populations & samples objectives: students should know the difference between a population and a...
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Populations & SamplesObjectives:
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
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GATE Frame: Populations & Samples
Population
Sample/Participants
A population is any entire collection of people, animals, plants or objects which demonstrate a phenomenon of interest.
A sample is a subset of the population; the group of participants from which data is collected.
Eligible
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.
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Parameters & StatisticsA 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.
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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.
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Sample Estimates of Population Parameters
Sample Statistic(point estimate)
Combine with measure of
variability of the point estimate
Population Parameter
Construct a range of values with an associated probability of containing the
true population value
L = lower valueU = upper value
€
L ≤ μ ≤ U
€
x
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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
75
n=100
€
x = 72
We draw our sample, and the mean HR is 72 bpm
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Standard ErrorIf 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:
PopulationN=1000
75
n=100
€
x = 71n=10
0
€
x = 72n=10
0
€
x = 78
n=100
€
x = 74n=10
0
€
x = 77n=100
€
x = 75
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Standard Error
PopulationN=1000
75
The mean of the means of all possible samples of size 100 would exactly equal the population mean:
All possible samples of size n=100
75x The standard
deviation of the means of all
possible samples is the standard error
of the mean
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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%
-1.96 SE
2.5%
+1.96 SE
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In addition, if we constructed 95%
confidence intervals around each individual sample mean:
95% of the intervals will
contain the true population mean.
PopulationMean
Sample Means
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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:
n
sdSE
Note: You will need to calculate the standard error in this course.
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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