- See your Course Syllabus for the reading assignments.
- Work through the Lesson 4 online notes that follow.
- Complete the Practice Questions and Lesson 4 Assignment.

After successfully completing this lesson, you should be able to:

- Distinguish between a population, sample, and sampling frame.
- Interpret and identify the factors that affect the margin of error.
- Identify types of probability samples and judgment samples.
- Apply the "Difficulties and Disasters" in sampling to real world problems.
- Identify all steps used and issues addressed by the Gallup Poll.

- sample surveys
- experiments
- observational studies
- case studies
- unit (sampling unit)
- population
- sample
- sampling frame
- census
- margin of error (ME)
- sample size (
*n)* - probability sampling
- judgment sampling
- simple random sample
- stratified sampling
- cluster sampling
- systematic sampling
- voluntary sample
- haphazard (convenience) sample
- gallup poll
- nonresponse (no response or voluntary response)
- random-digit dialing
- selection bias
- sample percent
- population percent

*Section 4.1. Chapter 4 in Textbook *

**Overview**

In this lesson, we will add to our knowledge base by explaining ways to obtain appropriate samples for statistical studies.

The following research strategies are described in this section of the textbook.

**Sample Surveys****Experiments****Observational Studies****Meta-Analyses**(also covered in Chapter 25--not required for the course)**Case Studies**

**Terms Used with Sample Surveys (Chapter 4 Section 4.2 in Textbook)**

It is first necessary to distinguish between a **census** and a **sample survey**. A **census** is a collection of data from every member of the population, while a **sample survey** is a collection of data from a subset of the population. A **sample survey** is a type of **observational study**. Obviously, it is much easier to conduct a sample survey than a census. The remaining sections of this lesson (Chapter 4) will discuss issues about **sample surveys**.

Of the many terms that are used with **sample surveys**, the following four need the most clarification because of how they are connected to each other.

**Sampling Unit**: The individual person or object that has the measurement (observation) taken on them / it**Population**: The entire group of individuals or objects that we wish to estimate some characteristic's (variable's) value**Sampling Frame**: The list of the sampling units from which those to be contacted for inclusion in the sample is obtained. The sampling frame lies between the population and sample. Ideally the sampling frame should match the population, but rarely does because the population is not usually small enough to list all members of the population.**Sample**: Those individuals or objects who provided the data collected

*Figure 4.1 Relationship between Population, Sampling Frame and Sample*

**Example 4.1. Who are those angry women? **

(Streitfield, D., 1988 and Wallis, 1987)

Recalling some of the information from **Example 2.1** in **Lesson 2**, in 1987, Shere Hite published a best-selling book called *Women and Love: A Cultural Revolution in Progress*. This 7-year research project produced a controversial 922-page publication that summarized the results from a survey that was designed to examine how American women felt about their relationships with men. Hite mailed out *100,000 *fifteen-page questionnaires to women who were members of a wide variety of organizations across the U.S. Questionnaires were actually sent to the leader of each organization. The leader was asked to distribute questionnaires to all members. Each questionnaire contained 127 open-ended questions with many parts and follow-ups. Part of Hite's directions read as follows: "Feel free to skip around and answer only those questions you choose." Approximately 4500 questionnaires were returned.

In **Lesson 2**, we determined that the

**population**was all American women.**sample**was the 4,500 women who responded.

It is also easy to identify that the **sampling unit** was an American woman. So, the key question is "What is the **sampling frame**?" Most people think the **sampling frame** was the 100,000 women who received the questionnaires. However, this answer is not correct because the **sampling frame** was the list from which the 100,000 who were sent the survey was obtained. In this instance, the **sampling frame** included all American women who had some affiliation with an organization. There is no statistical term to attach to the 100,000 women who received the questionnaire. However, *if the response rate had been 100%*, the sample would have been the 100,000 women who responded to the survey.

You should also remember that ideally the sampling frame should include the entire population. If this is not possible, the sampling frame should appropriately represent the desired population. In this case, the sampling frame of all American women who were "affiliated with some organization" did not appropriately represent the population of all American women. In **Lesson 2**, we called this problem **selection bias**.

