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Transcript
```Statistics for Business and
Economics
Module 1:Probability Theory and Statistical Inference
Spring 2010
Lecture 1
Priyantha Wijayatunga, Department of Statistics, Umeå University
[email protected]
These materials are altered ones from copyrighted lecture slides (© 2009 W.H.
Freeman and Company) from the homepage of the book:
The Practice of Business Statistics Using Data for Decisions :Second Edition
by Moore, McCabe, Duckworth and Alwan.
Introduction
 What is statistics? Use of data
 Population and sample
 Types of variables
 Graphical display of data
 Numerical summeries of data
 Obtaining data
What is statistics? Use of data
Science of Statistics provides collection of methods and tools for collection,
organization, analysis, interpretion and presention of data that are usually
numerical figures.
The goal of statistics is to gain understanding of the phenomenon from data.
Therefore we use data to understand the phenomenon
Collection (design): planning how to gather data
Presentation (description): summerizing data
Analysis and interpretation (inference): making prediction based on data
Population and sample
Popuation is the collection of all the individuals/items under study.
Sample is a subcollection (a part) individuals/items of the population.
Eg: If we study the life age structure of the Swedes then
Population: all the Swedes
A sample: all the Swedes living in Umeå
Parameter: a parameter is a numerical summary of the population.
Statistic: a numerical summary of a data sample.
– when parameters are not known they are estimated by corresponding
sample statistics
Population versus sample

Population: The entire group
of individuals in which we are
interested but can’t usually
assess directly.

Sample: The part of the
population we actually examine
and for which we do have data.
How well the sample represents
the population depends on the
sample design.
Example: All humans, all
working-age people in
California, all crickets
Population
Sample

A parameter is a number
describing a characteristic of
the population.

A statistic is a number
describing a characteristic of a
sample.
Variables
In a study, we collect information—data—from individuals/items. They
can be people, animals, plants, or any object of interest.
A variable is any characteristic of an individual/item.
A variable usually varies among individuals/items.
Example: age, height, blood pressure, ethnicity, leaf length, first language
The distribution of a variable tells us what values the variable takes and
how often it takes these values.
Two types of variables

Variables can be either quantitative…

Something that takes numerical values for which arithmetic operations
such as adding and averaging make sense

Example: How tall you are, your age, your blood cholesterol level, the
number of credit cards you own

… or categorical (qualitative).

Something that falls into one of several categories. What can be counted
is the count or proportion of individuals/items in each category.

Example: Your blood type (A, B, AB, O), your hair color, your ethnicity,
whether you paid income tax last tax year or not
How do you know if a variable is categorical or quantitative?
 What are the n individuals/units in the sample (of size “n”)?
 What is being recorded about those n individuals/units?
 Is that a number ( quantitative) or a statement ( categorical)?
Categorical
Quantitative
Each individual is
assigned to one of
several categories.
Each individual is
attributed a
numerical value.
Individuals
in sample
DIAGNOSIS
AGE AT DEATH
Patient A
Heart disease
56
Patient B
Stroke
70
Patient C
Stroke
75
Patient D
Lung cancer
60
Patient E
Heart disease
80
Patient F
Accident
73
Patient G
Diabetes
69
Two types of variables
1.
Qualitative (Categorical) : Measuremnet scale is a set of categories.
1. Nominal: unordered categories
[discrete]
(residence
{Umeå, Stockholm,Uppsala} )
2. Ordinal: ordered categories
[discrete]
( income
{Lower, Middle, Upper} )
2.
Quantitative: Measurement scale has numerical values and they represent the
magnitude of the variable
1. Interval: has a numerical distance between values
(prison term
{<5, 5–10, 10–15, 15–20, 20>}
or number of times married
{0, 1, 2, 3, 4, >5})
2. Continuous: Weight
Variables are either
1. Discrete: has a set of separate numbers (number of children)
(values cannot be subdivided) or
2. Continuous: has infinite continuum of real numbers (weight)
(between any two possible values, there áre many possible values)
Displaying distributions/data with graphs
(Graphical summary of data)


Ways to chart categorical data

Bar graphs

Pie charts
Ways to chart quantitative data

Histograms

Interpreting histograms

Stemplots

Stemplots versus histograms

Time plots
Ways to chart categorical data
Because the variable is categorical, the data in the graph can be
ordered any way we want (alphabetical, by increasing value, by year,
by personal preference, etc.)

Bar graphs
Each category is
represented by
a bar.

Pie charts
The slices must
represent the parts of one whole.
Bar graphs
Each category is represented by one bar. The bar’s height shows the count (or
sometimes the percentage) for that particular category.
