer the parameters of
a population from statistics known about a sample drawn from
the population
Greek letters usually refer to population characteristics:
population standard deviation = σ,
population mean = µ
Roman letters usually refer to sample characteristics:
sample standard deviation = s, sample mean =
The formula for the variance in a population is:
The formula for the variance in a sample is:
Frequency Distributions vs.
Probability Distributions
•
•
The general way in
which statistical
hypothesis testing is
done is to compare
obtained frequencies
to theoretical
probabilities.
Compare the
probability
distribution for
number of heads in
two, four and 12 coin
flips vs. an actual
example of coin
flipping. (from D.
Lane, “History of
Normal Distribution”
Tails, tails
Formula for binomial probabilities
Tails, heads or
Heads, tails
Heads,
heads
Comparing Obtained to Theoretical
Outcomes
•
•
If you did a sample experiment and you got, say, two
heads in two flips 90% of the time you would say that
there was a very strong difference between the obtained
frequencies and the expected frequencies, or between
the obtained frequency distribution and the probability
distribution
Over time, if we were to carry out lots and lots of coinflipping experiments, the occasions when we got 90%
occurrence of two heads in two flips would start to be
balanced out by results in the opposite direction, and
eventually with enough cases our obtained frequency
distribution would start to look like the theoretical
probability distribution. For an infinite number of
experiments, the frequency and probability distributions
would be identical.
Significance of Sample Size
• Central Limit Theorem: the larger the
sample size, the greater the probability that
the obtained sample mean will approximate
the population mean
•
Another way to put it is that the larger the
sample size, the greater the likelihood that the
sample distribution will approximate the shape
of a normal distribution for a variable with that
mean and standard deviation
Reasoning about Populations from
Sample Statistics
•
•
•
•
•
Parameters are fixed values and are generally
unknown
Statistics vary from one sample to another, are known
or can be computed
In testing hypotheses, we make assumptions about
parameters and then ask how likely our sample
statistics would be if the assumptions we made were
true
It’s useful to think of an hypothesis as a prediction
about an event that will occur in the future, that we
state in such a way that we can reject that prediction
We might reason that if what we assume about the
population and our sampling procedures are correct,
then our sample results will usually fall within some
specified range of outcomes.
Reasoning about Populations from
Sample Statistics
•
If our sample results fall outside this range into a
critical region, we must reject our assumptions.
• For example, if we assume that two populations, say
males and females, have the same views on
increasing the state sales tax but we obtain results
from our randomly drawn samples indicating that
their mean scores on the attiitude toward sales tax
measure are so different that this difference falls into
the far reaches of a distribution of such sample
differences, we would have to reject our assumption
that the populations do not differ. But our decision
would have a lot to do with how we defined the “far
reaches” of this distribution, called the “critical
region.”
Reasoning about Populations from
Sample Statistics, cont’d
• We can say that we have carried out
statistical hypothesis testing if
•
•
•
•
We have allowed for all potential outcomes of our
experiment or survey results ahead of the test
We have committed beforehand to a set of
procedures or requirements that we will use to
determine if the hypothesis should be rejected and
We agree in advance on which outcomes would mean
that the hypothesis should be rejected
Probability theory lets us assess the risk of error and
take these risks into account in making a
determination about whether the hypothesis should
be rejected
Types of Error
•
Error risks are of two types:
• Type I error, also called alpha (α) error, is the risk of
•
•
rejecting the null hypothesis (H0:hypothesis of no
difference between two populations, or no difference
between the sample mean and the population mean)
when it is in fact true. (we set our confidence level too
low)
Type II error, or beta (β) error, is the risk of failing to
reject a null hypothesis when it is in fact false. (set our
confidence level too high)
When we report the results of our test, it is often
expressed in terms of the degree of confidence we have
in our result (for example, we are confident that there
is less than a 5% or 2.5% or 1% probability that the
result we got was obtained by chance and that in fact
we should fail to reject the null hypothesis. This is
usually referred to as the confidence level or the
significance level.
Why We are Willing to Generalize
from Sample Data
•
•
•
Why should we generalize on the basis of limited
information?
•
•
Time and cost factors
Inability to define a population and list all of its elements
•
To survey the full-time faculty at UPC, for example, you might
obtain a list of all the faculty, number them from one to N, and
then use a random number table to draw the numbered cases
for your sample.
