
Know the symbols and the meanings
We can always easily calculate/know a statistic
 Often we don’t know (and will never know) the
value of the parameter(s); we would have to
take a census... Is that even possible?


Pew Research Center surveyed 1650 adult
internet users to estimate the proportion of
American internet users between 18 & 32 years
old. Common wisdom hold that this group of
young adults is among the heavier users of the
internet. The survey found that 30% of the
respondents in the sample were between ages 18
& 32.

Identify the population, sample, parameter, &
statistic (‘estimator’)?

This example... proportions... we have ‘means’
too...

Drawing conclusions about a population on the
basis of observing only a small subset of that
population (i.e., a sample); more on this a little
later...

Always involves some uncertainty

Does a given sample represent a particular
population accurately?

Is the sample systemically ‘off?’ By a lot? By a
little?

Recall, bias is…

Being systematically ‘off;’ scale 5 pound heavy;
clock 10 minutes fast, etc.

Textbook explains two different types of bias:
measurement bias & sampling bias

Don’t need to know if a particular bias is
measurement or sampling; just need to know the
concept of bias

Let’s discuss…
•
People who choose themselves by responding
to a general appeal
•
Biased because people with strong opinions,
especially negative opinions, are most likely
to respond
•
Often very misleading
•
With a partner, come up with one real-life
example of voluntary response sampling with
which you are familiar; 30 seconds & share
out
All
of our examples discussed
make our sample statistic
systematically ‘off;’ in other
words, bias
Worthless
data
Convenience Sampling: Choosing
individuals who are easiest to
reach.
With a partner, think of a real-life
example of convenience sampling.
30 seconds, then share out.
Situational examples
Bias; worthless data
 Convenience
sampling & voluntary
sampling are both bias sampling
methods
 How
do we minimize (we can’t
eliminate; just minimize as much as
possible) bias when sampling/taking
samples?
 Let
impersonal chance/randomness do
the choosing; more on this...
 (remember,
in voluntary response people
chose to respond; in convenience sampling,
interviewer made choice; in both situations,
personal choice creates bias)
 Simple
Random Sample (SRS) – type of
probability or random sample; this type of
sampling is without replacement
 SRS
 The
– chance selects the sample
use of chance selecting the sample is the
essential principle of statistical sampling
 Several
different types of random sampling,
all involve chance selecting the sample
 Choosing
samples by chance gives all
individuals an equal chance to be chosen
 We
will focus on Simple Random Sampling
(SRS)
 SRS
ensures that every set of n individuals
has an equal chance to be in the
sample/actually selected
Easiest ways to use chance/SRS:
• Names in a hat
• Random
digits generator in
calculator or Minitab
• Random
digits table
Random digits table: table of random digits,
long string of digits 0, 1, ..., 9 in which:
•
Each entry in the table is equally likely to be
0–9
•
Entries are independent. Knowing the digits
in one point of the table gives no information
about another part of the table
•
Table in rows & columns; read either way
(but usually rows); groups & rows – no
meaning; just easier to read
0
– 9 equally likely
00 – 99 equally likely
000 – 999 equally likely
Joan’s small accounting firm serves
30 business clients. Joan wants to
interview a sample of 5 clients to
find ways to improve client
satisfaction. To avoid bias, she
chooses a SRS of size 5.

Enter table at a random row

Notice her clients are numbered (labeled) with 2digits numbers (if this isn’t already done, you must
label your list), so we are going to go by 2-digit
number in table

Ignore all 2-digit number that are beyond 30 (our
data is numbered from 01 to 30)

