STAT 350 (Spring 2017) Homework 11 (20 points + 1 point BONUS).docx

STAT 350 (Spring 2017) Homework 11 (20 points + 1 point
BONUS) 1
Practice Problems: 12.5 (p. 588), 12.9 (p.588)
(4 pts.) 1. For each of the following graphs, identify the form,
direction (if possible) and relative
strength. In addition, state if you think that there is an
association between X and Y. No
explanation is required.
a) b)
c) d)
BONUS) 2
(14 pts.) 2. Deep-water (>300m) wave forecasts are important
for large cargo ships. One
method of prediction suggests that the wind speed (x, in knots)
is linearly related to the wave
height (y, in feet). A random sample of buoys was obtained, and
the wind speed and wave
height was measured at each. The summary data is shown

below.
n = 20, SXX = 91.75, SYY = 15.952, SXY= 36.4, x̄ = 9.25, ȳ =
1.68
The scatter plot of the data is shown below:
(2 pts.) a) Find the estimated regression line for the regression
of Wave Height as a function of
Wind Speed.
(1 pt.) b) Does the y-intercept have any physical meaning?
(1 pt.) c) How much change in wave height is expected when
the wind speed increases by one
knot? Please explain your answer.
(1 pt.) d) What is the expected value of wave height when the
wind speed is 8.6 knots (10
mph)?
(6 pts.) e) Complete the following ANOVA table.
Source of variation Degrees of Freedom Sum of squares Mean
square
Regression
Error
Total

(1 pt.) f) What is the estimated variance?
(1 pt.) g) What is the proportion of the wave height that is
explained by wind speed?
(1 pt.) h) From the information in the previous parts of this
question, do you believe that there is
an association between wave height and wind speed? Please
explain your answer. No
additional calculations are required.
BONUS) 3
(2 pts.) 3. Some physicians use the cholesterol ratio (CR = total
cholesterol/HDL cholesterol) as
a measure of a patient’s risk of heart disease. In addition, the
triglyceride concentration (TG)
is associated with coronary artery disease in many patients. In a
study of the relationship
between these two variables, a random sample of adults was
obtained, and the triglyceride
level denoted as x1 in mg/dL and cholesterol ratio (y) was
obtained for each person. The
scatterplot and regression line of ln(triglyceride level - 129)
denoted as x2 vs. cholesterol ratio
is below.
The ANOVA summary table is

Source of Variation Sum of Squares Degrees of freedom Mean
Square
Regression 103.16 1 103.16
Error 3.20 23 0.14
Total 106.36 24
(1 pt.) a) What is the coefficient of determination?
(1 pt.) b) Do you think that an increase in the triglyceride level
causes an increase in the
cholesterol level? Please explain your answer.
(1 pt.) BONUS: Why do you think that they had to take the
logarithm of the triglyceride level?
Additional Problems: Note, the book gives the sums so that you
can use the computing method
to calculate the estimated value of the parameters, not SXY and
SXX.
12.19, 12.21, 12.23
TESTS OF
SIGNIFICANCE
Module 6
1

SOME KEY INGREDIENTS FOR
INFERENTIAL STATISTICS
2
INFERENTIAL STATISTICS
• Allow us to draw conclusions about theoretical
principles that go beyond the group of participants in a
particular study
3
THE NORMAL CURVE
• Normal Distribution
• histogram or frequency distribution that is
a unimodal, symmetrical, and bell-shaped
• a mathematical distribution
• Researchers compare the distributions of
their variables to see if they approximately
follow the normal curve.
4
WHY THE NORMAL CURVE IS
COMMONLY FOUND IN

NATURE
• A person’s ratings on a variable or performance on a task is
influenced by a number of random factors at each point in
time.
• These factors can make a person rate things like stress levels
or
mood as higher or lower than they actually are, or can make a
person perform better or worse than they usually would.
• Most of these positive and negative influences on
performance or ratings cancel each other out.
• Most scores will fall toward the middle, with few very low
scores
and few very high scores.
• This results in an approximately normal distribution
(unimodal,
symmetrical, and bell-shaped).
5
THE NORMAL CURVE AND THE
PERCENTAGE OF SCORES BETWEEN THE
MEAN AND 1 AND 2 STANDARD
DEVIATIONS FROM THE MEAN
• There is a known percentage of scores that fall below any
given
point on a normal curve.

