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Sampling, Normal Curve,
and Hypotheses
in Quantitative Research
Types of Sampling
?Simple Random Sample
?Stratified Random Sample
?Cluster sampling
?Systematic
?Convenience
Simple Random Sample
Every subset of a specified size n from the population has an equal chance of being selected.
Stratified Random Sample
The population is divided into two or more groups called strata, according to some criterion,
such as geographic location, grade level, age, or income, and subsamples are randomly selected
from each strata.
Cluster Sample
The population is divided into subgroups (clusters) like families. A simple random sample is
taken of the subgroups and then all members of the cluster selected are surveyed.
Systematic Sample
Every kth member ( for example: every 10th person) is selected from a list of all population
members.
Convenience Sample
Selection of whichever individuals are easiest to reach.
It is done at the “convenience” of the researcher.
Now you decide:
• including 5 people from every sports
team on a collegiate campus
• including every teacher from 4
elementary schools chosen from a
group of 11 elementary schools in a
school district with 45 elementary
schools total
• including 25 employees whose names were
drawn from a hat 250 school employees
• including all people who attend parent-teacher
conferences
• including every 20th student from a list of
2000 students in a particular high school
Errors in Sampling
Non-Observation Errors
? Sampling error: naturally occurs
? Coverage error: people sampled do not match the
population of interest
? Underrepresentation
? Non-response: won’t or can’t participate
As the researcher, you will never eliminate ALL
elements of BIAS…but it is your job to minimize
the impact of bias on your research project by
carefully planning out your research design.
The Normal Curve
The Normal Distribution:
The Normal curve is a mathematical abstraction
which conveniently describes (“models”) many
frequency distributions of scores in real-life.
length of time before someone
looks away in a staring contest:
length of pickled gherkins:
Francis Galton (1876) ‘On the height and weight of boys aged 14, in town and
country public schools.’ Journal of the Anthropological Institute, 5, 174-180:
Francis Galton (1876) ‘On the height and weight of boys aged 14, in town and
country public schools.’ Journal of the Anthropological Institute, 5, 174-180:
Height of 14 year-old children
16
country
town
14
10
8
6
4
2
0
51
-5
2
53
-5
4
55
-5
6
57
-5
8
59
-6
0
61
-6
2
63
-6
4
65
-6
6
67
-6
8
69
-7
0
frequency (%)
12
height (inches)
Frequency of different wand lengths
An example of a normal distribution – the length of
Sooty’s magic wand…
Length of wand
Properties of the Normal Distribution:
1. It is bell-shaped and asymptotic at the extremes.
2. It’s symmetrical around the mean.
3. The mean, median and mode all have same value.
4. It can be specified completely, once mean and SD
are known.
5. The area under the curve is directly proportional
to the relative frequency of observations.
e.g. here, 50% of scores fall below the mean, as
does 50% of the area under the curve.
e.g. here, 85% of scores fall below score X,
corresponding to 85% of the area under the curve.
Relationship between the normal curve and the
standard deviation:
frequency
All normal curves share this property: the SD cuts off a
constant proportion of the distribution of scores:-
68%
95%
99.7%
-3
-2
-1
mean
+1
+2
+3
Number of standard deviations either side of mean
About 68% of scores fall in the range of the mean plus and minus 1 SD;
95% in the range of the mean +/- 2 SDs;
99.7% in the range of the mean +/- 3 SDs.
e.g. IQ is normally distributed (mean = 100, SD = 15).
68% of people have IQs between 85 and 115 (100 +/- 15).
95% have IQs between 70 and 130 (100 +/- (2*15).
99.7% have IQs between 55 and 145 (100 +/- (3*15).
68%
85 (mean – 1 SD)
115 (mean + 1 SD)
We can tell a lot about a population just from knowing
the mean, SD, and that scores are normally distributed.
If we encounter someone with a particular score, we can
assess how they stand in relation to the rest of their
group.
e.g. someone with an IQ of 145 is quite unusual (3 SDs
above the mean).
IQs of 3 SDs or above occur in only 0.15% of the
population [ (100-99.7) / 2 ].
Population ? all possible values
Sample ? a portion of the population
Statistical inference ? generalizing from a
sample to a population with calculated
degree of certainty
Two forms of statistical inference
? Hypothesis testing
? Estimation
Parameter ? a characteristic of population, e.g.,
population mean µ
Statistic ? calculated from data in the sample, e.g.,
sample mean ( )
P-hat ? a sample proportion, symbolized by
Distinctions Between Parameters and Statistics
(Vocabulary Review)
Parameters
Statistics
Source
Population
Sample
Notation
Greek (e.g., µ) Roman (e.g., xbar)
Vary
No
Yes
Calculated
No
Yes
Hypothesis Testing Steps
A.Null and alternative hypotheses
B.Significance level
C.Test statistic
D.P-value and interpretation
General Example:
A criminal trial is an example of hypothesis testing
without the statistics.