Ignore duplicates

Continue until we have 5 distinct 2-digit numbers
chosen & identify who those clients are
1.
Label... Assign a numerical label to every
individual
2.
Random Digits Table (or Minitab or names
in hat)... Select labels at random
3.
Stopping Rule ... Indicate when you should
stop sampling
4.
Identify Sample ... Use labels to identify
subjects/individuals selected to be in the
sample
 Be
certain all labels have the same # of
digits if using RDT(ensures individuals
have the same chance to be chosen)
 Use
shortest possible label, i.e., 1 digit
for populations up to 10 members (can
use labels from 0 to 9), 2 digits for
populations from 11 – 100 members
(can use labels from 00 to 99), etc. -this is just a good standard of
practice...
 Label
students; what labels should we use?
Label candy; what labels should we use?
 SRS
of 2 students using Random Digits Table;
enter table on line ___
 SRS
of 2 students using my Minitab (will my
Minitab be different from your Minitab?)
 Should
we allow duplicates? When should we
and when should we not?
 To
take a SRS, we need a list of our
population
 Is
this always possible?
 Could
we get a list of everyone in the United
States? In Santa Clarita? COC students?
 Randomization
is great to help with
voluntary response bias and convenience bias
 But
it doesn’t help at all with other possible
bias... such as wording, non-response, undercoverage, etc.
 Let’s
discuss other sources of potential bias
which could make our statistic/our data
worthless
 Most
samples suffer from some degree of
under coverage (another type of bias)
 What
is bias again?? ....
 ...
Bias is systemically favoring a particular
outcome
 Under
coverage occurs when a group(s) is
left out of the process of choosing the
sample somehow/entirely
Talk in your groups and come up with an
example of under coverage (some groups
in population are left out of the process
of choosing the sample)
 Another
source of bias in many/most sample
surveys is non-response, when a selected
individual cannot be contacted or refuses to
cooperate
 Big
problem; happens very often, even with
aggressive follow-up
 Almost
impossible to eliminate non-response; we
can just try to minimize as much as possible
 Note:
Most media polls won’t/don’t tell us the
rate of non-response
Response bias...occurs when
respondents are untruthful,
especially if asked about illegal or
unpopular beliefs or behaviors
Can you come up with some situations
where people might be less than
truthful?
Who won the “First Lady Debate?”
http://www.youtube.com/watch?v=EohGmGQUhA
http://perezhilton.com/tv/JIMMY_KIMMEL_LIV
E_Who_Won_The_Presidential_Debate_BEFOR
E_It_Even_Happened/?id=79b7cb451ec00#.Vb
-lN_NViko
http://www.mrctv.org/videos/kimmel-publicweighs-first-lady-debate
Should we ban disposable diapers?
A survey paid for by makers of disposable diapers
found that 84% of the sample opposed banning
disposable diapers. Here’s the actual question:
It is estimated that disposable diapers account for
less than 2% of the trash in today’s landfills. In
contract, beverage containers, third-class mail and
yard wastes are estimated to account for about 21%
of the trash in landfills. Given this, in your opinion,
would it be fair to ban disposable diapers?

Remember our survey ...
 Possible
activity: Sampling Distribution on
football lovers...

Even if we use SRS/probability sampling, and we
are very careful in reducing bias as much as
possible, the statistics we get from a given
sample will likely be different from the statistics
we get from another sample.

Statistics vary from sample to sample

We can improve our results (our sample statistic
can get closer to what our population parameter
actually is; our variability decreases) by
increasing our random sample size

Remember samples vary; parameters are fixed

Even if we use SRS/probability sampling, and we
are very careful in reducing bias as much as
possible, the statistics we get from a given
sample will likely be different from the statistics
we get from another sample.

Statistics vary from sample to sample

We can improve our results (our sample statistic
can get closer to what our population parameter
actually is; our variability decreases) by
increasing our random sample size

Remember samples vary; parameters are fixed
 Statistics
(like sample proportions) vary from
sample to sample
A
larger n (sample size) decreases standard
deviation (variability)
A
smaller n (sample size) increases standard
deviation (variability)
But remember no matter how much we
increase/how large our sample is, a large
sample size does not ‘fix’ underlying
issues, like bad wording, under coverage,
convenience sampling, etc.
Who
carried out the survey?
How was the sample selected/was
the sample representative of the
population?
How large was the sample?
What was the response rate?
How were the subjects contacted?
When was the survey conducted?
What was the exact question
asked?
 True
value of population parameter (p or μ)
is like a bull’s eye on a target
 Sample
statistic ((𝑥, 𝑝, etc.) is like an arrow
fired at the target; sometimes it hits the
bull’s eye and sometimes it misses
 Keep
in mind… we (very) often don’t know
how close to the bull’s eye we are…
When we take many samples from a population (sampling
distribution), bias & variability can look like the following:
- Bias, aim, accuracy
- Variability, precision, standard deviation
AND most of the time,
we don’t even know
where the target is!
... We are trying to
estimate where the
target is...estimation
method...