Accidents involving Firestone tire models
Bar graph sorted by rank (Pareto Chart)
 Easy to analyze
Sorted alphabetically
 Much less useful
Pie charts
Each slice represents a piece of one whole. The size of a slice depends on what
percent of the whole this category represents.
Child poverty before and after government
intervention—UNICEF, 1996
What does this chart tell you?
•The United States has the highest rate of child
poverty among developed nations (22% of under 18).
•Its government does the least—through taxes and
subsidies—to remedy the problem (size of orange
bars and percent difference between orange/blue
bars).
Could you transform this bar graph to fit in 1 pie
chart? In two pie charts? Why?
The poverty line is defined as 50% of national median income.
Ways to chart quantitative data

Histograms and stemplots
These are summary graphs for a single variable. They are very useful to
understand the pattern of variability in the data.

Line graphs: time plots
Use when there is a meaningful sequence, like time. The line connecting
the points helps emphasize any change over time.
Histograms
The range of values that a
variable can take is divided
into equal size intervals.
The histogram shows the
number of individual data
points that fall in each
interval.
Example: Histogram of the
December 2004 unemployment
rates in the 50 states and
Puerto Rico.
Interpreting histograms
When describing the distribution of a quantitative variable, we look for the
overall pattern and for striking deviations from that pattern. We can describe
the overall pattern of a histogram by its shape, center, and spread.
Histogram with a line connecting
each column  too detailed
Histogram with a smoothed curve
highlighting the overall pattern of
the distribution
Most common distribution shapes

Symmetric
distribution
A distribution is symmetric if the right and left
sides of the histogram are approximately mirror
images of each other.

A distribution is skewed to the right if the right
side of the histogram (side with larger values)
extends much farther out than the left side. It is
skewed to the left if the left side of the histogram
Skewed
distribution
extends much farther out than the right side.
Complex,
multimodal
distribution

Not all distributions have a simple overall shape,
especially when there are few observations.
Outliers
An important kind of deviation is an outlier. Outliers are observations
that lie outside the overall pattern of a distribution. Always look for
outliers and try to explain them.
The overall pattern is fairly
symmetrical except for 2
states clearly not belonging
to the main trend. Alaska
and Florida have unusual
representation of the
elderly in their population.
A large gap in the
distribution is typically a
sign of an outlier.
Florida
How to create a histogram
It is an iterative process – try and try again.
What bin size should you use?

Not too many bins with either 0 or 1 counts

Not overly summarized that you loose all the information

Not so detailed that it is no longer summary
 rule of thumb: start with 5 to10 bins
Look at the distribution and refine your bins
(There isn’t a unique or “perfect” solution)
Histogram of Drydays in 1995
IMPORTANT NOTE:
Your data are the way they are.
Do not try to force them into a
particular shape.
It is a common misconception
that if you have a large enough
data set, the data will eventually
turn out nice and symmetrical.
Stemplots (Stem–and–leaf plots)
How to make a stemplot:
1) Separate each observation into a stem, consisting of
all but the final (rightmost) digit, and a leaf, which is
that remaining final digit. Stems may have as many
digits as needed, but each leaf contains only a single
digit.
2) Write the stems in a vertical column with the smallest
value at the top, and draw a vertical line at the right
of this column.
3) Write each leaf in the row to the right of its stem, in
increasing order out from the stem.
STEM
LEAVES
Stemplot

To compare two related distributions, a back-to-back stemplot with
common stems is useful.

Stemplots do not work well for large datasets.

When the observed values have too many digits, trim the numbers
before making a stemplot.

When plotting a moderate number of observations, you can split
each stem.
Stemplots of the December 2004
unemployment rates in the 50 states.
(b) uses split stems.
Stemplots versus histograms
Stemplots are quick and dirty histograms that can easily be done by
hand, therefore very convenient for back of the envelope calculations.
However, they are rarely found in scientific or laymen publications.
Line graphs: time plots
In a time plot, time always goes on the horizontal, x axis.
We describe time series by looking for an overall pattern and for striking
deviations from that pattern. In a time series:
A trend is a rise or fall that
persists over time, despite
small irregularities.
A pattern that repeats itself
at regular intervals of time is
called seasonal variation.
Retail price of fresh oranges
over time
Time is on the horizontal, x axis.
The variable of interest—here
“retail price of fresh oranges”—
goes on the vertical, y axis.
This time plot shows a regular pattern of yearly variations. These are seasonal
variations in fresh orange pricing most likely due to similar seasonal variations in
the production of fresh oranges.
There is also an overall upward trend in pricing over time. It could simply be
reflecting inflation trends or a more fundamental change in this industry.
