Transcript Lecture Notes - Webpages at SCU

Psychology 215:
Statistics for Social Science
Kate Bezrukova
[email protected]
Introduction
Introduction
• Our Increasingly Quantitative World!!!
• Data are not just numbers! but numbers
that carry information
• Purposes:
– producing trustworthy data
– analyzing data to make their meaning clear,
and
– drawing practical conclusions from data
Basic Idea of Statistics:
To make inferences about a population using data from only
a sample
population
inference about population
(using statistical tools)
sample of data
Overview of Course
• Methods of data collection
– summarizing data
• Visualizing data
– correlation and association
• Standard scores and Normal Curve
– statistical inference
• Statistical tests
– necessary skills
Variables
Statistics vs. Parameters
•
sample – statistics
c – carries uncertainty
•
population – parameter
 – carries no uncertainty
is there any interest in a group larger than one you
have?
1. if “yes” - sample
2. if “no” - population
Terms
• Data - numbers collected for purpose in a
particular context
– Case/observational unit
• Variability – fundamental principle.
• Variables – any characteristics of a person/thing
that can be assigned a number or category
– measurement (continuous) variables – assumes a
range of numerical values
– categorical variables – simply records a category
designation.
Types of Variables
Variables = Aspect of a testing condition that can
change or take on different
characteristics with different conditions.
• Dependent Variable (DV)
• Independent Variable (IV)
• Extraneous Variable
– Confounded Variables
Graphs
Distributions
• data distribution – pattern of
variability.
– the center of a distribution
– the ranges
– the shapes
• simple frequency distributions
• grouped frequency distributions
– midpoint
Graphic Presentation of Data
the frequency polygon
(quantitative data)
the histogram
(quantitative data)
the bar graph
(qualitative data)
Distributions
• Bell-Shaped (also
known as symmetric”
or “normal”)
• Skewed:
– positively (skewed to
the right) – it tails off
toward larger values
– negatively (skewed to
the left) – it tails off
toward smaller values
Measures of Central Tendency
Mean
c=
S
c
Sc
N
an instruction “to add”
mean
characteristics:
1.
2.
c
a score of observations
N
number of observations
S (c c ) = 0
2
(c-c
)
S
= minimum, “least squares”
Median & Mode
• MEDIAN divides a distribution of scores
(always arrange in order!) into two parts
that are equal in size
• median location:
– if uneven N – actual score
– if even N – mean of the middle two scores
• MODE is the most common value, i.e., the
most frequently occurring score.
Simple Frequency Distributions
raw-score distribution
name
Student1
Student2
Student3
Student4
Student5
Student6
Student7
Student8
X
20
23
15
21
15
21
15
20
frequency distribution
Mean
f
X
3
2
2
1
15
20
21
23
c=
Sfc
N
Measures of Variability
Measures of Spread: Range
is simply a numerical distance b/w the
highest score and the lowest score
– how many hours did you sleep last night?
– how much sleep do you usually get on a
typical night?
Interquartile range
tells the range of scores that make up
the middle 50 percent of the
distribution.
• IQR = 75th percentile – 25th percentile
• a location value (.25 x N)
– similar to finding the median
• interpretation: “the middle 50 percent of
the scores have values from XXX to
YYY”
Deviation Scores
= raw score – mean
for samples: X – X
for populations: X – 
1.
2.
3.
if X > X, positive deviation scores
if X < X, negative deviation scores
if X = X, deviation scores = 0
interpretation: the number of points that a particular
score deviates from the mean (S (X – X) = 0).
Standard Deviation
s - is used to describe the variability of population (s is
parameter)
s - is used to estimate s from a sample of the
population (s is statistics)
S - is used to describe the variability of a sample when
we have no desire to estimate s. (S – statistics)
–
Choosing the correct SD:
1.
2.
–
how the data was gathered -- was sampling used?
generalization – purpose of the data?