Random sampling: every member of the population has an
equal chance of being selected for the sample
Theoretically, to do this requires that you have a list of all
the members of the population
•
•
Random sampling can be done with and without replacement
SPSS will draw a random sample for you from your list of
cases (Data/Select Cases/Random Sample of Cases) of a
desired size
Normal Distribution, a Review
•
•
The normal curve is an
essential component of
decision-making by which you
can generalize your sample
results to population
parameters
Notion of the “area under the
normal curve” the area
between the curve and the
baseline which contains 100%
of the cases
Characteristics of the normal curve,
cont’d
•
•
Constant proportion of the area under
the curve will lie between the mean
and a given point on the baseline
expressed in standard score units (Zs),
and this holds in both directions (both
above and below the mean). That is,
for any given distance in standard
(sigma) scores the area under the
curve (proportion of cases) will be the
same both above and below the mean
The most commonly occurring scores
cluster around the mean, where the
curve is the highest, while the
extremely high or extremely low scores
occur in the tails and become
increasingly rare (the height of the
curve is lower and in the limit
approaches the baseline. The total area
(sum of individual probabilities) sums
to 1.0
Table of the Area under the Normal
Curve
•
•
•
Tables of the Area under the
Normal Curve are available in
your supplemental readings,
p. 469 in Kendrick and p. 299
in Levin and Fox, and can be
found on the Web.
You can use this table to find
out the area under the
normal curve (the proportion
of cases) which theoretically
are likely to fall between the
population mean and some
score expressed in standard
unit or Z scoress.
For example, let’s find what
proportion of cases in a
normal distribution would lie
between the population mean
and a standard score of 2.2
(that is, a score on the
variable that is 2.2 standard
deviations above the meanalso called a Z score)
Z Scores and the Normal Table
•
In the normal table you look up the Z score of 2.2 and
to the right of that you will find the proportional “area
between the mean and Z”, which is .4861. Thus
48.61% of the cases in the normal distribution lie
between the mean and Z=2.2.
• What proportion of cases lie below this? Add 50% to
48.61% (because 50% of the cases lie below the
mean).
• What proportion of cases lie above this? 100%
(100% of cases) minus 50% + 48.61%, or 1.39% of
cases
• What proportion of cases lie between -2.2 and +2.2?
• Some Tables will express the values in percentages,
some in proportions.
Using the Mean and Standard Deviation to find
Where a Particular Value Might Fall
60
•
•
2119.733
What percent of vehicles have weights
between those two values, assuming a
random, representative sample and no
measurement error? 68.26
What would be the weight of a vehicle
that was two standard deviations above
the mean? Two standard deviations
below the mean? What percent of
vehicles have weights between those
two values, assuming a random,
representative sample and no
measurement error? 95.44%
50
40
30
20
10
Std. Dev = 849.83
Mean = 2969.6
0
N = 406.00
.0
00
52
.0
00
48
.0
00
44
.0
00
40
.0
00
36
.0
00
32
.0
00
28
.0
00
24
.0
00
20
.0
00
16
.0
00
12
0
0.
80
•
Let’s consider the “vehicle weight”
variable from the cars. sav file. From a
previous analysis we learned that the
distribution looked like this histogram
on the right and it had the sample
statistics reported in the table,
including a mean of 2969.56 and a
standard deviation of 849.827. What
would be the weight of a vehicle that
was one standard deviation above the
mean? Add one sd to the mean, and
you get 3819.387 One standard
deviation below the mean?subtract one
sd from the mean, and you get
Statistics
Vehicle Weight (lbs.)
Vehicle Weig ht (lbs.)
N
Mean
Std. Deviation
Variance
Skewness
Std. Error of Skewness
Kurtosis
Std. Error of Kurtosis
Range
Percentiles
Valid
Missing
25
50
75
406
0
2969.56
849.827
722206.2
.468
.121
-.752
.242
4408
2222.25
2811.00
3614.75
Z Scores
•
•
•
The Z score expresses the relationship between the mean
score on the variable and the score in question in terms of
standardized units (units of the standard deviation)
Thus from the calculations we just did we can say that the
value of vehicle weight of 3819.387 has a Z score of +1 and
the weight of 2119.733 has a Z score of -1
Turning the question around, suppose we wanted to know
where in the distribution we would find a car that weighed
4500 pounds. To answer that question we would need to find
the Z score for that value. The computing formula for finding
a Z score is
Thus, the z score for the vehicle weight 4500 pounds (X) is
4500-2969.56 (the mean)/849.827 (the standard deviation),
or Z=1.80. What about a 1000-lb car? (Z=-2.31)
How to Interpret and Use the Z Score: Table
of the Area under the Normal Curve
•
•
Suppose we know that a vehicle weight has a Z score
of +1 (is 1SD above the mean). Where does that
score stand in relation to the other scores?