We want: low bias & low variability (good aim &
good precision)

Properly chosen statistics computed from
random samples of sufficient size will (hopefully)
have low bias & low variability (good aim &
precision)

Hits the bull’s eye on the target (even though we
don’t know where the target is)

Can’t eliminate bias & variability (bad aim &
precision); can just do all that we can to reduce
bias & variability
 How
good are our estimators?
The
sampling distribution of a
statistic is the distribution of
values taken by the statistic in all
possible samples of the same size
from the same population.
 The
population we will consider is the scores
of 10 students on an exam as follows:
 The
parameter of interest is the mean score
in this population.
 The
sample is a SRS drawn from the
population.

Use RDT (enter at random line) to draw a SRS of size
n = 3 from this population. Calculate the mean ( x )
of the sample score. This statistic is an estimate of
the population parameter, μ.

Repeat this process 5 times. Plot your 5 xs on the

board. We are constructing a sampling distribution.

What is the approximate value of the center of our
sampling distribution? What
is the shape? Based on
this sampling distribution, what do you think our true
population parameter, μ, is?

Now let’s calculate our true population parameter, μ.

Also, remember the Reece’s Pieces simulation ...

If we use good, sound statistical practices, we
can get pretty good estimates of our population
parameters based on samples

Also, our estimates get better and better if ...

Our estimates get better and better as our ‘n’
increases... not our population size, but our
sample size, n.

In other words, the precision (variability; how much

So, an estimator based on a sample size of 10 is just
as precise taken from a population of 1000 people as
it is taken from a population of a million.

Let’s explore this idea more…
it fluctuates) of an estimator (a sample statistic, like
a sample proportion or a sample mean) does not
depend on the size of the population; it depends only
on the sample size.
Consider 1000 SRS’s of n = 100 for proportion of
U.S. adults who watched Survivor Guatemala
Discuss in your groups some observations you
have about this distribution; share out.
n = 100 
(1000 SRS for both
sampling distributions)
 n = 1000
What do you notice?
Given that sampling randomization is used properly,
the larger the SRS size ( n ), the smaller the spread
(the more tightly clustered; the more precise) of
the sampling distribution.
Center doesn’t change significantly (good estimator;
unbiased estimator)
Shape doesn’t change significantly
Spread (range, standard deviation) does change
significantly; spread is not an unbiased estimator

A SRS of 2,500 from U.S. population of 300
million is going to be just as accurate/same
amount of variability/precision as that same size
SRS of 2,500 from 750,000 San Francisco
population

Both just as precise (given that population is
well-mixed); equally trustworthy; no difference
in standard deviation, how ‘spread out’
distribution is

Not about population size; it’s about sample
size (n)
 As
long as the population is large relative to
the sample size, the precision has nothing to
do with the size of the population, but only
with the size of the sample.
 Increased
sample size improves precision/
reduces variability
 Surveys
based on larger sample sizes have
smaller standard error (SE) and therefore
better precision (less variability)
 Standard
Deviation (or standard error) =
p(1  p)
n
 Standard
Deviation (or standard error) =
p(1  p)
n
 Let’s
try the formula: Let’s say that 20,000
students attend COC; and 20% of them say
their
 favorite ice cream is vanilla. If we
were to take a SRS of 100 students and ask
them if vanilla is their favorite flavor of ice
cream, our standard deviation would be...
 Standard
Deviation (or standard error) =
p(1  p)
n
 Now
let’s say that 20,000 students attend
COC; and 20% of them say their favorite ice
cream
 is vanilla. If we were to take a SRS of
1,000 students and ask them if vanilla is their
favorite flavor of ice cream, our standard
deviation would be...
 Standard
Deviation (or standard error) =
p(1  p)
n
 Now
let’s say that 20,000 students attend
COC; and 20% of them say their favorite ice
cream
 is vanilla. If we were to take a SRS of
2,000 students and ask them if vanilla is their
favorite flavor of ice cream, our standard
deviation would be...
 Standard
Deviation (or standard error) =
p(1  p)
n
 Increased
sample size means we have more
information about our population so our
variability/standard
deviation is going to be

less.
 Increased
sample size improves precision/
reduces variability
 Surveys
based on larger sample sizes have
smaller standard error (SE) and therefore
better precision (less variability)
 Trade-offs…
 Cost
increases, time-consuming, etc.

