Chapter 3
Generalization: How broadly do
the results apply?
Generalization
So far we’ve studied significance and
estimation.
 Once we make a conclusion from a test
of significance or construct a confidence
interval, how broadly do these apply or to
what population can I generalize these
results?
 This generalization is the topic for this
section.

Generalization
Sometimes this generalization is difficult
and sometimes it is not.
 Generalizing to a larger population is valid
only when the sample is representative.
 Unfortunately, biased sampling methods
are common.

Section 3.1
Introduction to sampling from a
finite population
Notation Check
Statistics
Parameters
𝑥

(x-bar) Sample
Average or Mean
 𝑝 (p-hat) Sample
Proportion
𝜇 (mu) Population
Average or Mean
 𝜋 (pi) Population
Proportion
Statistics summarize a sample and parameters
summarize a population
Sampling Hope College students
Suppose we want to know the proportion of
Hope students that watched the Super Bowl.
Or the average number of traffic tickets
Hope students have received.
 The population of interest is all Hope
students.
 A census will get this information from all
Hope students.
 What if you don’t have time/money to
interview all students?

Sampling
We can take a sample of Hope students and
find the proportion of those in our sample
that watched the Super Bowl or mean
number of traffic tickets they have received.
 Using these statistics we can make
inferences to the parameters.
 How well will these statistics represent our
parameters of interest?
 The key to this question is how the sample
is selected from the population.

Random Sampling
 Getting
a random sample is key to making a
good inference. This can be tough; we don’t
live in a random world. For example, the
people you see on a daily basis can be very
different from the people others near to you
see on a daily basis.
 When samples are not random or
representative their results can be misleading.
Biased Sampling
ESPN Top 10: What is college
basketball's fiercest rivalry?
Connecticut vs. Tennessee (Women)
Duke vs. North Carolina
Hope vs. Calvin
Illinois vs. Missouri
Indiana vs. Purdue
Louisville vs. Kentucky
Penn vs. Princeton
Philadelphia's Big 5
Oklahoma vs. Oklahoma State
Xavier vs. Cincinnati
http://proxy.espn.go.com/chat/sportsnation/polling?event_id=1194
ESPN Top 10: What is college
basketball's fiercest rivalry?
75.1% Hope vs. Calvin
9.3% Duke vs. North Carolina
5.4% Indiana vs. Purdue
5.2% Philadelphia's Big 5
1.7% Penn vs. Princeton
1.5% Oklahoma vs. Oklahoma State
0.7% Louisville vs. Kentucky
0.6% Connecticut vs. Tennessee (Women)
0.3% Illinois vs. Missouri
0.3% Xavier vs. Cincinnati
Total Votes: 46,084
2012 State ACT Results


New York ranked 6th with an average
of 23.3.
Michigan ranked 45th with an average
of 20.1.
2011 State SAT Results
New York ranked 45th with an
average of 1466.
th
 Michigan ranked 6 with an average
of 1762.

???
ACT
SAT
MI
NY
100% 29%
4%
90%
Random Sample
To have a random sample, you can’t have
people self-select themselves into the
sample. (Basketball poll)
 You can’t choose a convenient sample
that is clearly not representative of the
population. (ACT vs. SAT)

Random Sample
A simple random sample is the easiest
way to ensure that your sample is
unbiased.
 A sampling method is biased if statistics
from samples consistently over or underestimate the population parameter.

Simple Random Sample
A simple random sample is like drawing
names out of a hat.
 Technically, a simple random sample is
a way of randomly selecting members of a
population so that every sample of a
certain size from a population has the
same chance of being chosen.

Sampling
Every simple random sample gives us
different values for the statistics.
 There is variability from sample to sample
(sampling variability).
 If we take repeated simple random
samples of Hope students, each sample
will consist of different students. We will
get different means or proportions each
time we do this. However …

Sampling
The sample means or proportions will
center around the population mean or
proportion if the sampling method is
unbiased (like a simple random sample).
 Our sampling variability will decrease
when we take larger and larger sample
sizes.

Exploration 3.1A: Sampling Words
We need to sample from a population of
interest if it is very large or is difficult to
measure every single member of the
population.
 If we were interested in High School GPA
for Hope students we would not need to
sample. The registrar’s office has all that
information. If we were interested in
something that has not already been
collected, we might want to sample.