Formulas:
•
•
deviation-score formula
raw-score formula
Deviation-Score Method
s=
where
2
)
S (XN
S=
s = standard deviation of population
S = standard deviation of sample
N = number of scores
1.
2.
3.
4.
2
)
S (X- X
find a deviation score for each raw score
square the deviation score
add them up
divide this sum by N and find a square
root
N
Raw-Score Method
s=
2
2
X)
(S
SX -
N
N
where S X2 = sum of the squared scores
(S X)2 = square of the sum of the raw scores
N = number of scores
1.
2.
3.
4.
square the sum of the raw scores and divide by N
square each score and add them up
subtract 1 from 2
divide this difference by N and find a square root
Standard Deviation (Sample)
S=
S (X- X
N-1
S=
)2
S
S=
2
2
X)
(S
SX -
2
2
fX)
(
S
fX -
N
N-1
SfX2 -squaring, multiplying, summing
- (SfX)2 - multiplying, summing, squaring
-
N
N-1
Variance
• the number before taking the square root
2
2
S(X-)
S
(x-x)
2=
s2 =
S
N
N-1
•
•
•
•
analysis of variance
N – population
N – 1 – sample
S(x-x)2 – sum of squares
Other Descriptive Statistics
Combination Statistics: z scores
• standard scores are usually used to compare two scores
from different distributions
• positive z scores = raw scores > mean
• negative z scores = raw scores < mean
• the absolute value of the z score |z score| tells the
number of standard deviations the score is from the
mean
X-X
z=
S
Sz = 0 because S (X – X) = 0
How much difference is there?
Effect Size Index “d”
• describes the size of the difference between
two distributions
|1- 2|
d=
s
always a positive number!
small effects
medium effects
large effects
d = .20
d = .50
d = .80
“huge”, “half the size of small”,
“somewhat larger than”,
spooled = (s12+ s22) / 2]
“intermediate between”
Descriptive statistics report:
Boxplot
- minimum score
- maximum score
- lower quartile
- upper quartile
- median
- mean
- the skew of the distribution:
positive skew: mean > median & high-score whisker is longer
negative skew: mean < median & low-score whisker is longer
Report:
1) boxplot (draw)
2) effect size index (calculate)
3) story (write): a) central tendency, b) form of the distributions,
c) overlap of distributions, d) interpretation of the effect size index
EXAMPLE: A descriptive statistics
report on “Marriage Ages”
1.
The graph shows boxplots of marriage ages of women and men.
+++++++++insert your boxplot here+++++++++
2.
3.
4.
5.
The mean age of women is 35 years old; the median is 33 years
old. The mean age of men is 38.64 years old; the median is 34
years old.
The marriage ages for both women and men are positively
skewed. More women and men get married at the younger age*.
Although the two distributions overlap, the middle 50% of the men
are somewhat older than the middle 50% of the women.
The difference in means produces an effects size index of .32.
This value is somewhat larger than a small effect size.
Correlation and Regression
Correlation Coefficient
a correlation coefficient (r) provides a quantitative way to express the
degree of linear relationship between two variables.
• Range: r is always between -1 and 1
• Sign of correlation indicates direction:
- high with high and
low with low -> positive
- high with low and
low with high -> negative
- no consistent pattern -> near zero
• Magnitude (absolute value) indicates
strength (-.9 is just as strong as .9)
.10 to .40 weak
.40 to .80 moderate
.80 to .99 high
1.00 perfect
Pearson product – moment correlation
coefficient: Formulas
Invented by Carl Pearson and used to describe the strength of the linear
relationship between two variables that are both either ratio or interval
variables.