In a trial a jury must decide between two hypotheses. The
null hypothesis is
H0: The defendant is innocent
The alternative hypothesis or research hypothesis is
H1: The defendant is guilty
The jury does not know which hypothesis is true. They
must make a decision on the basis of evidence presented.
In the language of statistics convicting the defendant is
called rejecting the null hypothesis in favor of the
alternative hypothesis. That is, the jury is saying that
there is enough evidence to conclude that the defendant
is guilty (i.e., there is enough evidence to support the
alternative hypothesis). We say, “We reject the null.”
If the jury acquits it is stating that there is not enough
evidence to support the alternative hypothesis. Notice that
the jury is not saying that the defendant is innocent, only
that there is not enough evidence to support the
alternative hypothesis. That is why we never say that we
accept the null hypothesis…we say, “We fail to reject the
null.” (Although non-stats people often do this wrong!)
Specific Example:
Crazy guy…but good at explanations!
Another Specific Example:
A department store manager determines that a new
billing system will be cost-effective only if the mean
monthly account is more than $170.
What null and alternative hypotheses
can we write for this situation?
The system will be cost effective if the mean account
balance for all customers is greater than $170.
We express this belief as a our research hypothesis, that is:
H1: µ > 170 (this is what we want to determine)
Thus, our null hypothesis becomes:
H0: µ < 170 (we assume is true until proven otherwise) Interpretation P-value answer the question: What is the probability of the observed test statistic … when H0 is true? Thus, smaller and smaller P-values provide stronger and stronger evidence against H0 Small P-value ? strong evidence for HA Interpreting the p-value… The smaller the p-value, the more statistical evidence exists to support the alternative hypothesis. •If the p-value is less than 1%, there is overwhelming evidence that supports the alternative hypothesis. •If the p-value is between 1% and 5%, there is a strong evidence that supports the alternative hypothesis. •If the p-value is between 5% and 10% there is a weak evidence that supports the alternative hypothesis. •If the p-value exceeds 10%, there is no evidence that supports the alternative hypothesis. We observe a p-value of .0069, hence there is overwhelming evidence to support H1: > 170.
11.38
Interpreting the p-value…
Overwhelming Evidence
(Highly Significant)
Strong Evidence
(Significant)
Weak Evidence
(Not Significant)
No Evidence
(Not Significant)
0
.01
.05
.10
p=.0068
11.39
Conclusions of a Test of
Hypothesis…
If we reject the null hypothesis, we conclude that there is enough
evidence to infer that the alternative hypothesis is true.
If we fail to reject the null hypothesis, we conclude that there is
not enough statistical evidence to infer that the alternative
hypothesis is true. This does not mean that we have proven that
the null hypothesis is true!
11.40
Prior to testing, you would decide on a level of
significance…
Your computed “p-value” will indicate whether you
should reject the null or fail to reject the null.
Let’s examine some p-values and make decisions:
If p = .45, we would __________________.
If p = .20, we would __________________.
If p = .09, we would __________________.
If p = .01, we would __________________.
If p = .009, we would __________________.
In summary…
*Sampling critically important to your study.
*Null and alternative hypotheses are the
foundation of research investigations.
*Interpreting the p-value provides evidence as to
whether you have “statistically significant”
evidence to support your claim or not.
Correlation &
Regression
Correlation
Finding the relationship between two
quantitative variables without being
able to infer causal relationships
Correlation is a statistical technique
used to determine the degree to which
two variables are related
Scattergram (or scatterplot)
• Rectangular coordinate
• Two quantitative variables
• One variable is called independent or
criterion (X) and the second is called
dependent or predictive (Y)
• Points are not joined
• No frequency table
Y
*
*
*
X
Example
Wt. 67 69 85 83 74 81 97 92 114 85
(kg)
SBP 120 125 140 160 130 180 150 140 200 130
mmHg)
Wt. 67 69 85 83 74 81 97 92 114 85
(kg)
SBP 120 125 140 160 130 180 150 140 200 130
SBP(mmHg)
(mmHg)
220
200
180
160
140
120
100
wt (kg)
80
60
70
80
90
100
110
120
Scatter diagram of weight and systolic blood
pressure
Scatter plots
The pattern of data is indicative of the type of
relationship between your two variables:
? positive relationship
? negative relationship
? no relationship
Positive relationship
18
16
14
Height in CM
12
10
8
6
4
2
0
0
10
20
30
40
50
Age in Weeks
60
70
80
90
Negative relationship
Reliability
Age of Car
No relation
An Example
A familiar statement from parents to children:
If you want to get ahead, stay in school.