A time plot can be used to compare two or more
data sets covering the same time period.
1918 influenza epidemic
# Cases # Deaths
Date
1918 influenza epidemic
800 700
700 600
600 500
500 400
400 300
300 200
200 100
100
0
0
wek
ee1
we k 1
wek
ee3
we k 3
wek
ee5
we k 5
wek
ee7
we k
7
wek
we ee9k
ewk 9
ee11
we k 1
ewk 1
e1
we ek3
ewk 13
e1
we ek5
ewk 15
ee17
k
17
0
# deaths reported
800
10000
9000
10000
8000
9000
7000
8000
6000
7000
5000
6000
4000
5000
3000
4000
3000
2000
2000
1000
1000 0
we
0
0
130
552
738
414
198
90
56
50
71
137
178
194
290
310
149
Incidence
36
531
4233
8682
7164
2229
600
164
57
722
1517
1828
1539
2416
3148
3465
1440
# cases diagnosed
week 1
week 2
week 3
week 4
week 5
week 6
week 7
week 8
week 9
week 10
week 11
week 12
week 13
week 14
week 15
week 16
week 17
1918 influenza epidemic
# Cases
# Cases
# Deaths
# Deaths
The pattern over time for the number of flu diagnoses closely resembles that for the
number of deaths from the flu, indicating that about 8% to 10% of the people
diagnosed that year died shortly afterward from complications of the flu.
Scales matter
Death rates from cancer (US, 1945-95)
Death rates from cancer (US, 1945-95)
Death rate (per
thousand)
250
200
150
100
250
Death rate (per thousand)
How you stretch the axes and choose your
scales can give a different impression.
200
150
100
50
50
0
1940
1950
1960
1970
1980
1990
0
1940
2000
1960
1980
2000
Years
Years
Death rates from cancer (US, 1945-95)
250
Death rates from cancer (US, 1945-95)
220
Death rate (per thousand)
Death rate (per thousand)
200
150
100
50
0
1940
1960
Years
1980
2000
A picture is worth a
thousand words,
200
BUT
180
160
There is nothing like
hard numbers.
 Look at the scales.
140
120
1940
1960
1980
Years
2000
Describing distributions/data with
numbers (Numerical summary of data)

Measures of center (central location): mean, median, mode

Comparing mean, median and mode

Measures of spread: range, interquartile range, standard
deviation,

Five-number summary and boxplots

Choosing measures of center and spread
Measure of center : the mean
The mean or arithmetic average
To calculate the average, or mean, add
all values, then divide by the number of
individuals. It is the “center of mass.”
Sum of heights is 1598.3
divided by 25 women = 63.9 inches
58 .2
59 .5
60 .7
60 .9
61 .9
61 .9
62 .2
62 .2
62 .4
62 .9
63 .9
63 .1
63 .9
64 .0
64 .5
64 .1
64 .8
65 .2
65 .7
66 .2
66 .7
67 .1
67 .8
68 .9
69 .6
x1  x2  ...  xn
x
n
Example: Mean earnings
of Black females
1 n
x   xi
n i 1
262,934
x
 \$17,528.93
15
Measure of center: the median
The median is the midpoint of a distribution—the number such
that half of the observations are smaller and half are larger.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
25 12
6.1
1. Sort observations by size.
n = number of observations
______________________________
2.a. If n is odd, the median is
observation (n+1)/2 down the list
 n = 25
(n+1)/2 = 26/2 = 13
Median = 3.4
2.b. If n is even, the median is the
mean of the two middle observations.
n = 24 
n/2 = 12
Median = (3.3+3.4) /2 = 3.35
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
1
2
3
4
5
6
7
8
9
10
11
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
Measure of center : the Mode
The mode is the most frequent observation
Data: 5, 7, 6, 7, 6, 6, 4, 6, 5, 4
Mode is 6
Comparing the mean, the median and the mode
The mean and the median are the same only if the distribution is
symmetrical. The median is a measure of center that is resistant to skew
and outliers. The mean and mode are not.
Mean, median and mode
for a symmetric distribution
Mean
Median
Mode
Left skew
Mean
Median
Mode
Mean, median and mode
for skewed distributions
Mean
Median
Mode
Right skew
Mean and median of a distribution with outliers
Percent of people dying
x  3.4
x  4.2
Without the outliers
With the outliers
The mean is pulled to the
The median, on the other hand,
right a lot by the outliers
is only slightly pulled to the right
(from 3.4 to 4.2).
by the outliers (from 3.4 to 3.6).
Impact of skewed data
Mean and median of a symmetric
Disease X:
x  3.4
M  3.4
Mean and median are the same.
