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Populations & Samples
Objectives:

Students should know the difference between a population and a
sample

Students should be able to demonstrate populations and samples
using a GATE frame

Students should know the difference between a parameter and a
statistic

Students should know the main purpose for estimation and
hypothesis testing

Students should know how to calculate standard error
GATE Frame:
Populations & Samples
A population is any entire collection of
people, animals, plants or objects which
demonstrate a phenomenon of interest.
Population
A sample is a subset of the population;
the group of participants from which
data is collected.
In most situations, studying an entire
population is not possible, so data is
collected from a sample and used to
estimate the phenomenon in the
population.
Eligible
Sample/
Participants
Parameters & Statistics
A population value is called a parameter.
A value calculated from a sample is called a statistic.
Note: A sample statistic is a point estimate of a
population parameter.
Estimating Population Parameters
Confidence intervals (CI) are ranges defined by lower
and upper endpoints constructed around the point
estimate based on a preset level of confidence.
Hypothesis Testing is used to determine probabilities
of obtaining results from a sample or samples if the
result is not true in the population.
Sample Estimates of
Population Parameters
Sample Statistic
(point estimate)
Population Parameter
L  U
x
Combine with
measure of variability
of the point estimate
L = lower value
U = upper value

Construct a range of values with an associated
probability of containing the true population
value
What is Standard Error?
Suppose a population of 1000 people has a mean heart rate of
75 bpm (but we don’t know this). We want to estimate the
HR from a sample of 100 people drawn from the population:
Population
N=1000
n=100
x  72
  75

We draw our sample, and the
mean HR is 72 bpm
Standard Error
If we draw another sample, the mean will probably be a little different
from 72, and if we draw lots of samples we will probably get lots of
estimates of the population mean:
Population
N=1000
  75
n=100
n=100
n=100
x  72
x  71
x  78
n=100

x 75
n=100
x 74
n=100
x  77
Standard Error
The mean of the means of all possible samples of size 100 would exactly
equal the population mean:
Population
N=1000
All possible samples of size
n=100
  75
x  75
The standard deviation of
the means of all possible
samples is the standard
error of the mean
Sample Representativeness
The sample means will follow a normal distribution, and:
95% of the sample means will be between the population mean and
±1.96 standard errors.
95% of sample
means
2.5%
2.5%
-1.96 SE

+1.96 SE
In addition, if we
constructed 95%
confidence intervals
around each
individual sample
mean:
95% of the intervals
will contain the true
population mean.
Population
Mean

Sample
Means
Why is This Important and Useful?
We rarely have the opportunity to draw repeated samples from a
population, and usually only have one sample to make an inference
about the population parameter:
The standard error can be estimated from a single sample, by dividing the
sample standard deviation by the square root of the sample size:
sd
SE 
n
Note: You will need to calculate the standard error in this course.
Standard Error and
Confidence Intervals
The sample SE can then be used to construct an interval around the
sample statistic with a specified level of confidence of containing the true
population value:
The interval is called a confidence interval
The Most Commonly Used Confidence Intervals:
90% = sample statistic + 1.645 SE
95% = sample statistic + 1.960 SE
99% = sample statistic + 2.575 SE