Let’s think of the distribution image again. Recall that
we said that half of the cases fall below the mean,
and that 34.13% of the cases fall between the mean
and one SD below it, and 34.13% of the cases fall
between the mean and one sd above it. So if a vehicle
weight has a Z score of +1, what proportion of cases
are above and what percent are below it? Let’s look
at the next slide
Table of the Area under the Normal
Curve, continued
Consider the z score of 1.00. .3413 of scores lie between z and
the mean; .1587 of scores lie above a z of 1.00, and .8413 lie
below it.
Now suppose z was -1.0.
.3413 of scores would
still lie between z and
the mean; what percent
of scores would lie above
it and below it?
Remember that the
normal distribution is
symmetrical
Sampling Distribution of Sample Means
and the Standard Error of the Mean
•
•
The characteristics of populations, or parameters, are usually not
known. All we can do is estimate them from sample statistics.
What gives us confidence that a sample of, say, 100 or 1000
people permits us to generalize to millions of people?
The key concept is the notion that theoretically we could draw all
possible samples from the population of interest and that for the
sample statistics that we collect, such as the mean, there will be a
sampling distribution with its own mean and standard deviation.
In the case of the mean, this is called the sampling distribution of
sample means, and its mean is represented as µ
•
•
Characteristics: (1) approximates a normal curve (2) its mean is equal to
the population mean (3) its standard deviation is smaller that that of the
population (sample mean more stable than scores which comprise it)
We can also estimate the standard deviation of the sampling
distribution of sample means, which would give us an indicator of
the amount of variability in the distribution of sample means. This
value, known as the standard error of the mean, is represented
by the symbol σ Basically, it tells you how much statistics can be
expected to deviate from parameters when sampling randomly from the
population
Estimating the Standard Error
•
The standard error of the mean is hypothetical and
unknowable; consequently we estimate it with
sample statistics using the formula: standard
deviation of the sample divided by the square root
of the sample size. (SPSS uses N in the
denominator; Levin and Fox advocate N-1 for
obtaining an unbiased estimate of the standard
error Makes little difference with large N)
As you will quickly note, the standard error is very sensitive to sample size,
such that the larger the sample size, the smaller the error. And the smaller
the error, the greater the homogeneity in the sampling distribution of
sample means (that is, if the standard error is small relative to the range,
the sample means aren’t all over the place). The standard error is of
importance primarily because it is used in the calculation of other inferential
statistics and when it is small it increases the confidence you can have that
your sample statistics are representative of population parameters.
Finding Z Scores and the Standard
Error with SPSS
•
Let’s calculate Z scores and standard errors for the
variables companx1 and companx3 in the Lesson3.sav data
set
•
•
•
•
•
•
•
Go to Anayze/Descriptive Statistics/ Descriptives
Move the variables companx3 (difficulty understanding the
technical aspects of computers) and companx1(fear of making
mistakes) into the Variables window using the black arrow
Click on “Save standardized values as variables” (this creates
two new variables whose data are expressed in Z scores rather
than raw scores)
Click options and check “S. E. Mean” as well as mean, s, etc
Click Continue and then OK
Go to the Output viewer to see descriptive statistics for the
variables
Go to the Data Editor and note the new variables which have
been added in the right-most columns
Compare S.E.s, Raw Scores to Z
Scores
Note that the standard errors of the two variables are about the
same although the range is larger for “difficulty understanding”
Descriptive Statistics
Fear of Making Mistakes
Difficulty Understanding
Valid N (listwise)
Raw scores
N
Statistic
9
9
9
Minimum
Statistic
2.00
1.00
Maximum
Statistic
5.00
5.00
Mean
Statistic
Std. Error
3.7778
.3643
2.3333
.3727
Std.
Deviation
Statistic
1.09291
1.11803
Z scores.
