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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: wont or cant 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.

Lets 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 ...
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