Definitional Formula
S ( zx zy )
r=
N
where
r = Pearson product-moment correlation coefficient
zx = a z score for variable X
zy = a z score for variable Y
N = number of pairs of X and Y values
Computational Formulas
raw-score formula
The rule of thumb
Correlation coefficient should be based on at least 30 observations
Pearson product – moment correlation
coefficient: Blanched Formula
r=
ΣΧΥ - (Χ)(Υ)
N
(Sx) (Sy)
where
X and Y are paired observations
XY = product of each X value multiplied by its paired Y value
X =mean of variable X
Y =mean of variable Y
Sx =standard deviation of X distribution
Sy =standard deviation of Y distribution
N = number of paired observations
Correlation Coefficient: Limitations
1. Correlation coefficient is appropriate measure of
association only when relationship is linear
2. Correlation coefficient is appropriate measure of
association when equal ranges of scores in the sample
and in the population (truncated range)
3. Correlation doesn't imply causality
–
–
–
Using U.S. cities a cases, there is a strong positive correlation
between the number of churches and the incidence of violent
crime
Does this mean churches cause violent crime, or violent crime
causes more churches to be built?
More likely, both related to population of city (3d variable -lurking or confounding variable)
Coefficient of Determination
r2 tells the proportion of variance that two
variables in a bivariate distribution have in
common.
Introduction to Linear Regression
linear regression is the statistical procedure of
estimating the linear relationship between Y and X
Y = a + bX,
where X and Y are variables representing scores on the Y and X
axes,
b = slope of the regression line
a = intercept of the line with Y axes
positive slope -- if the highest point on the line is to the right of the
lowest point
negative slope – if the highest point on the line is to the left of the
lowest point
Regression Coefficients
Sy
b = r Sx, where
r = correlation coefficient for X and Y
Sy = standard deviation of the Y variable
Sx = standard deviation of the X variable
a = Y – bX, where
Y = mean of the Y scores
b = regression coefficient
X = mean of the X scores
Introduction to Probability
What are the odds that a card drawn at random
from a deck of cards will be an ace?
the probability is 4/52
Theoretical Distributions and
Theoretical Probabilities
• probabilities range from .000 to 1
• the expression p = .077 means that there are 7.7 chances in 100 of
the event to occur.
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2
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Ace 2
3
4
5
6
7
8
9
10
J
Q
K
Theoretical and Empirical
Distributions
• theoretical distributions
(logic, math)
• empirical distributions
(observation)
The Standard Normal Distribution
1. the mean, median and mode
are the same score on the X
axis where the curve peaks
2. the area to the left or right of
the line = 50% of the total
area
3. the tails of the curve are
asymptotic to the X axis –
they never cross the axis but
continue in both directions
indefinitely
4. the inflection points are
where the curve is the
steepest (-1s and +1s)
The Normal Distribution table
(Table C in Appendix C)
The Table C can be used to determine areas
(proportions) of a normal distributions and obtain the
probability figures.
• Column A contains a z score
• Column B -- the area b/w the mean and the z-score
• Column C -- the area beyond the z score
• all the proportions hold for - z because the curve is
symmetrical
EXAMPLE: The College Frisbee Golf Course
On the average, it takes 27 throws to complete the
College Frisbee Golf Course. The standard deviation
about this mean is 4.
Questions that we can answer:
1. finding the proportion of a population that has
scores of a particular size (e.g., what proportion of
population would be expected to score 22 or less?)
– interpretation: there are XX.XX chances in 100
that players would have 22 or less throws.
2. finding the score that separates the population
into two proportions (e.g., if 950 students played
and a prize was given for scoring 20 or less, how
many would get prizes?)
– interpretation: XX of the 950 people would get
prizes.
Example: The College Frisbee Golf
Course. Cont’d
3.
finding the extremes scores in a Population (e.g.,
suppose an experimenter wanted to find some very
good Frisbee players to use in an experiment. She
decided to use the top 10 percent from the CFGC.
What is the cut-off score?)
• interpretation: in order to be considered as a very
good Frisbee player, you need to have a XXX score
or higher.
4. finding the proportion of the population between
two scores (e.g., what proportion would score
between 25 and 30?)
• interpretation: there are XX.XX chances in 100 that
players would score between 25 and 30.