• 50% of scores fall above the mean and 50% of scores fall
below the
mean.
• 34% of scores fall between the mean and 1 standard deviation
above
the mean.
• 34% of scores fall between the mean and 1 standard deviation
below
the mean.
• 14% of scores fall between 1 standard deviation above the
mean and
2 standard deviations above the mean.
• 14% of scores fall between 1 standard deviation below the
mean and
2 standard deviations below the mean.
• 2% of scores fall between 2 and 3 standard deviations above
the
mean.
• 2% of scores fall between 2 and 3 standard deviations below
the
mean.
6
THE NORMAL CURVE
TABLE AND Z SCORES

• A normal curve table shows the percentages
of scores associated with the normal curve.
• The first column of this table lists the Z score
• The second column is labeled “% Mean to Z” and gives
the percentage of scores between the mean and that
Z score.
• The third column is labeled “% in Tail.”
.
7
Z % Mean to Z % in Tail
.09 3.59 46.41
.10 3.98 46.02
.11 4.38 45.62
SAMPLE AND
POPULATION
• Population
• entire set of things of interest
• e.g., the entire piggy bank of pennies
• e.g., the entire population of individuals in the US
• Sample

• the part of the population about which you actually
have information
• e.g., a handful of pennies
• e.g., 100 men and women who answered an online
questionnaire about health care usage
8
WHY SAMPLES INSTEAD OF
POPULATIONS ARE STUDIED
• It is usually more practical to obtain
information from a sample than from the
entire population.
• The goal of research is to make
generalizations or predictions about
populations or events in general.
• Much of social and behavioral research is
conducted by evaluating a sample of
individuals who are representative of a
population of interest.
9
METHODS OF SAMPLING
• Random Selection

• method of choosing a sample in which each individual
in the population has an equal chance of being
selected
• e.g., using a random number table
• Haphazard Selection
• method of selecting a sample of individuals to study by
taking whoever is available or happens to be first on a
list
• This method of selection can result in a sample that is not
representative of the population.
10
STATISTICAL TERMINOLOGY
FOR SAMPLE AND
POPULATIONS
• Population Parameters
• mean, variance, and standard deviation of a
population
• are usually unknown and can be estimated from
information obtained from a sample of the
population
• Sample Statistics
• mean, variance, and standard deviation you

figure for the sample
• calculated from known information
11
PROBABILITY
• Expected relative frequency of a particular outcome
• outcome
• term used for discussing probability for the result of an
experiment
• expected relative frequency
• number of successful outcomes divided by the number of total
outcomes you would expect to get if you repeated an experiment
a
large number of times
• long-run relative-frequency interpretation of probability
• understanding of probability as the proportion of a particular
outcome that
you would get if the experiment were repeated many times
12

STEPS FOR FIGURING
PROBABILITY
• Determine the number of possible successful
outcomes.
• Determine the number of all possible outcomes.
• Divide the number of possible successful outcomes by
the number of all possible outcomes.
13
FIGURING PROBABILITY
• You have a jar that contains 100 jelly beans.
• 9 of the jelly beans are green.
• The probability of picking a green jelly bean would be
9 (# of successful outcomes) or 9%
100 (# of possible outcomes)
14
RANGE OF PROBABILITIES
• Probability cannot be less than 0 or greater than 1.
• Something with a probability of 0 has no chance of
happening.

• Something with a probability of 1 has a 100% chance of
happening.
15
P
• p is a symbol for probability.
• Probability is usually written as a decimal, but can also
be written as a fraction or percentage.
• p < .05
• the probability is less than .05
16
PROBABILITY, Z SCORES,
AND THE NORMAL
DISTRIBUTION
• The normal distribution can also be thought of as a
probability distribution.
• The percentage of scores between two Z scores is the
same as the probability of selecting a score between
those two Z scores.
17