Life was good and easy with the Normal
distribution

Could easily calculate probabilities

Good working model

If we could use the Normal distribution with
sampling distributions for proportions, life would
be great 

Guess what? We can. Meet the Central Limit
Theorem
 Has
many versions (one for proportions, one
for means, etc.)
 Let’s
 To
discuss proportions for now
use CLT with proportions, three conditions
must be met
 Random;
samples collected randomly from
population
 Large
sample; np ≥ 10 and n(1 – p) ≥ 10;
proportion of expected successes and failures
at least 10
 Big
population (population at least 10 times
sample size)

20,000 students attend COC; and we know that
20% of them say their favorite ice cream is
vanilla. We want to find out... IF we asked a SRS
of 100 students if vanilla is their favorite ice
cream, what’s the probability that half or more
of them would say ‘yes.’?

Let’s check conditions to see if we can use the
CLT to calculate that probability.
Random?
 Large sample: np ≥ 10 and n(1 – p) ≥ 10 ?
 Big population: population is at least 10 times
the sample size?


20,000 students attend COC; and we know that
20% of them say their favorite ice cream is
vanilla. We want to find out... IF we asked a SRS
of 15 students if vanilla is their favorite ice
cream, what’s the probability that half or more
of them would say ‘yes.’?

Let’s check conditions to see if we can use the
CLT to calculate that probability.
Random?
 Large sample: np ≥ 10 and n(1 – p) ≥ 10 ?
 Big population: population is at least 10 times
the sample size?


20,000 students attend COC; and we know that
20% of them say their favorite ice cream is
vanilla. We want to find out... IF we asked a SRS
of 5,000 students if vanilla is their favorite ice
cream, what’s the probability that half or more
of them would say ‘yes.’?

Let’s check conditions to see if we can use the
CLT to calculate that probability.
Random?
 Large sample: np ≥ 10 and n(1 – p) ≥ 10 ?
 Big population: population is at least 10 times
the sample size?

It’s important to check conditions before we
calculate probabilities to make sure that
what we calculate is accurate, valuable,
meaningful information (not nonsense)
 Random
 Large
sample: np ≥ 10 and n(1 – p) ≥ 10
 Big population: population is at least 10
times the sample size

the examples have described situations in which
we know the value of the population parameter,
p.

Very unrealistic

The whole point of carrying out a survey (most of
the time) is that we don’t know the value of p,
but we want to estimate it

Think about the elections... Parties are taking a
lot of sample surveys (polls) to see who is
‘leading’
 Took
a random sample of 2,928 adults in the
US and asked them if they believed that
reducing the spread of AIDS and other
infectious diseases was an important policy
goal for the US government.
 1,551responded
 Spiral
‘yes;’ 53%
back for a moment... Random? Large
sample? Big population?
 Random?
Large sample? Big population?
(So, if we wanted to find a probability using
the CLT, we could...)
 These
are the exact conditions we must
check to create a confidence interval as well
 More
on confidence intervals in a few...

The above percentage just tells us about OUR
sample of those specific 2,928 people.

What about another sample? Would we get a
different % of yes’s?

What about the percentage of all adults in the
US who believe this? How much larger or smaller
than 53% might it be? Do we think a majority
(more than 50%) of Americans share this belief?

We don’t know p, population parameter; we do
ˆ for this sample; it’s 53%
know p

We don’t know p, population parameter; we do
know p
ˆ for this sample; it’s 53%. We also know:

Our estimate (53%) is unbiased; remember
sampling distributions? (maybe not exactly = p;
maybe just a little low or a little high)

Standard error (typical amount of variability) is
pˆ (1  pˆ )
0.53(1  0.53)
about

 0.0092  0.9%

n

2928
Because we have a ‘large sample,’ the
probability
distribution of our p
ˆ s is close to

Normally distributed & centered around the true
population parameter.