Samples, Sampling Distributions,
and Confidence Intervals
Sampling
The essential idea of sampling is to learn about
the whole by studying a part
Two important terms:
1.
2.
•
population – the entire group of people or objects
sample -- a (typically small) part of the population
Biased Sampling -- a tendency to systematically
overrepresent / underrepresent certain segments
of the population
– convenience samples
– voluntary response
Random Samples
• Random sample – any method that allows every
possible sample of size N an equal chance to be
selected.
• A method of getting a random sample -- a table of
random numbers (Table B in Appendix C):
– each position is equally likely to be occupied by any one of
the digits 0,1,2,3,4,5,6,7,8,9
– the occupant of any one position has no impact on the
occupant of any other position
Sampling Distributions: Describing
the Relationships b/w X and 
• expected value is the mean of a
sampling distribution
• standard error is the standard
deviation
• the sampling distribution of the
mean:
– every sample is drawn randomly from
a specific population
– the sample size (N) is the same for
all samples
– the number of samples is very large
– the mean (X) is calculated for each
sample
– the sample means are arranged into
a frequency distribution
Central Limit Theorem
• The sampling distribution of the mean
approaches a normal curve as N increases.
• If you know , and s for the population:
– the mean of the sampling distribution (expected
value) = ,
– the standard deviation of the sampling distribution
(standard error) = s.
symbols:
sx – the standard error of the mean
E (X) – the expected value of the mean
Calculating the Standard Error
of the Mean
Determining Probabilities About
Sample Means
z=
X–
sx
A test of computer anxiety has a population mean of 65 and a
standard deviation of 12. A random sample of size 36 is drawn from
the population with a sample mean of 68. What is the probability of
selecting a sample with a smaller mean?
1.
2.
Compute the standard error of the mean: 12/ SQRT(36) = 12/6 = 2
Compute a standard score for the mean of 68: Z = (68 - 65)/2 = 3/2 = +1.5
3.
The area between the mean and a Z-score of +1.5 is .4332
4.
Thus, the probability of a score below 68 is .5000 + .4332 = .9332
The t distribution
• when you do not know s and you don’t have population data to
calculate it – use s as an estimate of s.
• t distribution – depends on the sample size with different
distributions for each N.
– degree of freedom: df = N – 1, ranges from 1 to ∞
– t values are similar to the z scores used with the normal curve
– as df increases, the t distribution approaches the normal distribution
 Table D in Appendix C:
– first column shows the degree of freedom
– the top row is for confidence intervals
– each column is associated with a percent of probability
Confidence Intervals
• a confidence interval establishes an interval within which a
population parameter is expected to lie (with a certain degree of
confidence)
• confidence intervals for population means produce an interval
statistic (lower and upper limits) that is destined to contain  95
percent of the time
LL = X – t(sx)
UL = X + t(sx)
where X is the mean of the sample from the population
t is a value from the t distribution table
sx is the standard error of the mean, calculated from a
sample
EXAMPLE: Confidence Intervals
•


•
There are several published tests of self-efficacy. Suppose that the
population mean for a particular test is 20. A therapy class taught by
a graduate student had just finished the term. A psychologist wanted
to determine the effect of the course on students’ sense of selfefficacy. The following statistics were produced. Use a 95 percent
confidence interval to analyze the data. Write a conclusion about the
effect of the class.
N = 36
SX = 684
SX2 = 13,136
Interpretation: The population mean for the test is 20, and the 95
percent confidence interval about the mean of all those taking the
workshop is entirely below 20. You can conclude (with 95 percent
confidence) that the workshop was not sufficient to increase selfefficacy.
Hypothesis Testing: OneSample Designs
Hypothesis Testing
•


•
the Frito-Lay® company claims that bags of Doritos®
tortilla chips contain 269.3 grams. When you buy a bag
of chips, how much do you get?
0 - a parameter that describes a population (company
standard)
1 - a parameter that describes a population of actual
weights
there are three possible relationships b/w 1 and 0:
1.