**Chapter 4** of your text also lists three difficulties that are possible when samples are obtained for surveys. These three difficulties, which happen to be possible with this example, include:

**Using the wrong sampling frame**. We just discussed this problem in the

preceding paragraph. This problem is also called selection bias.**Not reaching the individuals selected.**Because the questionnaire was sent to leaders of organizations, there is no guarantee that these questionnaires actually reached the women who were supposed to be in the sample.**Getting****"no response****" or a****"volunteer response.****"**In**Lesson 2**, we learned that this survey has a problem with**nonresponse bias**because of the low response rate. This problem can also be called "**no response**" or "**volunteer response**."

Sample surveys are generally used to estimate the percentage of people in the population that have a certain characteristic or opinion. If you follow the news, you will probably recall that most of these polls are based on samples of size 1000 to 1500 people. So, why is a sample size of around 1000 people commonly used in surveying? The answer is based on understanding what is called the **margin of error**.

**The margin of error:**

- measures the accuracy of the percent estimated in the survey
- is calculated using a formula that includes the sample size (
*n*)

For a sample size of *n* = 1000, the **margin of error is \(\frac {1}{\sqrt{n}}=\frac{1}{\sqrt{1000}}=0.03\)** , or about 3%.

Even though you will not be asked to calculate a **margin of error** in this course, you should remember the margin of error formula and that the **margin of error** formula depends only on the size of the sample. The size of the population is not used in the calculation of the **margin of error**. So, a percentage estimated by a *selected* sample size will have the same **margin of error (accuracy)**, regardless of whether the population size is 5,000 or 5 billion. It also helps that pollsters believe that an accuracy of ± 3% is reasonable with surveys.

So what does the margin of error represent? The following statement represents the **generic interpretation of a margin of error**.

**Generic Interpretation**: If one obtains many samples of the same size from a defined population, the difference between the sample percent and the true population percent will be within the margin of error, at least 95% of the time.

**Key Features of the Interpretation of the Margin of Error**

- Statistical theory is often based on what would happen if the survey were repeated many times. So, even though a pollster usually obtains only one sample, the pollster must remember that the margin of error interpretation is based on doing the survey repeatedly under identical conditions.
- The margin of error represents the largest distance that would occur between the
**sample percent**, which is the percent obtained by the poll, and the true**population percent**, which is unknown because we have not sampled the entire population. - In statistics, when talking about the margin of error, it is just not possible to say that we are 100% certain that with all samples the difference between the sample percent and the population percent will be within the margin of error. So, statisticians work with reasonable conditions so that one can say that at least 95% of the time, the difference between the sample percent and the population percent will be within the margin of error.

**Example 4.2. Margin of Error**

Suppose a recent poll based on 1000 Americans finds that 55% approve of the president's current educational plan. Since the sample size is 1000, the margin of error is about 3%. These poll results suggest that 55% ± 3% of all Americans approve of the president's current economic plan. What is the correct interpretation of this margin of error?

**Margin of Error Interpretation**

The difference between our sample percent and the true population percent will be within 3%, at least 95% of the time. This means that we are almost certain that 55% ± 3% or (52% to 58%) of all Americans approve of the president's current educational plan. **Because the range of possible values from this poll all fall above 50%**, we can also say that we are pretty sure that a majority of Americans support the president's current educational plan. If any of the range of possible values would have been 50% or less, then we would not have been able to say that the majority supported the plan. The range of values (52% to 58%) is called a** 95% confidence interval**. We will go into further detail about confidence intervals in **Lesson 7**.

There is a predictable relationship between sample size and margin of error. The numbers found in **Table 4.1** help to explain this relationship.