True, unknown population parameter probably
centered around 0.53; Normally distributed;
standard error (SD; amount of variability in
sample statistic) = 0.009; so ...
About 68% of the data is as close or closer than 1
standard error away from the unknown
population parameter, p
 95% of the data is as close or closer than 2
standard errors away from the unknown
population parameter, p
 99.7% of the data is as close or closer than 3
standard errors away from the unknown
population parameter, p


True, unknown population parameter probably
centered around 0.53; Normally distributed; standard
error (SD) = 0.009

So we can be highly confident, 99.7% confident, that
the true, unknown population proportion, p, is
between
0.53 + (3)(0.009)
to
0.53 – (3)(0.009)

This is a confidence interval; we are 99.7% confident
that the the interval from about 50.3% to 55.7%
captures the true, unknown population proportion of
Americans who believe that reducing the spread of
AIDS and other infectious diseases is an important
policy goal for the US government.
 True,
unknown population parameter probably
centered around 0.53; Normally distributed;
standard error (SD) = 0.009
 What
if we wanted to construct a confidence
interval in which we are 95% confident?

What proportion of us have at least one tattoo? So our
sample statistic, our p
ˆ=

If we were to ask another group of COC students, we would
get another (likely different) p
ˆ

445 Math 075
 students were asked this last Spring; 133/445 =
0.299 = 29.9% had at least one tattoo

Remember, larger n, generally less variation; but still
value (unbiased estimator)
centered at same

We want to be able to say with a high level of certainty what
proportion of all COC students have at least one tattoo. But
we don’t know the true, unknown population parameter, p.

We don’t know p (population parameter)

ˆ (sample statistic); actually we have 2
We do know p
sample statistics – our class and the Math 075 data

Our estimators are unbiased (what does that mean?)

 check conditions for each of our samples:
Let’s
Random selection; Large sample; Big population

Can we use either sample statistic (either pˆ )? If so,
which should we use?

Calculate our standard deviation (our standard error)


Our distribution is ≈ Normal (because our
conditions are met), centered around p
ˆ ; 68%
with 1 SD; 95% within 2 SDs; 99.7% within 3 SDs

Let’s create a 95% confidence interval ...


We are 95% confident that the interval from
_____ to _____ captures the true, unknown
population parameter, p, the proportion of all
COC students that have at least one tattoo.

This is a confidence interval with a 95%
confidence level

Our distribution is ≈ Normal (because our
conditions are met), centered around p
ˆ ; 68%
with 1 SD; 95% within 2 SDs; 99.7% within 3 SDs

Let’s create a 99.7% confidence interval ...


We are 99.7% confident that the interval from
_____ to _____ captures the true, unknown
population parameter, p, the proportion of all
COC students that have at least one tattoo.

This is a confidence interval with a 99.7%
confidence level

Our distribution is ≈ Normal (because our
conditions are met), centered around p
ˆ ; 68%
with 1 SD; 95% within 2 SDs; 99.7% within 3 SDs

Let’s create a 68% confidence interval ...


We are 68% confident that the interval from
_____ to _____ captures the true, unknown
population parameter, p, the proportion of all
COC students that have at least one tattoo.

This is a confidence interval with a 68%
confidence level
 Our
distribution is ≈ Normal (because our
ˆ ; 68%
conditions are met), centered around p
with 1 SD; 95% within 2 SDs; 99.7% within 3
SDs
 lengths of our
did you notice about the
confidence intervals as we changed from 68%
confident to 95% confident to 99.7%
confident?
 What
 More
on this a little later...

Statistical inference provides methods for
drawing conclusions about a population based on
sample data

Methods used for statistical inference assume
that the data was produced by properly
randomized design

Confidence intervals, are one type of inference,
and are based on sampling distributions of
statistics. The other type of inference we will
learn and practice is hypothesis testing (more on
this later).

Estimator ± margin of error (MOE)

Our estimator we just used was our sample
proportion, our p
ˆ

Our margin of error we just used was our standard
error, our standard deviation,


Margin of error tells us amount we are most likely
‘off’ with our estimate

Margin of error helps account for sampling variability
(NOT any of the bias’ we discussed...voluntary
response, non-response, et.)
 that
the mean temperature in Santa Clarita
in degrees Fahrenheit is between -50 and
150?
 that
the mean temperature in Santa Clarita
in degrees Fahrenheit is between 70 and
70.001?
 that
the mean temperature in Santa Clarita
in degrees Fahrenheit is between -50 and
150?
 that
the mean temperature in Santa Clarita
in degrees Fahrenheit is between 70 and
70.001?
 In
general, large interval  high confidence
level; small interval  lower confidence
level
 99%
confidence level (or 99.7%)
 95%
confidence level
 90%
confidence level
 Typically
we want both: a reasonably high
confidence level AND a reasonably small
interval; but there are trade-offs; more on
this in a little bit
 Will
we ever know for sure if we captured
the true unknown population parameter p?
No. Actual p is unknown.
 Interpretation
of a confidence interval:
“I am ___% confident that the interval
from _____ to _____ captures the true,
unknown population proportion of
(context).”

