Copyright © 2012 by Nelson Education Limited. Chapter 5 Introduction to inferential Statistics: Sampling and the Sampling Distribution 5-1

Copyright © 2012 by Nelson Education Limited.

Chapter 5Introduction to inferential Statistics:

Sampling and the Sampling Distribution

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• The basic logic and terminology of inferential statistics

• Random sampling• The sampling distribution

In this presentation you will learn about:

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– Population and Sample– Parameter and Statistic– Representative and EPSEM

Basic Logic and Terminology of

Inferential Statistics

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• Problem: The populations we wish to study are almost always so large that we are unable to gather information from every case.

Population and Sample

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• Solution: We choose a sample -- a carefully chosen subset of the population – and use information gathered from the cases in the sample to generalize to the population.

Population and Sample (continued)

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• Statistics are mathematical characteristics of samples.

• Parameters are mathematical characteristics of populations.

• Statistics are used to estimate parameters.

PARAMETER

STATISTIC

Parameter and Statistic

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• Sample must be representative of the population.– Representative: The sample has the same characteristics

as the population.• How can we ensure a sample is representative?

– Samples drawn according to the rule of EPSEM (equal probability of selection method), in which every case in the population has the same chance of being selected for the sample are likely to be representative.

Representative and EPSEM

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• The most basic EPSEM sampling technique is simple random sampling (SRS).

• There are variations of this technique, such as sstratified random sampling and cluster sampling.– The textbook, however, considers only simple random

sampling since it is the most straightforward application of EPSEM.

Random Sampling Techniques

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• To draw a simple random sample, we need:– A list of the population.– A method for selecting cases from the

population so each case has an equal probability of being selected (EPSEM).

Simple Random Sampling (SRS)

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• You want to know what % of students at a large university work during the semester.

• Draw a sample of 500 from a list of all students at the university (N = 20,000).

• Assume the list is available from the Registrar.

Simple Random Sampling (SRS): An Example

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• How can you draw names so every student has the same chance of being selected?–Each student has a unique 6 digit Student ID

Number that ranges from 000000 to 999999.• Use a Table of Random Numbers (or a computer

program) to select 500 ID numbers with 6 digits each. • Each time a randomly selected 6-digit number matches

the ID of a student, that student is selected for the sample.

Simple Random Sampling (SRS): An Example (continued)

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• Below is an excerpt from a Table of Random Numbers:

• To select a random sample from a numbered population list, begin anywhere in the Table and select groups of digits the same length as the largest number on your population list.

– Since the population list numbers in our example are in the 100,000s (i.e. Student ID Numbers range from 000000 to 999999), we select digits in groups of six.

• Every time a number selected from the Table corresponds to the identification number of a case on the population list, that case is selected for the sample.

• Continue the process until the desired number of cases (500 in our example) is selected for the sample.

– Disregard duplicate numbers.– Ignore cases in which the randomly selected number does not match an identification number on the population list.

• e.g. ignore cases in which no student ID matches the randomly selected number.


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• For example, if we decide to begin from the left of the top row of the Table, and work across the row, we take the first set of six digits (i.e., 104801, as boxed in the table below). – If this number (104801) corresponds to the identification number of a case on the

population list (i.e., a Student's ID Number), that case is selected for the sample.• Next, we take the second set of six digits (501101).

• If this number corresponds to a Student's ID Number, that case is selected for the sample.

• We continue this process until 500 names are selected for the sample.


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• Part of the list of 500 selected students might look like this:

ID # Student501101 Andrea Oja691791 Sasha Albright255958 Keisha Evans


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• After questioning each of these 500 students, you find that 368 (74%) work during the semester.


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• In the research situation above, identify each the following:– Population – Sample– Statistic– Parameter

Applying Logic and Terminology of Inferential

Statistics

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• Population = All 20,000 students.• Sample = The 500 students selected

and interviewed.• Statistic =74% (% of sample that held

a job during the semester).• Parameter = % of all students in the

population who held a job.

Applying Logic and Terminology of Inferential

Statistics (continued)

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• The single most important concept in inferential statistics.

• Definition: The theoretical, probabilistic distribution of a statistic for all possible samples of a given size (n).

• The sampling distribution is a theoretical concept.

The Sampling Distribution

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• Every application of inferential statistics involves 3 different distributions:1. Population: empirical;

unknown2. Sampling Distribution:

theoretical; known3. Sample: empirical;

known• Information from the sample

is linked to the population via the sampling distribution.

Population

Sampling Distribution

Sample

The Sampling Distribution (continued)

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1. Normal in shape.

2. Has a mean equal to the population mean.

3. Has a standard deviation (called the standard error) equal to the population standard deviation, σ, divided by the square root of n.

The Sampling Distribution: Properties

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• Tells us the shape of the sampling distribution and defines its mean and standard deviation. – If repeated random samples of size n are

drawn from a normal population with mean µ and standard deviation σ, then the sampling distribution of sample means will be normal with a mean µ and a standard deviation (standard error) of σ/n

The Sampling Distribution: First Theorem

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• Again, if we begin with a trait that is normally distributed across a population (height, weight) and take an infinite number of equally sized random samples from that population, the sampling distribution of sample means will be normal.

The Sampling Distribution: First Theorem (continued)

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• If repeated random samples of size n are drawn from any (e.g., non-normal) shaped population with mean µ and standard deviation σ, then as n becomes large the sampling distribution of sample means will approach normality, with a mean µ and standard deviation of σ/n – Again, for any trait or variable, even those that

are not normally distributed in the population (income, wealth) as sample size grows larger, the sampling distribution of sample means will become normal in shape.

The Sampling Distribution: Second (Central Limit) Theorem

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• The importance of the Central Limit Theorem is that it removes the constraint of normality in the population.

The Sampling Distribution: Second (Central Limit) Theorem

(continued)

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The Sampling Distribution: Theorem Proofs

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The Sampling Distribution: Theorem Proofs (continued)

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Proof: population mean, 5, equals mean of the sampling distribution, 5, and population standard deviation divided by the square root of n, 2.236/2, is equal to standard deviation of the sampling distribution, 1.581.


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1. Definition: Distribution of a statistic for all possible outcomes of a certain size.

• The statistic for example can be a mean or a proportion.

2. Shape: Normal• Since the sampling distribution is normal, we can use Appendix A

to find areas.

3. Central tendency and dispersion: Mean is same as the population mean; standard deviation of sampling distribution – the standard error – is equal to the population standard deviation divided by the square root of n.

Sampling Distribution: Key Points

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• Role of sampling distribution in inferential statistics: Links the sample with the population:

Population

Sampling Distribution

Sample

Sampling Distribution: Key Points (continued)

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Three Distributions and their Symbols

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Copyright © 2012 by Nelson Education Limited. Chapter 5 Introduction to inferential Statistics: Sampling and the Sampling Distribution 5-1