NORMAL CURVES, SAMPLES AND
POPULATIONS, AND PROBABILITY IN
RESEARCH ARTICLES
• Normal curve is sometimes mentioned in
the context of describing a pattern of
scores on a particular variable.
• Probability is discussed in the context of
reporting statistical significance of study
results.
• Sample selection is usually mentioned in
the methods section of a research article.
18
DEGREES OF FREEDOM
(DF)
• The number by which you divide to get the estimated
population variance
• Number of scores free to vary when estimating a
population parameter
• If you know the mean of the population and all but one
of the scores in the sample, you can figure out the
score you don’t know.
• Once you know the mean, one of the scores in the

sample is not free to have any possible value and the
degrees of freedom then would = N – 1
• Degrees of freedom is the number of scores in the sample
minus 1.
19
INTRODUCTION TO HYPOTHESIS
TESTING
20
HYPOTHESIS TESTING
• A systematic procedure for deciding whether
the results of a research study supports a
hypothesis that applies to a population
• hypothesis
• a prediction intended to be tested in a research study
• can be based on informal observation or theory
• theory
• a set of principles that attempts to explain one or more
facts, relationships, or events
• usually gives rise to various specific hypotheses that can
be tested in research studies
21

HYPOTHESIS TESTING
• Researchers want to draw conclusions about a particular
population.
• e.g., babies in general
• Conclusions will be based on results of studying a sample.
• e.g., one baby
The Core Logic of Hypothesis Testing
• Researchers must spell out in advance what would have to
happen in order to
allow them to conclude that their hypothesis was supported.
• They then conduct their experiment.
• Then they figure the probability of getting their particular
experimental result if
their hypothesis was not true.
• They answer the question:
• What is the probability of getting our research results if the
opposite of
what is predicted were true?
• If it is highly unlikely that we would get our research results
if the opposite of

what we are predicting were true:
• We can reject the opposite prediction.
• If we reject the opposite prediction, we can accept our
prediction.
22
THE HYPOTHESIS-TESTING PROCESS
(A 5 STEP PROCESS USING SPSS)
• Step 1: Restate the question as a research hypothesis
and a null hypothesis about the population.
• Step 2: Determine the confidence interval (.05 or .01)
• Step 3: Run the test using functions in SPSS
• Step 4: Identify the number under ‘Sig 2-tail’ and
compare that to your confidence interval
• Step 5: Decide whether to reject the null hypothesis.
If p (Sig 2-tail) is < or = to the confidence interval, than
you reject the null hypothesis
Always tell your reader what that means in regards to
your research study!
23

IMPLICATIONS OF REJECTING OR
FAILING TO REJECT THE NULL
HYPOTHESIS
• Rejecting the null hypothesis says that your results support
the research hypothesis.
• The results never prove the research hypothesis or show that
your
hypothesis is true.
• You will either “reject the null hypothesis” or “fail to reject
the null hypothesis”
• You will NEVER “prove the null hypothesis” or “prove the
research hypothesis”
• Research studies and their results are based on the probability
or chance of
getting your result if the null hypothesis were true.
• When the results are not extreme enough to reject the null
hypothesis, you do not say that the results support the null
hypothesis.
• You say that the results are not statistically significant, or that
the
results are inconclusive.
• We are basing research on probabilities, and the fact that we
did not find a
result in this study does not mean that the null hypothesis is
true.
24

DECISION ERRORS
• When the right procedures lead to the
wrong decisions
• In spite of calculating everything correctly,
conclusions drawn from hypothesis testing
can still be incorrect.
• This is possible because you are making
decisions about populations based on
information in samples.
• Hypothesis testing is based on probability.
25
TYPES OF ERRORS
Type I
• Rejecting the null hypothesis when
the null hypothesis is true
• You find an effect when in
fact there is no effect.
• A Type I error is a serious error as
theories, research programs,
treatment programs, and social
programs are often based on

conclusions of research studies.
• The chance of making a Type I
error is the same as the
significance level.
• If the significance level was
set at p < .01, there is less
than a 1% chance that you
could have gotten your result
if the null hypothesis was true.
• To reduce the chance of
making a Type I error,
researchers can set a very
stringent significance level
(e.g., p < .001).
Type II
• With a very extreme significance level,
there is a greater probability that you
will not reject the null hypothesis when
the research hypothesis is actually
true.
• concluding that there is no
effect when there is actually an
effect
• The probability of making a
Type II error can be
reduced by setting a very
lenient significance level
(e.g., p < .10).