*Table 4.1. Calculated Margins of Error for Selected Sample Sizes*

Sample Size (n) | Margin of Error (M.E.) |
---|---|

200 | 7.1% |

400 | 5.0% |

700 | 3.8% |

1000 | 3.2% |

1200 | 2.9% |

1500 | 2.6% |

2000 | 2.2% |

3000 | 1.8% |

4000 | 1.6% |

5000 | 1.4% |

From this table, one can clearly see that as sample size increases, the margin of error decreases. In order to add additional clarity to this finding, the information from **Table 4.1** is also displayed in** Figure 4.2.**

*Figure 4.2 Relationship Between Sample Size and Margin of Error*

In **Figure 4.2**, you again find that as the sample size increases, the margin of error decreases. However, you should also notice that the amount by which the margin of error decreases is substantial between samples sizes of 200 and 1500. This implies that the accuracy of the estimate is strongly affected by the size of the sample. In contrast, the margin of error does not substantially decrease at sample sizes above 1500. Therefore, pollsters have concluded that it is not worth it to spend additional time and money for samples that contain more than 1500 people.

Sampling Methods can be classified into one of two categories:

**Probability Sampling**: Sample has a known probability of being selected**Judgment Sampling**: Sample does not have known probability of being selected

**Probability Sampling**

In probability sampling it is possible to both determine which sampling units belong to which sample and the probability that each sample will be selected. The following sampling methods, which are listed in **Chapter 4**, are types of **probability sampling**:

**Simple Random Sampling (SRS)****Stratified Sampling****Cluster Sampling****Multistage Sampling****Random-Digit Dialing****Systematic Sampling**

Of the five methods listed above, students have the most trouble distinguishing between **stratified sampling** and **cluster sampling**.

**Stratified Sampling** is possible when it makes sense to partition the population into groups based on a factor that may influence the variable that is being measured. These groups are then called strata. An individual group is called a stratum. With **stratified sampling **one should:

- partition the population into groups (strata)
- obtain a simple random sample from each group (stratum)
- collect data on each sampling unit that was randomly sampled from each group (stratum)

**Stratified sampling** works best when a heterogeneous population is split into fairly homogeneous groups. Under these conditions, stratification generally produces more precise estimates of the population percents than estimates that would be found from a simple random sample. **Table 4.2** shows some examples of ways to obtain a stratified sample.

*Table 4.2. Examples of Stratified Samples*

Example 1 | Example 2 | Example 3 | |

Population | All people in U.S. | All PSU intercollegiate athletes | All elementary students in the local school district |

Groups (Strata) | 4 Time Zones in the U.S. (Eastern,Central, Mountain,Pacific) | 26 PSU intercollegiate teams | 11 different elementary schools in the local school district |

Obtain a Simple Random Sample | 500 people from each of the 4 time zones | 5 athletes from each of the 26 PSU teams | 20 students from each of the 11 elementary schools |

Sample | 4 × 500 = 2000 selected people | 26 × 5 = 130 selected athletes | 11 × 20 = 220 selected students |

**Cluster Sampling** is very different from Stratified Sampling. With **cluster sampling** one should

- divide the population into groups (clusters).
- obtain a simple random sample of so many clusters from all possible clusters.
- obtain data on every sampling unit in each of the randomly selected clusters.

It is important to note that, unlike with the strata in stratified sampling, the clusters should be microcosms, rather than subsections, of the population. Each cluster should be heterogeneous. Additionally, the statistical analysis used with cluster sampling is not only different, but also more complicated than that used with stratified sampling.

*Table 4.3. Examples of Cluster Samples*

Example 1 | Example 2 | Example 3 | |

Population | All people in U.S. | All PSU intercollegiate athletes | All elementary students in a local school district |

Groups (Clusters) | 4 Time Zones in the U.S. (Eastern,Central, Mountain,Pacific.) | 26 PSU intercollegiate teams | 11 different elementary schools in the local school district |

Obtain a Simple Random Sample | 2 time zones from the 4 possible time zones | 8 teams from the 26 possible teams | 4 elementary schools from the l1 possible elementary schools |

Sample | every person in the 2 selected time zones | every athlete on the 8 selected teams | every student in the 4 selected elementary schools |

Each of the three examples that are found in **Tables 4.2** and **4.3 **were used to illustrate how both stratified and cluster sampling could be accomplished. However, there are obviously times when one sampling method is preferred over the other. The following explanations add some clarification about when to use which method.