Exploration 3.1A: Sampling Words
That being said, in this activity we will be
using the words in the Gettysburg Address
as our population.
 There are fewer than 300 in this speech and
we could easily look at the entire speech to
find out average word length, proportion of
words that contain an e, etc.
 We will be sampling from this speech not to
get information from the population, but to
help us learn some things about sampling.

Only picture of Lincoln at Gettysburg
(Edward Everett spoke for over two hours. Lincoln
followed with his two-minute speech.)
Exploration 3.1A
Select what you think is a representative
sample of 10 words from the Gettysburg
(pg 3-10). Record your words in table in
question 2.
 Make dotplots of both average length and
proportion containing e on the board.
 Only work through question 22.
 HW: Exercises 3.1.3 and 3.1.4

Review of Section 3.1
A sampling method is biased if statistics
from samples consistently over or underestimate the population parameter.
 A simple random sample is the easiest
way to insure that your sample is
unbiased.
 Therefore, if we have a simple random
sample, we can infer our results to the
population from which is was drawn.

Review of Section 3.1

We saw biased and unbiased sampling in
the Gettysburg Address exploration. We
also saw that:
◦ When we increase sample size, the variability
of our sampling distribution decreases.
◦ This variability can be predicted.
◦ Changing the population size has no effect on
variability.
Population
distribution of
word lengths
Distribution of average
word length from samples
of size 20
Section 3.2: Inference for a
Single Quantitative Variable
Using methods similar to what we did in the last
section, we will see how a null distribution for a
single quantitative variable can be obtained and even
predicted.
Example 3.2: Estimating Elapsed
Time
Estimating Time
Does it ever seem that time drags or flies
by?
 Students in a stats class (for their project 2)
collected data on students’ perception of
time
 Subjects were told that they’d listen to music
and asked questions when it was over.
 Played 10 seconds of the Jackson 5’s “ABC”
and asked how long they thought it lasted
 Can students accurately estimate the length?

Hypotheses
Null Hypothesis: People will accurately
estimate the length of a 10 second-song
snippet, on average. (μ = 10 seconds)
Alternative Hypothesis: People will not
accurately estimate the length of a 10
second-song snippet, on average. (μ ≠ 10
seconds)
Estimating Time
A convenience sample of 48 students on
campus were subjects and song length
estimates were recorded.
Dot Plot
Collection 2
 The average estimate was 13.71 sec and
the standard deviation was 6.50 sec.

5
volume = "low"
10
15
20
Estimate
25
30
Skewed, mean, median
The distribution obtained is not
symmetric, but is right skewed.
 When data are skewed right, the mean
Collection 2
gets pulled out to the right while the Dot Plot
median is more resistant to this.

5
volume = "low"
10
15
20
Estimate
25
30
Mean v Median
The mean is 13.71 and the median is 12.
 How would these numbers change if on
of the people that gave an answer of 30
seconds actually said 300 seconds?
Collection 2
 The standard deviation is 6.5 sec. Is it
resistant to outliers?

Dot Plot
5
volume = "low"
10
15
20
Estimate
25
30
Population?
One way to develop a null distribution is
to draw samples from some population
that we think our population of time
estimates might look like under a true
null.
 Under the null the mean is 10 sec.
 We might assume the population is
skewed and has a standard deviation
similar to what we found.

Simulation-based Inference
We have a possible population data set
similar to what we need.
 Let’s go and get that data.
 Then go to the One Mean applet and
develop a null distribution.
 Find out where our actual mean of
13.71sec is located.
 And finally see how a t-distribution could
predict all this.

T-distribution
The t-distribution is very similar to a
normal distribution, but with slightly
“heavier” tails.
 The t-statistic is the standardized statistic
we use with a single quantitative variable
and can be found using the formula:
𝑥−𝜇
𝑡= 𝑠
𝑛

Validity Conditions

The theory-based test for a single mean
requires either:
◦ The sample size is at least 20.
◦ If the sample size is less than 20 the sample
distribution is not skewed.

Let’s use the theory-based applet to run
this test and find a confidence interval.
(We first need to get the data.)
Estimating Time
Formulate Conclusions.
 Based on our small p-value, we can
conclude that people don’t accurately
estimate the length of a 10-second song
snippet and in fact they overestimate it.
 To what larger population can we make
our inference?
Estimating Time

Collection 2
We are 95% confident that the average
estimate of a 10 second song is between
Dot Plot
11.823 and 15.597 seconds.
5
volume = "low"
10
15
20
Estimate
25
30
Exploration 3.2: Sleepless Nights?

Page 3-32