Chapter 18 Sampleing Distribution Models Reading Guide Ap Statistics

Part 1

This chapter is broken into two parts.  In this chapter we will learn about a very important concept in statistics.  In fact, this concept forms the foundation of most of the topics we will cover for the rest of this course.

Like the title of the chapter says, we will now start to look at models that are created from samples, or "sampling distribution models".  Part 1 will focus on proportions.

Before we go any farther, you need to be prepared to take notes and don't forget that you should also be reading your text book to get a deeper understanding of these topics.  Part 1 is covered on pages 410 to 416 in your book.

Sampling Distributions for Proportions

We should recall that a proportion can summarize the results of a categorical variable.  But just collecting one sample proportion does not tell the entire story of the population.  This is important to recognize, because the purpose of the science of statistics is to provide a way to make prediction about populations using samples.

An amazing outcome in statistics is that if you collect multiple sample proportions (I mean lots and lots of samples) and put the results of those sample proportions into their own distribution (histogram) you will find that the distribution takes on a Normal shape.  In fact, not only would the shape of the distribution of sample proportions take on a bell-shaped curve or Normal shaped curve, the center of the distribution would be located at the true population proportion.

The following video demonstrates how collecting multiple samples and putting them into their own "sampling distribution" creates a brand new Normal distribution that we can use.  The applet in the video uses samples that collect means instead of proportions, but the concept applies to both means and proportions.

The video also gives the properties of a sampling distribution.  (You should probably copy these down.)

Sampling Distributions

Again, even though the video uses samples of means instead of proportions, the same properties can be applied for sampling distributions of proportions.

So, if we are going to use a normal model created from these samples, we need to define the mean of the sampling distribution model and the standard deviation of the sampling distribution model.

For proportions, the mean of the sampling distribution for proportions is equal to the population proportion and the standard deviation of the sampling distribution for proportions is given by the formula:

A sampling distribution of proportions can now be defined by the notation

Assumptions & Conditions

Before we are allowed to use any sampling distribution model there are a few things we need to check.  These things are called "assumptions and conditions".

Here are those assumptions and conditions:

1. The Randomization Condition - We are supposed to RANDOMLY collect our sample so that it REPRESENTS the population.  A lot of times you will be told that the sample was collected at random or was "randomly selected" and that would be enough to satisfy this condition.  But if you are not sure that the samples were randomly collected you must "ASSUME" that the sample represents the population to satisfy this condition.

2. The 10% Condition - If a sample is collected without replacement (and most samples do not replace a respondent or test subject back into the population), then technically we don't have independent events.  Therefore we must check the sample size and make sure that it is less than 10% of the population size.  As long as this is true, then this condition will be satisfied.

3. The Success/Failure Condition - Even though a sample size needs to be less than 10% of the population, we still need a large enough sample size to get accurate results.  That's why this condition needs to be checked.  As long as  np > 10 and nq > 10, then we are "ok" and this condition will be satisfied.  In other words, as long as we have at least 10 successes and at least 10 failures, our sample size is large enough.

How Can You Use These Sampling Distributions?

This video shows you how to find probabilities using a sampling distribution for proportions.

Sampling Distribution for Proportions

This marks the end of  Part 1 for Chapter 18.  Part 2 starts below.

Part 2

At this point we will shift our focus from proportions to means.

The Central Limit Theorem

In part 1 of this chapter there was a video that introduced THE CENTRAL LIMIT THEOREM.

There are two properties of the central limit theorem that we need to remember:

1.  The mean of a sampling distribution will be equal to the population mean.

2. A sampling distribution can be approximated by a Normal model. (Bell-Shaped)

Just like with sampling distributions for proportions, when working with sample MEANS, you need to define the mean of your Normal model and the standard deviation of your Normal model.

**The mean of the Normal model for means is equal to the population mean.  And is denoted by . . . **

**The Standard Deviation for the sampling distribution of means is given by the formula . . . **

Assumptions and Conditions for Means

Just as in part 1, we must check certain assumptions and conditions before we are allowed to use a sampling distribution for means.  These are similar to the assumptions and conditions for proportions, but slightly different.

1. Randomization Condition - This is the same for both means and proportions, so just scroll up to see the details for this condition again.

2. Independence Assumption or 10% Condition - You must assume that the observations are independent, but if you cannot make that assumption, you must check the 10% condition above.

3. Large Enough Condition - We will talk about this condition more in Chapter 24, but for now we will just assume that our sample is large enough.

How Can I Use the Central Limit Theorem for Means?

The following video should help answer that question.

Sampling Distributions for Means

One Last Big Topic

We don't always have the population standard deviation or the population proportion.  If we don't have them, then we can use the sample proportion or sample standard deviation to estimate the standard deviation of the sampling distribution.  This is called STANDARD ERROR.  Read pgs 424 and 425, in your textbook, for more information.

Try It Yourself

Ch 18 Try It Yourself

Ch 18 Try It Yourself

True or False Practice

Any Questions

Post any questions you have from this chapter below or feel free to answer other people's questions.  This is your opportunity to create a quality learning community for the class.

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Chapter 18 Sampleing Distribution Models Reading Guide Ap Statistics

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