Underlying this nagging parental advice is the following claimed
empirical relationship:
+
LEVEL OF EDUCATION =====> LEVEL OF SUCCESS IN LIFE
Suppose we collect data through by means of a survey asking
respondents (say a representative sample of the population aged 3555) to report the number of years of formal EDUCATION they
completed and also their current INCOME (as an indicator of
SUCCESS). We then analyze the association between the two
interval variables in this reformulated hypothesis.
+
LEVEL OF EDUCATION =========> LEVEL OF INCOME
(# of years reported)
($000 per year)
Since these are both continuous variables, we analyze their association
by means of a scattergram or scatterplot.
Data collected from two different societies:
Years of Education versus Yearly Income
An Example (cont.)
Note that the two scattergrams are drawn with the
same horizontal and vertical scales to facilitate
comparison between the two charts.
Both scattergrams show a clear positive association
between the two variables, i.e., the plotted points in
both form an upward-sloping pattern running from
Low – Low to High – High.
At the same time there are obvious differences
between the two scattergrams (and thus between the
relationships between INCOME and EDUCATION in
societies A and B).
Questions For Discussion
In which society, A or B, is the hypothesis most powerfully confirmed?
In which society, A or B, is there a greater incentive for people to stay in school?
Which society, A or B, does the U.S. more closely resemble?
How might we characterize the difference between societies A and B?
An Example (cont.)
We can visually compare and contrast the nature of
the associations between the two variables in the two
scattergrams by drawing a number of vertical strips
in each scattergram.
Points that lie within each vertical strip represent
respondents who have (just about) the same value
on the independent (horizontal) variable of
EDUCATION.
Within each strip, we can estimate (by “eyeball
methods”) the average magnitude of the dependent
(vertical) variable INCOME and put a mark at the
appropriate level.
Average Income for Selected Levels of Education
We can connect these marks to form a line of averages that is
apparently (close to being) a straight line.
An Example (cont.)
Now we can assess two distinct characteristics of the
relationships between EDUCATION and INCOME in
scattergrams A and B.
How much the does the average level of INCOME change
among people with different levels of education?
How much dispersion of INCOME there is among people with
the same level of EDUCATION?
An Example (cont.)
In both scattergams, the line of averages is upward-sloping, indicating a
clear apparent positive effect on EDUCATION on INCOME.
But in the scattergram for society A, the upward slope of the line of
averages is fairly shallow.
The line of averages indicates that average INCOME increases
by only about $1000 for each additional year of EDUCATION.
On the other hand, in the scattergram for society B, the upward
slope of the line of averages is much steeper.
The graph in Figure 1B indicates that average INCOME
increases by about $4000 for each additional year of
EDUCATION.
In this sense, EDUCATION is on average more “rewarding” in
society B than A.
An Example (cont.)
There is another difference between the two scattergrams.
In scattergram A, there is almost no dispersion within each vertical strip (and
almost no dispersion around the line of averages as a whole).
In scattergram B, there is a lot of dispersion within each vertical strip (and
around the line of averages as a whole).
We can put this point in simpler language.
In society A, while additional years of EDUCATION produce rewards in
terms of INCOME that are modest (as we saw before), these modest rewards are
essentially certain.
In society B, while additional years of EDUCATION produce on average much
more substantial rewards in terms of INCOME (as we saw before), these large
expected rewards are highly uncertain and are indeed realized only on average.
For example, in scattergram B (but not A), we can find many pairs of cases such
that one case has (much) higher EDUCATION but the other case has (much)
higher INCOME.
An Example (cont.)
This means that in society B, while EDUCATION has a big impact on
EDUCATION, there are evidently other (independent) variables (maybe
family wealth, ambition, career choice, athletic or other talent, just
plain luck, etc.) that also have major effects on LEVEL OF INCOME.
In contrast, in society A it appears that LEVEL OF EDUCATION
(almost) wholly determines LEVEL OF INCOME and that essentially
nothing else matters.
Another difference between the two societies is that, while both
societies have similar distributions of EDUCATION, their INCOME
distributions are quite different.