2.
3.
1 = 0
1 < 0
1 > 0
Outline
1.
2.
gather sample data from the population you are
interested in and calculate a statistic
recognize two logical possibilities for the population:
1.
2.
3.
4.
H0: a statement specifying an exact value for the parameter of
the population
H1: a statement specifying all other values for the parameter
using a sampling distribution that assumes H0 is
correct, find the probability of the statistic you
calculated
if the probability is very small, reject Ho and accept H1.
If the probability is large, retain both H0 and H1,
acknowledging that the data do not allow you to reject
H0
Establishing a significance level:
Setting Alpha (a)
• significance level is the choice
of a probability level
• rejection region
• critical values: For example,
t.05 (14 df) = 2.145
– the sampling distribution that
was used (t)
 a level (.05)
– degree of freedom (14)
– critical value from the table
(2.145, for a two-tailed test)
The one-sample t test
X – 0
t=
sx
; df = N – 1 , where
X is the mean of the sample
0 is the hypothesized mean of the
population
sx is the standard error of the mean
EXAMPLE 1: “Doritos bags”,
one-sample t-test
1.
2.
3.
4.
the population mean is that of all bags; the sample
mean (N = 8) is 270.675 grams
H0: the mean weight of Doritos bags is 269.3 grams,
the weight claimed by the Frito-Lay company.
H1: the mean weight of Doritos bags is not 269.3
grams
the t distribution for 7 df (N-1) (Table D in Appendix C)
if the probability is low – reject H0, meaning that the
mean of Doritos bags is NOT 269.3 grams
if the probability is high – reject H1, meaning that the
mean weight of Doritos bags could be 269.3 grams
EXAMPLE 2: “Misinformation test”
one-sample t-test
• A psychologist who taught the introductory
psychology course always gave her class a
"misinformation test" on the first day of class.
Her test contained some commonly held
incorrect beliefs about psychology. Over the
years the mean number of errors on this test had
been 21.50. The data for this year follow.
Analyze the data with a t test calculate the effect
size index, and write a conclusion.
 X = 198
X2 = 4700
N = 11
Effect size index: How big is a
difference?
d =
where
|X – 0|
s
X is the mean of the sample
0 is the mean specified by the null hypothesis
s is the standard deviation of the null hypothesis
population
• small effect
d = .20
• medium effect
d = .50
• large effect
d = .80
Type I and Type II errors
Four possibilities:
1.
The H0 is true but test rejects it ( Type I error)
2.
The H0 is false but test accepts it (Type II error)
3.
The H0 is true and test accepts it ( correct decision )
4.
The H0 is false and test rejects it (correct decision)
•
•
the probability of Type I error is denoted by a (alpha)
the probability of Type II error is denoted by b (beta)
One- and two-tailed tests
1.
a two-tailed test of
significance:
•
2.
the sample is from a
population with a mean less
than/greater than that of the
Ho
a one-tailed test of
significance:
•
•
the sample is from a
population with a mean less
than that of the Ho
the sample is from a
population with a mean
greater than that of the Ho
a, Type I error and p-value??
a is the probability of a Type I error (e.g., a =
.05)
a Type I error refers to when we mistakenly
reject Ho.
p-value is the probability of obtaining the
sample statistic actually obtained, if Ho is
true
Testing the statistical significance
of correlation coefficients
• definition formula:
r-r
t = sr ; sr = √(1-r2)/(N-2)
• working formula:
N-2
t = (r)√ 1-r2 ; df = N – 2, where N = number of pairs
• Two uses of the t distribution:
– to test a sample mean against a hypothesized population mean
– to test a sample correlation coefficient against a hypothesized
population correlation coefficient of .00
Hypothesis Testing: TwoSample Designs
Logic
1.
begin with two logical possibilities:
•
•
2.
3.
4.
assume that Treatment A has no effect (that is, assume Ho)
decide on an alpha level (e.g., a = .05).
choose an appropriate inferential statistical test:
•
•
•
5.