**With Example 1**: Stratified sampling would be preferred over cluster sampling, particularly if the questions of interest are affected by time zone. Cluster sampling really works best when there are a reasonable number of clusters relative to the entire population. In this case, selecting 2 clusters from 4 possible clusters really does not provide much advantage over simple random sampling.**With Example 2**: Either stratified sampling or cluster sampling could be used. It would depend on what questions are being asked. For instance, consider the question "Do you agree or disagree that you receive adequate attention from the team of doctors at Sports Medicine when injured?" The answer to this question would probably not be team dependent, so cluster sampling would be fine. In contrast, if the question of interest is "Do you agree or disagree that weather affects your performance during an athletic event?" The answer to this question would probably be influenced by whether or not the sport is played outside or inside. Consequently, stratified sampling would be preferred.**With Example 3**: Cluster sampling would probably be better than stratified sampling if each individual elementary school appropriately represents the entire population. Stratified sampling could be used if the elementary schools had very different locations (i.e., one elementary school is located in a rural setting while another elementary school is located in an urban setting.) Again, the questions of interest would affect which sampling method should be used.

**Judgment Sampling**

The following sampling methods that are listed in your text are types of **judgment sampling**:

**volunteer samples****haphazard (convenience) samples**

Since **judgment sampling** is based on human choice rather than random selection, statistical theory cannot explain what is happening. In your textbook, the two types of judgment samples listed above are called "sampling disasters."

**Section 4.2. Article: "How Polls are Conducted"**

The article is exceptional and provides great insight into how major polls are conducted. When you are finished reading this article you may want to go to the Gallup Poll Web site, http://www.gallup.com, and see the results from recent Gallup polls.

It is important to be mindful of margin or error as discussed in this article. We all need to remember that public opinion on a given topic cannot be appropriately measured with one question that is only asked on one poll. Such results only provide a snapshot at that moment under certain conditions. The concept of repeating procedures over different conditions and times leads to more valuable and durable results. Within this section of the article, there is also an error: "in 95 out of those 100 polls, his rating would be between 46% and 54%." This should instead say that in 95 out of those 100 polls, the true population percent would be within the confidence interval calculated. In 5 of those surveys, the confidence interval would not contain the population percent.

Answer the following Practice Questions to check your understanding of the material in this lesson.

*Come up with an answer to these questions by yourself and then click the icon on the left to reveal the answer.*

1. Which of the following is not an example of probability sampling?

- a. simple random sampling
- b. cluster sampling
- c. convenience sampling
- d. stratified sampling

2. Which of the following surveys would have the smallest margin of error?

- a. a sample size of n = 1,600 from a population of 50 million
- b. a sample size of n = 500 from a population of 5 billion
- c. a sample size of n = 100 from a population of 10 million

3. Suppose a recent survey finds that 80% of Penn State students prefer that fall semester begins after Labor Day. The results of this survey were based on opinions expressed by 200 Penn State students. Which of the following represents the calculation of the margin of error for this survey?

- a. 200
- b. 1/200
- c. \(1/ \sqrt{200}\)
- d. \(\sqrt{200}\)

4. Suppose a margin of error for a poll is 4%. What is the correct interpretation of the margin of error for this poll? In about 95% of all samples of this size, the ________________.

- a. difference between the sample percent and the population percent will be within 4%.
- b. probability that the sample percent does not equal the population percent is 4%.
- c. probability that the sample percent does equal the population percent is 4%.
- d. difference between the sample percent and the population percent will exceed 4%.

5. In order to survey the opinions of its customers, a restaurant chain obtained a random sample of 30 customers from each restaurant in the chain. Each selected customer was asked to fill out a survey. Which one of the following sampling plans was used in this survey?

- a. cluster sampling
- b. stratified sampling