A is quite egalitarian with respect to INCOME, which ranges
only from about $40,000 to about $60,000, while B is
considerably less egalitarian with respect to INCOME, which
ranges from under to $10,000 to at least $100,000 — and
possibly higher.)
In summary, in society A the INCOME rewards of EDUCATION are
modest but essentially certain, while in society B the INCOME rewards
of EDUCATION are substantial on average but quite uncertain in
individual cases.
Correlation Coefficient: r
?
It is also called Pearson’s correlation
or product moment correlation
coefficient.
?
It measures the nature and strength
between two variables of
the quantitative type.
The sign of r denotes the nature of
association
while the value of r denotes the
strength of association.
?
If the sign is +, this means the
relation is direct (an increase in one
variable is associated with an
increase in the other variable and a
decrease in one variable is associated
with a decrease in the other
variable).
?
While if the sign is -, this means an
inverse or indirect relationship
(which means an increase in one
variable is associated with a decrease
in the other).
?
?
The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of the
association as illustrated
by the following diagram.
strong
-1
intermediate
-0.75
-0.25
weak
weak
0
indirect
perfect
correlation
intermediate
0.25
strong
0.75
1
Direct
no relation
perfect
correlation
If r = Zero this means no association or
correlation between the two variables.
If 0 < r < 0.25 = weak correlation. If 0.25 = r < 0.75 = intermediate correlation. If 0.75 = r < 1 = strong correlation. If r = l = perfect correlation. How to compute the simple correlation coefficient (r) r? x? y ? ? xy ? n 2 2 ? ? ? ? ( x) ( y) ? ? 2 2 ??x ? ?.? ? y ? ? ? ?? ? n n ? ?? ? This slide used for explanation only…not a required “understanding” slide. Example: A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in the following table . It is required to find the correlation between age and weight. serial No Age (years) Weight (Kg) 1 7 12 2 6 8 3 8 12 4 5 10 5 6 11 6 9 13 This slide used for explanation only…not a required “understanding” slide. These 2 variables are of the quantitative type, one variable (Age) is called the independent and denoted as (X) variable and the other (weight) is called the dependent and denoted as (Y) variables to find the relation between age and weight compute the simple correlation coefficient using the following formula: r ? x? y ? ? xy ? 2 ? ( x) ? ? ? x2 ? ? n ? n 2 ?? ( y) ? ?.? ? y 2 ? ?? n ?? ? ? ? ? This slide used for explanation only…not a required “understanding” slide. Serial n. Age (years) (x) Weight (Kg) (y) xy X2 Y2 1 7 12 84 49 144 2 6 8 48 36 64 3 8 12 96 64 144 4 5 10 50 25 100 5 6 11 66 36 121 6 9 13 117 81 169 Total ?x= 41 ?y= 66 ?xy= 461 ?x2= 291 ?y2= 742 This slide used for explanation only…not a required “understanding” slide. r? 41 ? 66 461 ? 6 ? (41) 2 ? ? (66) 2 ? ?291 ? ?.?742 ? ? 6 6 ? ?? ? r = 0.759 strong direct correlation This slide used for explanation only…not a required “understanding” slide. EXAMPLE: Relationship between Anxiety and Test Scores X2 Y2 Anxiety (X) Test score (Y) 10 8 2 1 5 6 ?X = 32 2 100 4 20 3 64 9 24 9 4 81 18 7 1 49 7 6 25 36 30 5 36 25 30 ?Y = 32 ?X2 = 230 ?Y2 = 204 ?XY=129 XY This slide used for explanation only…not a required “understanding” slide. Calculating Correlation Coefficient r? (6)(129) ? (32)(32) ?6(230) ? 32 ??6(204) ? 32 ? 2 2 774 ? 1024 ? ? ?.94 (356)( 200) r = - 0.94 Indirect strong correlation This slide used for explanation only…not a required “understanding” slide. exercise Multiple Correlation Tables Repeated correlations with multiple variables t-tests, & ANOVAs and their application to the statistical analysis of neuroimaging Adapted from Carles Falcon & Suz Prejawa Populations and samples Population ? z-tests Sample (of a population) ?t-tests NOTE: a sample can be 2 sets of scores, eg fMRI data from 2 conditions Comparison between Samples Are these groups different? Comparison between Conditions (fMRI) Reading aloud (script) Reading aloud vs vs “Reading” finger spelling (sign) Picture naming t-tests comp infer 12 95% CI 10 8 6 Left hemisph ... Purchase answer to see full attachment

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