6.
7.
Ho: A= no A -- treatment A does not have an effect, the difference
between population means is zero
H1: A= no A – treatment A does have an effect, the difference
between population means is not zero
test statistic (e.g., mean)
a sampling distribution of the test statistic (when Ho is true) (e.g.,
t distribution)
a critical value for the alpha level
calculate a test statistic using the sample data
compare a test statistic to the critical value from the sampling
distribution
write a conclusion
Independent - sample designs
H0: 1 = 2
H1: 1 = 2
t=
X1 – X2
sX1 – X2
where,
sX1 – X2 is the standard error of a difference
df = N1 + N2 – 2
assumptions:
1.
the DV scores are normally distributed and have equal
variances
2.
the samples are randomly assigned/selected
EXAMPLES: Two-sample t-tests,
independent design
Example “Achievement Test Scores” (equal
N's design)
"Achievement test scores are declining all
around us," brooded Professor Probity. "Not in
my class," vouchsafed Professor Paragon.
"Here are my final-exam scores for last year
and this year on the same exam. Run your own
t test on them. Calculate the effect size index."
What did Probity discover?
Example “Yummi Very Vanilla” (unequal N's
design)
An experimenter randomly divided a group of
volunteers into two groups. One group fasted
for 24 hours and the other for 48 hours. One
person in the 24-hour group dropped out of the
study. Scores below represent the number of
ounces of Yummi (Very Vanilla) consumed
during the first 10 minutes after the fast.
Perform a t test, calculate d, and write a
sentence summary of your results.
Paired (correlated) - samples
designs
• natural pairs – we do not assign the participants to one
group or the other
• matched pairs – we do assign the participants to one
group or the other
• repeated measures – more than one measure is taken
on each participant (e.g., before-and-after experiment)
X-Y
• t = sD
where,
• sD = √sX2 + sY2 – 2rXY (sX)(sY)
df = N -1, where N = number of pairs
EXAMPLE: “Rats Learning a Simple Maze”
two-sample t-test, paired design
•
A comparative psychologist was interested in the effect of X-irradiation
on learning. A group of rats learned a simple maze and were paired on
the basis of the number of errors. One group was X-irradiated; then both
groups learned a new, more complicated maze, and errors were
recorded. Identify the independent and dependent variables. Test for a
difference between the two groups. Find the effect size index. Write a
conclusion for the study.
Effect size index
• effect size index for independent samples:
when N1 is equal N2:
|X1 – X2|
s-hat = √N1(sX1 – X2)
d = s-hat
when N1 is not equal N2:
s-hat12(df1) + s-hat22(df2)
s-hat = √
df1 + df2
• effect size index for paired (correlated) samples:
|X – Y|
s-hatd = √N(sD), where N is the number
d = s-hatd
of pairs of participants
• interpretation of d:
 d = .20 – small effect
 d = .50 – medium effect
 d = .80 – large effect
Establishing a confidence interval
about a mean difference
• confidence interval for independent samples:
– LL = (X1 – X2) – t (sX1-X2)
– UL = (X1 – X2) + t (sX1-X2)
– interpretation: we can expect, with 95 percent confidence, that
the true difference between [use the terms of your experiment] is
between XX to AA.
• confidence interval for correlated samples:
– LL = (X – Y) – t (sD)
– UL = (X – Y) + t (sD)
Power
• Power = 1 – b
• Factors that affect the power:
– effect size
– the standard error of a difference:
• sample size
• sample variability
– alpha (a)
To allocate plenty of power, use large N’s!
Analysis of Variance: OneWay Classification
Example: “Social status and attitudes
toward fate”, one-way ANOVA
•
•
•
•
The following hypothetical data
are scores from a test which
measures a person's attitudes
toward fate. (An example of such
a test is Rotter's internal-external
scale.) A high score indicates that
the person views fate as being out
of his or her control. Low scores
indicate that he or she feels
directly responsible for what
happens. Test for differences
among the three social classes.