The lower the confidence level (say 10% confident), the
shorter the confidence interval (I am 10% confident that
the mean temperature in Santa Clarita is between 70.01
degrees and 70.02 degrees)

The higher the confidence level (say 99% confident), the
wider the confidence interval (I am 99% confident that the
mean temperature in Santa Clarita is between 40 degrees
and 100 degrees)

What else effects the length of the confidence interval?

Larger the n (sample size), shorter the confidence interval
(small MOE)
Smaller the n (sample size), longer the confidence interval
(large MOE)


Let’s look at how sample size (n) effects the length of the
confidence interval, specifically the margin of error

Remember, estimate ± margin of error

Margin of error

Let’s put some numbers in as a simple example...

Larger the n (sample size), shorter the confidence interval
(small MOE)

Smaller the n (sample size), longer the confidence interval
(large MOE)
 So,
if you want (need) high confidence level
AND small(er) interval (margin of error), it is
possible if you are willing to increase n
 Can
be expensive, time-consuming
 Sometimes
 In
not realistic (why?)
reality, you may need to compromise on the
confidence level (lower confidence level)
and/or your n (smaller n).

Alcohol abuse is considered by some as the #1
problem on college campuses. How common is it?

A recent SRS of 10,904 US college students
collected information on drinking behavior &
alcohol-related problems. The researchers
defined “frequent binge drinking” as having 5 or
more drinks in a row 3 or more times in the past 2
weeks. According to this definition, 2,486 students
were classified as frequent binge drinkers.

Based on these data, what can we say about the
proportion of all US college students who have
engaged in frequent binge drinking?

Let’s create a confidence interval so we can
approximate the true population proportion of
all US college students who engaged in frequent
binge drinking.

How confident do we want to be (i.e, what
confidence level do we want to use)?

We must check conditions before we calculate a
confidence interval...
Random?
 Large sample?
 Big population?

Perform Minitab calculations
 1 sample, proportion
 Options, 99 CL
 Data, summarized data, events, trials
 “+”, data labels

Always conclude with interpretation, in context
I am 99% confident that the interval from 21.8% to
24.9% contains the true, unknown population
parameter, p, the actual proportion of all US
college students who have engaged in frequent
binge drinking.

Let’ use our Math 075 statistic of 133/445
students had at least one tattoo (or 29.9%)

We already checked our conditions

Previously we created confidence intervals at
68%, 95%, and 99.7% confidence levels (applied
Empirical Rule... as an introduction to
confidence intervals)

Now let’s use Minitab and construct a 90%
confidence interval; an 80% confidence interval.
Be sure to practice interpreting these confidence
intervals as well.
 Often
researchers choose the margin of error
& confidence level they want ahead of
time/before survey
 So
they need to have a particular n to
achieve the MOE and the CL they want.
 p(1  p) 
MOE  z * 

n 

 Often
researchers choose the MOE
 A common CL is 95%, so z* ≈ 2
 Can solve for n & get formula in textbook
 0.5(1  0.5) 
m  2

n


A company has received complaints about its
customer service. They intend to hire a
consultant to carry out a survey of customers.
Before contacting the consultant, the company
president wants some idea of the sample size
that she will be required to pay for.
One critical question is the degree of satisfaction
with the company's customer service. The
president wants to estimate the proportion p of
customers who are satisfied. She decides that
she wants the estimate to be within 3% (0.03) at
a 95% confidence level.
 No
idea of the true proportion p of satisfied
customers; so use p = 0.5. The sample size
required is given by
 p(1  p) 
MOE  z * 

n 

 0.5(1  0.5) 
0.03  2

n


n
= 1111.11
 So
if the president wants to estimate the
proportion, p, of customers who are
satisfied, at the 95% confidence level, with a
margin of error of 3%, she would need a
sample size (an n) of at least 1,112.