Point Estimates, Confidence
Intervals
•
•
A point estimate is an obtained sample value such as a
mean, which can be expressed in terms of ratings,
percentages, etc. For example, the polls that are released
showing a race between two political candidates are based
on point estimates of the percentage of people in the
population who intend to vote for or at least favor one or the
other candidate
Confidence level and confidence interval
•
A confidence interval is a range that the researcher constructs
around the point estimate of its corresponding population
parameter, often expressed in the popular literature as a
“margin of error” of plus or minus some number of points,
percentages, etc. This range becomes narrower as the
standard error becomes smaller, which in turn becomes smaller
as the sample size becomes larger
Confidence levels
•
Confidence levels are usually expressed in terms
like .05, .01, etc. in the scholarly literature, and
5%, 1% etc in the popular press. They are also
called significance levels. They represent the
likelihood that the population parameter which
corresponds to the point estimate falls outside
that range. To turn it around the other way, the
represent the probability that if you constructed
100 confidence intervals around the point
estimate from samples of the same size, 95 (or
99) of them would contain the true percentage
of people in the population who preferred
Candidate A (or Candidate B)
Using the Sample Standard Error to
Construct a Confidence Interval
• Since the mean of the sampling distribution
of sample means for a particular variable
equals the population mean for that
variable, we can try to estimate how likely
it is that the population mean falls within a
certain range, using the sample statistics.
•
We will use the standard error of the mean from
our sample to construct a confidence interval
around our sample mean such that there is a
95% likelihood that the range we construct
contains the population mean
Calculating the Standard Error with
SPSS
•
Let’s consider the variable “vehicle weight” from the
Cars.sav data file.
• Let’s find the mean and the standard error of the
mean for “vehicle weight”
• Go to Analyze/Descriptive Statistics/ Frequencies,
then click the Statistics button and request the mean
and the standard error of the mean (S. E. Mean)
Statistics
Vehicle Weight (lbs.)
N
Valid
Missing
Mean
Std. Error of Mean
406
0
2969.56
42.176
Constructing the Confidence
Interval Upper and Lower Limits
•
•
Now let’s construct a confidence interval around the
mean of 2969.56 such that we can have 95%
confidence that the population mean for the variable
“vehicle weight” will fall within this range.
We are going to do this using our sample statistics
and the table of Z scores (area under the normal
curve). To obtain the upper limit of the confidence
interval, we take the mean (2969.56) and add to it (Z
X S.E)) where Z is the z-score corresponding to the
area under the normal curve representing the amount
of risk we’re willing to take (for example 5%, or, we
want to be 95% confident that the population mean
falls within the confidence interval) and S.E. is our
sample standard error (42.176)
Formulas for computing confidence intervals
around an obtained sample mean
 Formulas for Upper and Lower Limits
of the Confidence Interval
 Upper Limit: Sample Mean plus (Z times
the standard error), where Z
corresponds to the desired level of
confidence
 Lower Limit: Sample Mean minus (Z
times the standard error), where Z
corresponds to the desired level of
confidence
Consult the Table of the Area under
the Normal Curve
• In the normal table look until you find the
figure .025. You will note that this is onehalf of 5%. Since we have to allow for the
possibility that the population mean might
fall in either of the two tails of the
distribution we have to cut our risk area
under the normal curve in half, hence .025.
An area under the curve such that only
.025 percent of cases, in either tail of the
distribution, would fall that far from the
mean, corresponds to a Z of 1.96
Another normal table; Z corresponding
to 95% confidence interval (two tailed)
= 1.96 (this table shows area beyond Z)
Calculations for the Confidence
Interval
•
•
•
Now, compute the upper limit of the confidence
interval (this represents the largest value (CIu)that
you are able to say, with 95% confidence, that the
population mean vehicle weight could take. 2969.56
+ (1.96 (the Z score)(42.176 the standard error) =
3052
Now, compute the lower limit of the confidence
interval (this represents the lowest value that you are
able to say, with 95% confidence, that the population
mean vehicle weight could take (CIl). 2969.56 –
(1.96) (42.176) = 2886
Thus we can say with 95% confidence that the mean
vehicle weight in the population falls within the range
2886-3052
Constructing the Confidence
Interval in SPSS
•
•
•
Now run this analysis using
SPSS
Go to Analyze/ Descriptive
Statistics/ Explore. Put Vehicle
Weight into the Dependent List
box. Click on Statistics and
choose Descriptives and set the
confidence interval for the
mean at 95%. Click Continue
and then OK.
Compare your output Lower
Bound and Upper Bound to the
figures you did by hand.
Because you may not have
gotten all the significant digits
for the mean and standard
error your figures may be off a
tiny bit
Examining your SPSS Output
Here’s what your output should look like:
Descriptives
Now rerun the analysis to find
the confidence interval for the
99% level of confidence. But
before you do, consult the
table of the area under the
normal curve and see if you
can figure out what the value
of Z should be by which you
would multiply the S.E. of the
mean (hint: divide .01 by 2,
for each of the two tails of the
distribution)
Vehicle Weight (lbs.)