26
Special Note:
Decreasing the probability of a Type
I error increases the probability of a
Type II error.
The compromise is to use
standard significance levels of
p < .05 and p < .01.
HYPOTHESIS TESTS AS REPORTED
IN RESEARCH ARTICLES
• In research articles, for each result of interest, the
researcher usually says whether the result was
statistically significant.
• The researcher gives the symbol for the specific
method used in figuring out the probabilities.
• There will be an indication of significance level
(e.g., p < .05 or p < .01).
• Usually a two-tailed test is used; if this is not the
case, the researcher will generally specify that a
one-tailed test was used.
27

CHI-SQUARE TESTS AND STRATEGIES WHEN
POPULATION DISTRIBUTIONS ARE NOT NORMAL
28
CHI-SQUARE TESTS
• Are used when the variable of interest is a nominal variable
• The values of a nominal variable are categories.
• The scores of a nominal variable represent frequencies.
• The goal of hypothesis testing with ANOVA is to determine
whether the means of the sample differ more than you
would expect if the null hypothesis were true.
• Chi-square tests examine how well the observed
breakdown of people or observations over categories fits
an expected breakdown.
• Chi-square test of goodness of fit involves levels of a single
nominal variable.
• Chi-square test for independence is used when there are two
nominal variables each with several categories.
29

CHI-SQUARE TEST FOR GOODNESS
OF FIT
• A comparison of an observed frequency distribution to an
expected
frequency distribution
• You first figure a number for the amount of mismatch between
the
observed frequencies and the expected frequencies and then see
whether that number indicates a greater mismatch than you
would
expect by chance.
• The mismatch between observed and expected for any one
category is just the
observed frequency minus the expected frequency.
• The differences are not used directly because positive and
negative differences
would cancel each other out—that is why each difference is
squared.
• Expected frequencies for the different categories may not be
the same.
• A particular amount of difference between observed and
expected has a different
importance according to the size of the expected frequency.
• The mismatch between observed and expected for a particular
category is weighted
to take into account the expected frequency for that category.
• This is done by dividing your squared difference for a
category by the expected

frequency for that category.
• The weighting puts the squared difference onto a more
appropriate scale for
comparison.
• If the expected frequency for a particular category is 25, you
divide the squared difference by
25.
30
THE CHI-SQUARE DISTRIBUTION
• Estimating the distribution of chi-square statistics that
would arise by chance
• The exact shape of the chi-square distribution
depends on degrees of freedom, but they are all
skewed to the right because the chi-square statistic
cannot be less than 0 but can have very high values.
• The degrees of freedom for a chi-square test are the
number of categories that are free to vary, given the
total.
• For example, if there are two categories, there is one
degree of freedom.
• df = Ncategories – 1
31

THE CHI-SQUARE TEST FOR
INDEPENDENCE
• Used when there are two nominal
variables, each with several categories
• Contingency Table
• table in which the distributions of two nominal
variables are set up so that you have the
frequencies of their contributions as well as the
totals
32
INDEPENDENCE
• No relationship between the variables in
the contingency table
• It is important to determine whether the
lack of independence in the sample is
large enough to reject the null
hypothesis of independence of the
population.
33
ASSUMPTIONS FOR THE
CHI-SQUARE TESTS

• Each score must not have any special relationship to
any other score.
• You cannot use the chi-square tests if the scores are
based on the same people being tested more than
once.
34
STRATEGIES FOR HYPOTHESIS
TESTING WHEN POPULATION
DISTRIBUTIONS ARE NOT
NORMAL
• When the variables are quantitative but
the assumption that the population
follows a normal curve or the assumption
of equal variances is violated:
• You cannot use the t test or ANOVA.
• One common situation where assumptions
are clearly violated is when there is a floor or
ceiling effect or there are outliers (extreme
scores that can make the results statistically
significant even when there is no effect).
35
CHI SQUARE EXAMPLE
(SPSS)

36
Research Question: Do you agree or disagree that
people who break the law should be given stiffer
sentences?
• Step 1: Determine your hypotheses.
• H1 (RH): A person’s sex and race will influence their
preference for stiffer sentences for lawbreakers.
• H0 (NH): A person’s sex and race will not influence their
preference toward stiffer sentences for lawbreakers.
• Step 2: Confidence interval is 95% or .05
• Step 3:
• Sig 2-tail = .048
• Step 5: .048 = .05 which means that we reject the null.
We can conclude that the preference for stiffer
sentences for lawbreakers does have a
connection to respondent’s sex.
37
WHAT IF YOU HAVE VARIABLES
WHOSE VALUES ARE CATEGORIES?