… and a right-skewed distribution
Multiple myeloma:
x  3.4
M  2.5
The mean is pulled toward
the skew.
Measures of spread: range &
Minimum = 0.6
interquartile range
The first quartile, Q1, is the value in the
sample that has 25% of the data at or
below it ( it is the median of the lower
half of the sorted data, excluding M).
The third quartile, Q3, is the value in the
sample that has 75% of the data at or
below it ( it is the median of the upper
half of the sorted data, excluding M).
Range = maximum – minimum
Interquartile range = Q3 – Q1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
6
7
1
2
3
4
5
1
2
3
4
5
6
7
1
2
3
4
5
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
6.1
Q1= first quartile = 2.2
(n+1)x25%th data
M = median = 3.4
Q3= third quartile = 4.35
(n+1)x75%th data
Maximum = 6.1
Five-number summary and boxplot
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
Largest = max = 6.1
BOXPLOT
7
Q3= third quartile
= 4.35
M = median = 3.4
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
4
3
2
1
Q1= first quartile
= 2.2
Smallest = min = 0.6
0
Disease X
Five-number summary:
min Q1 M Q3 max
Calculating quartiles
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
Here the sample size is n=25
Q3= third quartile = (25+1)x75%th data = 4.35
M = median = (25+1)x50%th data= 3.4
Q1= first quartile = (25+1)x25%th data=2.2
Boxplots for skewed data
Years until death
Comparing box plots for a normal
and a right-skewed distribution
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Boxplots remain
true to the data and
depict clearly
symmetry or skew.
Disease X
Multiple Myeloma
Side-by-side boxplots
Side-by-side boxplots comparing the earnings of four
groups of hourly workers at National Bank
Suspected outliers
Outliers are troublesome data points, and it is important to be able to
identify them.
One way to raise the flag for a suspected outlier is to compare the
distance from the suspicious data point to the nearest quartile (Q1 or
Q3). We then compare this distance to the interquartile range
(distance between Q1 and Q3).
We call an observation a suspected outlier if it falls more than 1.5
times the size of the interquartile range (IQR) above the first quartile or
below the third quartile. This is called the “1.5 * IQR rule for outliers.”
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
7.9
6.1
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
8
7
Q3 = 4.35
Distance to Q3
7.9 − 4.35 = 3.55
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
Interquartile range
Q3 – Q 1
4.35 − 2.2 = 2.15
4
3
2
1
Q1 = 2.2
0
Disease X
Individual #25 has a value of 7.9 years, which is 3.55 years above
the third quartile. This is more than 3.225 years, 1.5 * IQR. Thus,
individual #25 is a suspected outlier.
Measure of spread: the standard deviation
The standard deviation “s” is used to describe the variation around the
mean. Like the mean, it is not resistant to skew or outliers.
1. First calculate the variance s2.
n
1
2
s2 
(
x

x
)

i
n 1 1
2. Then take the square root to get
the standard deviation s.
x
Mean
± 1 s.d.
1 n
2
s
(
x

x
)
 i
n 1 1
Calculations …
Women height (inches)
i
xi
x
(xi-x)
(xi-x)2
1
59
63.4
-4.4
19.0
2
60
63.4
-3.4
11.3
3
61
63.4
-2.4
5.6
4
62
63.4
-1.4
1.8
5
62
63.4
-1.4
1.8
6
63
63.4
-0.4
0.1
7
63
63.4
-0.4
0.1
8
63
63.4
-0.4
0.1
9
64
63.4
0.6
0.4
10
64
63.4
0.6
0.4
11
65
63.4
1.6
2.7
Degrees freedom (df) = (n − 1) = 13
12
66
63.4
2.6
7.0
s2 = variance = 85.2/13 = 6.55 inches squared
13
67
63.4
3.6
13.3
14
68
63.4
4.6
21.6
Sum
0.0
Sum
85.2
1
s
df
n
 (x
i
 x)
2
1
Mean = 63.4
Sum of squared deviations from mean = 85.2
s = standard deviation = √6.55 = 2.56 inches
Mean
63.4
Properties of standard deviation

s measures spread about the mean and should be used only when
the mean is the measure of center.

s = 0 only when all observations have the same value and there is
no spread. Otherwise, s > 0.

s is not resistant to outliers.

s has the same units of measurement as the original observations.
Software output for summary statistics:
Excel - From Menu:
Tools/Data Analysis/
Descriptive Statistics
Give common
statistics of your
sample data.
Minitab
Choosing measures of center and spread

Because the mean is not
Height of 30 Women
resistant to outliers or skew, use
69
it to describe distributions that are
68
fairly symmetrical and don’t have
 Plot the mean and use the
standard deviation for error bars.