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Rang e
Skewness
Kurtosis
Lower Bound
Upper Bound
Statistic
2969.56
2886.65
Std. Error
42.176
3052.47
2940.50
2811.00
722206.2
849.827
732
5140
4408
1392.50
.468
-.752
.121
.242
Finding the 99% confidence
interval
Interpolate in the
table to find the z
score corresponding
to the 99%
confidence interval
(that is, there’s only a
1% probability (.005
in each tail of the
distribution) that the
population parameter
falls outside the
stated range
Output for 99% Confidence Level
Your output for the 99% confidence level should like this
Descriptives
Vehicle Weight (lbs.)
Mean
99% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Rang e
Skewness
Kurtosis
Lower Bound
Upper Bound
Statistic
2969.56
2860.41
Std. Error
42.176
3078.71
2940.50
2811.00
722206.2
849.827
732
5140
4408
1392.50
.468
-.752
.121
.242
Were you able to figure out
the correct value of Z from
the normal table? You have
to interpolate between .0051
and .0049. Z equals
approximately 2.58. That is,
at a Z score of + or – 2.575,
only about .005 of the means
will fall above (or below) that
score in the sampling
distribution
Write a sentence reporting your findings, indicating how
confident you are (and what you are confident about),
e.g., we can state with XXX confidence that mean
population vehicle weight is between XXX and XXX
Some Commonly Used Z Values
•
A quick chart of Z values for “two-tailed” tests
•
Note that these values are appropriate when we want to put
upper and lower limits around our point estimate. On other
occasions we will not be interested in both “tails” of the
distribution but only in, say, the upper tail (the upper 5% of
the distribution) and so the Z score would be different.
Look up in the table to see what the z score corresponding
to the upper 5 percent of the sampling distribution would
be. For example, if we assume that the population mean is
zero and we get a sample value so much higher that the
corresponding Z score is in the upper 5 percent, we might
conclude that the sample score did not come from the
population whose mean is zero.
•
•
•
95% confidence level: Z = 1.96
99% confidence level: Z = 2.575
99.9 confidence level: Z = 3.27
Confidence Intervals and Levels for
Percentage and Proportion Data
•
•
•
Using a sample percentage as a point estimate we can
construct a confidence interval around the estimate
such that we can say with 95% confidence that the
corresponding population parameter falls within a
certain range
The first thing we need is an estimate of the standard
error of proportions, similar in concept to the
standard error of sample means in that there is a
sampling distribution of sample proportions which will
be normal
The standard error of proportions is the standard
deviation of the sampling distribution of sample
proportions
Computing Formula for Standard Error
of Proportions from Sample Data
•
The formula for computing the population standard
error of proportions, σp , from the sample data is
σp

p(1- p)
N
where p is the proportion of cases endorsing option A,
for example, those preferring candidate A rather than
B. (1-p) is also seen as q in some formulas. Thus the
standard error of proportions in a sample with 10 cases
where .60 of the respondents preferred Candidate A
and .40 preferred Candidate B is the square root of
(.60)(.40)/10 or .1549 (Note: example on p. 258 in
Kendrick has typo in it. Denominator P (1-P) should
equal .2281, not .2881. The correct standard error of
proportions for the example in the text is .0124)
σp

.60(1- .60)
N
Putting Confidence Intervals
Around a Sample Proportion
•
Continuing with our example, we want to find the
upper and lower bounds for a confidence interval
around the sample proportion p of .60 in favor of
Candidate A. What can we say with 95% confidence
is the interval within which the corresponding
parameter in the population will fall?
•
•
•
CIupper = P + (Z)( σ) or .60 +(1.96 X .1549) or .904
CIlower = P - (Z)( σ) or .60 – (1.96 X .1549) or .296
Thus the proportion in the population which favors
Candidate A could range anywhere from .296 to .904, and
we are 95% confident in saying that!!! What do you think
our problem is? How could we narrow the range?
Using Dummy Variables to Find
Confidence Intervals for Proportion
•
•
A dummy variable is one on which each case is coded for
either presence or absence of the attribute. For example,
we could recode the ethnicity data into the dummy variable
“whiteness” or “Chinese-ness” so that every case would
have either a 1 or a zero on the variable. All of the white
(or Chinese) respondents would get a 1 and the others
would get a zero on the variable.