Fill in:
Independent variable_________
Dependent variable_________
Lower Class
Middle Class
Upper Class
12
13
6
10
4
5
11
9
8
9
6
2
7
12
2
3
10
1
Analysis of Variance: One-Way
Classification
estimate of s2
F = estimate of s2
variation between treatment means
variation within treatments
1. F is a ratio of two estimates of the population variance
– a between-treatments variance over an error
variance.
2. when Ho is true, both variances are good estimators of
the population variance (ratio is about 1)
3. when Ho is not true, the between – treatments
variance over-estimates the population variance (ratio
is greater than 1)
New Terms
• Sum of squares (SS) – sum of squared
deviations. S(X – X)2
• Mean square (MS) is the ANOVA term for a
variance s-hat2
• Grand mean – is the mean of all scores
• tot – the symbol stands for all such numbers in
the experiment (SXtot)
• t – the symbol applies to a treatment group (SXt)
• K – is the number of treatments in the
experiment (number of levels of the IV)
• F test – is the result of an analysis of variance
Factorial ANOVA
Analysis of Variance: Factorial
Design
•
Suppose you wanted to see if the number of bedrooms as well as a house’s style
and the interaction between a style and number of bedrooms has an effect on the
house price.
traditional
1 year
modern
30 years
1 BR
250K
300K
275K
1 BR
279K
189K
234K
5 BR
500K
550K
525K
5 BR
500K
250K
375K
375K
425K
389.5K
219.5K
600
1 BR
600
1 BR
5 BR
5 BR
500
500
400
400
300
300
200
200
100
100
0
0
1
2
1 year
30 years
Factorial Design Notation
• shorthand notations for factorial designs:
“2 x 2”, “3 x 2” ,etc.
Factor A
B1
Factor B
1 BR
B2
5 BR
A1
A2
1 year
30 years
Cell A1B1
279K
Cell A2B1
189K
234K
Cell A1B2
500K
Cell A2B2
250K
375K
389.5K
219.5K
Advantages of two-way ANOVA:
• valuable resources can be spent
more efficiently by studying two
factors simultaneously rather
than separately
• we can investigate interactions
between factors
Exercise 1
• Identify the response variable, factors,
state the number of levels for each factor,
and the total number of observations (N).
(a) A study of productivity of tomato plant
compares four varieties of tomatoes and
two types of fertilizer. Five plants of each
variety are grown with each type of
fertilizer. The yield in pounds of tomatoes
is recorded for each plant.
Exercise 2
• Identify the response variable, factors, state the
number of levels for each factor, and the total
number of observations (N).
(b) A marketing experiment compares six
different types of packaging for a laundry
detergent. A survey is conducted to determine
the attractiveness of the packaging in four
different parts of the country. Each type of
packaging is shown to 30 different consumers in
each part of the country, who rate the
attractiveness of the product on a 1 to 10 scale.
Exercise 3
• Identify the response variable, factors,
state the number of levels for each factor,
and the total number of observations (N).
(c) To compare the effectiveness of three
different weight-loss programs, five men
and five women are randomly assigned to
each other. At the end of the program, the
weight loss for each of the participants is
recorded.
Exercise 4
• Numbers in the cell are means based on 8
scores for each cell.
• Data Set is an example of:
a. a simple ANOVA
b. a 2 x 2 factorial ANOVA
c. a 3 x 3 factorial ANOVA
Exercise 5
• The figure that shows an interaction is
Growth and age
• Imagine a two-way factorial design to study the
following scientific hypothesis: “Toddlers get
taller; adults don’t.” Here’s a quick summary:
Response: Height in inches
Factor 1: Age groups – 2-year-olds and
adults
Factor 2: Time – at the start of the study, and
three years later
Make a two-way table summarizing the results you
would expect to get from this study. Then draw
and label a graph showing the interaction
pattern.