• t Tests and the ANOVA require:
• the measured variable to have scores that are
quantitative
• e.g., ratings on a scale of stress that range from 0–10,
numerical scores on a test of intelligence, scores on a
measure of gastrointestinal symptoms
• the populations to follow a normal curve.
• Categorical variables require an alternative
hypothesis-testing procedure.
38
THE T TEST FOR INDEPENDENT
MEANS
39
T TESTS
• Hypothesis-testing procedure in which the population
variance is unknown
• compares t scores from a sample to a comparison
distribution called a t distribution
• t Test for a single sample
• hypothesis-testing procedure in which a sample mean is

being compared to a known population mean but the
population variance is unknown
• Works basically the same way as a Z test, but:
• because the population variance is unknown, with a t test
you have to estimate the population variance
• With an estimated variance, the shape of the distribution is
not normal, so a special table is used to find cutoff scores.
40
T TESTS FOR INDEPENDENT MEANS
• Hypothesis-testing procedure used for studies
with two sets of scores
• Each set of scores is from an entirely different group of
people and the population variance is not known.
• e.g., a study that compares a treatment group to a
control group
41
THE DISTRIBUTION OF
DIFFERENCES BETWEEN MEANS
• When you have one score for each person with two
different groups of people, you can compare the mean of

one group to the mean of the other group.
• The t test for independent means focuses on the difference
between the means of the two groups.
• The comparison distribution is a distribution of differences
between means.
• created by randomly selecting one mean from the distribution
of means
from the first group’s population
• randomly selecting one mean from the distribution of means
for the
second group’s population
• Subtract the mean from the second distribution of means from
the mean
from the first distribution of means to create a difference score
between
the two selected means.
• Repeat this process a large number of times and you will have
a
distribution of differences between means.
• Note that this is not the actual way a distribution of means is
created, but
conceptually this is what a distribution of means is.
42
THE LOGIC OF A
DISTRIBUTION OF

DIFFERENCES BETWEEN
MEANS
• The null hypothesis is that Population M1 = Population M2
• If the null hypothesis is true, the two population means from
which the samples are drawn are the same.
• The population variances are estimated from the sample
scores.
• The variance of the distribution of differences between
means is based on estimated population variances.
• The goal of a t test for independent means is to decide
whether the difference between means of your two actual
samples is a more extreme difference than the cutoff
difference on this distribution of differences between means.
43
MEAN OF THE DISTRIBUTION OF
DIFFERENCES BETWEEN MEANS
• With a t test for independent means, two populations
are considered.
• An experimental group is taken from one of these
populations and a control group is taken from the other
population.
• If the null hypothesis is true:

• The populations have equal means.
• The distribution of differences between means has a
mean of 0.
44
T TEST EXAMPLE (USING
SPSS)
• Step 1. RH (H1) – Group therapy increases ADL.
NH (H0) – Group therapy does not increase ADL
• Step 2. The level of confidence is .05
• Step 3. Conduct the test in SPSS.
45
• Step 4. P=.053 which rounds to .05
• Step 5. Compare these two numbers and apply the rule
that if P < or = confidence interval (.05 or .01) than you
reject the null. .053 = .05 so we reject the null.
By rejecting the null, we see that group therapy does in
fact work as it increases a patient’s active daily living
activities.
46

THE T TEST FOR
INDEPENDENT MEANS IN
RESEARCH ARTICLES
• When found in research articles, the results of these
tests are accompanied by reporting of the means
and sometimes the standard deviations.
• t = (dftotal) = x.xx, p < .01
47
INTRODUCTION TO ANALYSIS OF
VARIANCE
48
WHAT IF YOU HAVE MORE THAN TWO GROUPS OF
SCORES?
• Very often when doing research, you will have more
than one experimental group and one control group.
• e.g., one group receives treatment A, one group receives
treatment B, and one group does not receive any treatment
• The analysis of variance (ANOVA) procedure is a
statistical procedure that tests the variation among

the means of more than two groups.
49
BASIC LOGIC OF THE ANALYSIS
OF VARIANCE (ANOVA)
• The null hypothesis when using the ANOVA is that the
populations
being compared all have the same mean.
• The goal of hypothesis testing with ANOVA is to determine
whether
the means of the sample differ more than you would expect if
the
null hypothesis were true.
• With ANOVA, we are examining the variation among means.
• The ANOVA is about a comparison of the results of two
different
ways of estimating population variances.
50
• The analysis of variance or ANOVA is a technique
that resolves the short coming of the T-test. It
examines the means of sub groups in the sample
and analyzes the variances as well. That is, it
examines more than whether the actual values are
clustered around the mean or spread from it.

