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

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.

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.

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)

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

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

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

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

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

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

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.

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