Otherwise use the median in the
five number summary which can
be plotted as a boxplot.
Height in Inches
outliers.
67
66
65
64
63
62
61
60
59
58
Box Plot
Boxplot
Mean ±
+/-SD
SD
Mean
What should you use, when, and why?
Arithmetic mean or median?

Middletown is considering imposing an income tax on citizens. City Hall
wants a numerical summary of its citizens’ income to estimate the total tax
base.


Mean: Although income is likely to be right-skewed, the city government
wants to know about the total tax base.
In a study of standard of living of typical families in Middletown, a sociologist
makes a numerical summary of family income in that city.

Median: The sociologist is interested in a “typical” family and wants to
lessen the impact of extreme incomes.
Obtaining data

Observation versus Experiment and association versus causation

Population versus sample

Sampling methods

Simple random samples

Stratified samples

Probelms in sampling
Data
Available data are data that were produced in the past for some other
purpose but that may help answer a present question inexpensively.
The library and the Internet are sources of available data.
Government statistical offices are the primary source for demographic,
economic, and social data (visit the Fed-Stats site at www.fedstats.gov).
When data are not available conduct a statistical survey or experiment
Observation and experiment
Observational study: Record data on individuals without attempting to
influence the responses.
Example: Watch the behavior of consumers looking at store displays,
or the interaction between managers and employees.
Experimental study: Deliberately impose a treatment on individuals
and record their responses. Influential factors can be controlled.
Example: To answer the question “Which TV ad will sell more
toothpaste?” show each ad to a separate group of consumers and
note whether they buy toothpaste.
Observational studies are essential sources of data on a variety of
topics. However, when our goal is to understand cause and effect,
experiments are the only source of fully convincing data.
Association or cause–effect
If ”early retirement” and ”early death” are found to be related (associative or
causative or both)?
That is,

Does ”early retirement” causes ”early death”?

Does ”personal health” or similar thing causes the both simultaneously?

Are both happening?
Collecting data: statistical surveys
Sampling plans
Population size=N;
1.
2.
3.
4.
sample size=n;
k=N/n
Simple random sampling: subjects are selected with equal chances (n/N)
Systematic random sampling: selected the first one from first k subjects in the
population with equal chances and then select every kth subject until the
sample is completed
Stratified random sampling: divide population into several groups (strata) and
then select a random sample from each of them (Can be either proportional or
disproportional)
Cluster random sampling: divide the population into a large numebr of clusters
and then select a random sample of clusters. Use all the subjects in them
Sampling variability
Each time we take a random sample from a population, we are likely to
get a different set of individuals and a calculate a different statistic. This
is called sampling variability.
The good news is that, if we take lots of random samples of the same
size from a given population, the variation from sample to sample—the
sampling distribution—will follow a predictable pattern. All of
statistical inference is based on this knowledge.
The variability of a statistic is described by the spread of its sampling
distribution. This spread depends on the sampling design and the
sample size n, with larger sample sizes leading to lower variability.
 Large random samples are almost always close estimates of the true
population parameter. However, this only applies to random samples.
Remember the “QuickVote” online surveys.
They are worthless no matter how many people
participate because they use a voluntary
sampling design and not random sampling.
Bias
The study design is biased if it systematically favors certain outcomes
There are many types of biases: sampling bias, response bias, etc.
Managing bias and variability

To reduce bias, use random sampling. The values of a statistic
computed from an SRS neither consistently overestimate or
underestimate the value of a population parameter.

To reduce the variability of a statistic from an SRS, use a larger
sample. You can make the variability as small as you want by taking
a large enough sample.

Population size doesn’t matter: The variability of a statistic from a
random sample does not depend on the size of the population, as
long as the population is at least 100 times larger than the sample.
Practical note
Large samples are not always attainable.

Sometimes the cost, difficulty, or preciousness of what is studied limits
drastically any possible sample size

Blood samples/biopsies: No more than a handful of repetitions
acceptable. We often even make do with just one.

Opinion polls have a limited sample size due to time and cost of
operation. During election times though, sample sizes are increased
for better accuracy.
Caution about sampling surveys

Nonresponse: People who feel they have something to hide or
who don’t like their privacy being invaded probably won’t answer,
yet they are part of the population.

Response bias: Fancy term for lying when you think you should not
tell the truth, or forgetting. This is particularly important when the
questions are very personal (e.g., “How much do you drink?”) or
related to the past.

Wording effects: Questions worded like “Do you agree that it is
awful that…” are prompting you to give a particular response.
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