• What if we use religious affiliation as a variable?
• Ex: Religious affiliation is correlated with attitudes
towards affirmative action.
• We will want to know if Catholics are different than
Jews, Muslims than Catholics, Protestants than Muslims,
and so on…
• The ANOVA will allow us to see these differences
51
VARIATION BETWEEN SAMPLES DUE TO
CHANCE FACTORS
• Even with identical populations, there are slight differences
between samples.
• This variation is due to the same chance factors that influence
variability within the samples.
• The variation among the means of the samples is related
directly to the
amount of variation in each of the populations from which the
samples are
taken.
• For example, if you were studying the effect of different
exercise programs on
weight loss, you would have much more variation among the
means of these
samples if your samples consisted of men and women who
weighed between

110 and 350 lbs than if your groups were made up only of
women who
weighed between 200 and 250 lbs.
• Even if you have 3 identical populations, the sample means
from the
populations that each have small amount of variance will be
closer
together than samples from populations each with a large
amount
of variance.
52
WHEN THE NULL HYPOTHESIS IS NOT
TRUE
• If the null hypothesis is not true, the populations do not have
the same mean.
• The means of the populations are spread out for two different
reasons.
• chance factors that cause variation within the population
• the different treatments received by the groups (treatment
effect)
53
COMPARING WITHIN-GROUPS AND BETWEEN-GROUPS
ESTIMATES

OF POPULATION VARIANCE WHEN THE NULL
HYPOTHESIS IS TRUE
• The within-groups and between-groups estimates are both
based
on the chance variation within populations.
• They are estimates of the same population variance, so both
estimates
should be about the same.
• The ratio of the between-groups estimate to the within-groups
estimate
should be one to one.
• e.g., if the within-groups estimate is 59.6, the between-groups
estimate should
be about 59.6; the ratio should be about 1.
54
BETWEEN-GROUPS ESTIMATE OF THE POPULATION
VARIANCE WHEN THE RESEARCH HYPOTHESIS IS TRUE
• Between-groups estimate is influenced by:
• variation within the populations
• chance factors
• the variation of the means of the populations from one another
• treatment effect
• The within-groups estimate is only based on variation within
the

populations.
• The between-groups estimate should be larger than the within-
groups
estimate.
• The ratio of the between-groups estimate to the within-groups
estimate
should be greater than 1.
• If the between-groups estimate is 206.5 (variation due to
chance factors
and treatment effect) and the within-groups estimate is 59.6
(variation due
to chance factors), the ratio of the between-groups estimate to
the within-
groups estimate would be 3.46.
55
ANALYSIS OF VARIANCE
IN RESEARCH ARTICLES
• ANOVA results are reported by giving the F,
degrees of freedom, and significance level.
• F(dfBetween, dfWithin) = x.xx, p < .01
• Group means are usually given in a table.
• For a factorial analysis of variance, the F
ratios for any main effects and interaction
effects are reported in the text and the cell
means, and sometimes marginal means are

reported in a table.
56
ANOVA USING SPSS
• Step 1. One’s religion is correlated to one’s level of
education.
• Step 2. Confidence level is set at .05
• Step 3. Run tests in SPSS. See next slide.
57
58
Let’s examine these
charts one by one.
• Steps 4 & 5. If we use the rule than
we see that . 003 is < .05 which
means that we reject the null.
Remember our null states that there is
NO relationship between one’s

religion and level of education.
We can check this by looking at the
means. See next slide.
59
You can see here that the
significance level for the corrected
model F value (15.869) is
.000. What this means is that if
religion and education are unrelated
to each other in the population, we
might expect samples that would
generate this amount of explained
variance less than once in 1,000
samples (Babbie, 2014).
• What we see here in the mean column are the average
levels of educations- years of school completed for people in
each religion. We can quickly find the difference between

them in this one chart and one function of ANOVA. We can
see that Jewish respondents (mean = 16.07) have greater
than 2 years of education higher than any other group of
respondents.
60

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STAT 350 (Spring 2017) Homework 11 (20 points + 1 point BONUS).docx