Let’s create a dummy variable for the variable “Country of
origin” in the Cars.sav data set. The new dummy variable
will be “American in Origin.” If you look at the country of
origin variable in the Variable View you will see that it is
coded as 1 = American, 2 = European, and 3 = Japanese.
We are going to recode it into a new dummy variable where
all of the American cars get a “1” and the Japanese and
European cars all get a zero.
Creating a Dummy Variable
•
•
•
•
•
In SPSS go to Transform/Recode/Into Different
Variables
Move the “Country of Origin” variable into the
Numeric Variable window. In the Ouput Window give
the new variable the name “AmerOrig” and the label
“RecodedCountryofOriginVariable.” click Change
Click on the “Old and New Values” button and recode
the old variable such that a value of 1 from the old
equals 1 in the new, and values in the range 2-3 in
the old equal zero in the new.
Click Continue, then OK
Go to the Variable View window and create value
labels for the new variable where American = 1 and
non-American origin = zero
Entering Old and New Values and Value
Labels for the Dummy Variable
Compare Frequency Distributions of Old
Variable to New Dummy Variable
•
Obtain the frequency distributions for the old and
recoded variables to make sure that the dummy
variable was created correctly
Country of Origin
Valid
Missing
Total
American
European
Japanese
Total
System
Frequency
253
73
79
405
1
406
Percent
62.3
18.0
19.5
99.8
.2
100.0
Valid Percent
62.5
18.0
19.5
100.0
Cumulative
Percent
62.5
80.5
100.0
RecodedCountryofOriginVarialbe
Valid
Missing
Total
Non-American Origin
American Origin
Total
System
Frequency
152
253
405
1
406
Percent
37.4
62.3
99.8
.2
100.0
Valid Percent
37.5
62.5
100.0
Cumulative
Percent
37.5
100.0
Put Confidence Intervals around
the Proportion with SPSS
•
•
•
Go to Analyze/Descriptives/Explore. Click Reset to clear the window
and move the new recoded variable into the window
Click on the Statistics button and select Descriptives and set the
confidence interval to 95%. Click Continue and OK
From the output you will get the point estimate for the proportion of
American cars of .6247, and can say with 95% confidence that the
corresponding population parameter falls within a range of
approximately .57 to .67
Descriptives
RecodedCountry
ofOriginVarialbe
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Lower Bound
Upper Bound
Statistic
.6247
.5773
Std. Error
.02409
.6720
.6385
1.0000
.235
.48480
.00
1.00
1.00
1.0000
-.517
-1.741
.121
.242
A Word about the t Distribution
•
•
•
•
•
Levin and Fox and some other authors advocate using the t distribution to
construct confidence intervals and find significance levels when the sample
size is small. When it is large, say over 50, the Z and t distributions are
very similar. Think of t as a standard score like z.
When comparing an obtained sample mean to a known or assumed
population mean, t is computed as the sample mean minus the population
mean, divided by an unbiased estimate of the standard error (the sample
standard deviation divided by the square root of N-1)
The t table is entered by looking up values of t for the sample size minus
one (N-1) (also known as the “degrees of freedom” in this case) and the
significance level (area under the curve corresponding to the alpha level
(.05, 01, .005, or 1 minus the degree of confidence)).
Suppose we had a sample size of 31 and an obtained value of t of 2.75.
Entering the t distribution at df=30 and moving across we find that a t of
2.75 corresponds to a significance or alpha level (area in the tail of the
distribution) of only .01 (two-tailed-.005 in each tail), which means that
there is only 1 chance in 100 of obtaining a sample mean like ours given
the known population mean. (the t-test in practice is adjusted for whether it
is “one-tailed” or “two-tailed.”
Levin and Fox provide examples of setting confidence intervals for means
using the t distribution rather than z.
T Distribution
A Word about the Concept
“Degrees of Freedom” (DF)
•
•
In scholarly journals you will see references to ‘DF”
when the results of statistical tests are reported.
What does this mean?
Degrees of freedom generally is calculated as the
number of independent cases (which are free to vary)
in the sample from which we are computing a
statistic. For example, suppose we had the following
data: 5,6,7,8,9. And we calculated their mean as
35/5 = 7. We could change any four of those
numbers, e.g., we could change them to 1, 2, 3, and
4, but the fifth one would have to make their total
come out to 35 (would have to be 25) so that the
mean of the five numbers would remain 7. Thus the
degrees of freedom in computing the mean is n-1 or
in this case 4.