## Sampling Snoops

### Lisa Ove GibsonBay District Schools

#### Description

Students practice formulating a hypothesis and designing an experiment to test the hypothesis. Then they identify several sampling techniques they can use to test their hypotheses.

#### Objectives

The student identifies the common uses and misuses of probability or statistical analysis in the everyday world.

The student reads and interprets data displayed in a variety of forms including histograms.

The student formulates a hypothesis and designs an experiment.

The student evaluates the hypothesis by making inferences and drawing conclusions based on statistical results.

The student identifies and uses different types of sampling techniques (for example, random, systematic, stratified).

The student knows whether a sample is biased.

#### Materials

-Summary Sheet of Instruction for Sampling Snoops (see Associated File)
-Experimental Design Rubric (see Associated File)
-Histogram Activity Sheet (see Associated File)
-Answer Key for the Histogram Activity Sheet (see Associated File)
-Data Detective Diary #6 Reference Sheet (see Associated File)
-Frequency Table for Eighth Grade Students at Mystery Falls Middle School (see Associated File)
-Calculator Problem (see Associated File)
-Data Detective Diary (used throughout the Unit Plan: Statistical Sleuths)
-Classroom display board (Examples: chart paper, chalkboard, or overhead projector)
-Corresponding writing device
-[The American Heritage College Dictionary, Third Edition], Boston, New York: Houghton Mifflin Company, 2000, page 644

#### Preparations

1. Prepare a large writing space in the front of the classroom to record Detective Diary entries (#4, #5, and #6)and Comments/ideas generated during today's discussion. (Obtain appropriate writing tools).

2. Review concepts necessary to teach students how to read and interpret a histogram, but not how to construct a histogram. To see additional information about histograms, review the Weblinks provided in the Weblinks section of this document and review the Frequency Table for 8th Grade Students at Mystery Falls Middle School on page 6 in the Associated File.

3. Make sure students' detective diaries are available for today’s assignment.

4. Prepare copies of Experimental Design Rubric, Histogram Activity Sheet, Data Detective Diary #6 Reference Sheet, and OPTIONAL Frequency Table for Eighth Grade Students at Mystery Falls Middle School for each student. (See Associated File.)

5. Collect Detective Diaries at the end of Part II of today’s lesson. Allow yourself enough time to evaluate student responses to Detective Diary #6 before returning evaluated diaries to students. Make sure students have their diaries before beginning the next lesson, The Guise of a Graph Gumshoe.

#### Procedures

Disclaimer: The student will not read and interpret data displayed in a variety of ways. They will only read and interpret data displayed in a histogram and a table. Also, the student knows whether a sample is biased, and the student identifies and uses different types of sampling techniques will EACH be reviewed in this lesson; however, the student will not be formatively assessed during this lesson.

Part I:
1. REVIEW for previous lesson Designing Detectives. Use the Experimental Design Rubric to provide specific (verbal or written) feedback for each student regarding his or her hypothesis in Detective Diary Entry #3. (See page 2 of the Associated File.) Students' hypotheses should be plausible, and the design of the experiment to test the hypothesis should be reasonable (affordable and appropriate for an eighth grader to accomplish), clearly explained, and possible within the constraints of the scientific process.

2. REVIEW from previous lesson Designing Detectives. Conclude the discussion of experimental design from lesson #1, Designing Detectives. During this discussion include possible ways to collect statistical data for a math experiment: by researching available information from books, articles, the Internet, and/or professional resources; making observation; and creating and administering surveys using specific sampling techniques (random, systematic, stratified, etc.).

3. NEW MATERIAL: Explain random, systematic, and stratified sampling techniques. (Definitions are provided in the Unit Plan Associated File. Further information is available in Extensions.) Further explain that these sampling techniques are aimed at avoiding bias. In a random sample, each member of the population is selected entirely by chance and has an equal chance of being selected. If a systematic sample were conducted, it would represent every fifth, tenth, fifteenth, etc.) member of the population. A systematic sample is acquired by selecting one member of the population on a random basis and by choosing additional members at evenly spaced intervals until the desired number for the sample space has been collected. If a stratified sample is conducted, the entire population is divided into meaningful subgroups (strata), and then each group is randomly sampled (as described by the random sample above). For more detailed information about these sampling techniques, use the information provided by your textbook or any other classroom resource. Ask students to come up with two examples of their own for each sample type and add this information to their diaries.

4. Students record the definitions of random, systematic, and stratified sampling techniques in their Detective Diaries.

5. Explain the features of a biased sample. When the method used to acquire a sample results in a sample that is systematically different from the population, it is called a biased sample. Example: Remember question #5 from the first diagnostic assessment: Do you think the results of this experiment would have been biased if Marie had conducted her survey the second month of school instead of at the end of the year? Some students might have thought that the time of year that the sample was taken would make the sample biased; however, that is not the case. Technically, this is a biased sample because it is the method used to create the sample, not the actual make up of the sample itself that defines the bias. The statistics that were collected from Mystery Falls Middle School are an example of results from a biased sample because the only population that is represented is from one location only. (If you need more information about these sampling techniques, please refer to the Weblinks section of this lesson plan.)

6. Provide a copy of the Calculator Problem for each student. (See page 7 of the Associated File.) Read the problem aloud or ask students to read the prompt individually. Make sure students comprehend the actual problem. Allow a short class discussion about what students think in regards to using a calculator in math class.

7. Use the Experimental Design Rubric again for each student and explain that it will serve as the guidelines for Detective Diary #4. (See page 2 of Associated File.) Answer any questions students have regarding the rubric.

8. Write this entry in an area that all students can see for Detective Diary Entry #4: Formulate a hypothesis regarding the number of work hours per week saved/spent when the average eighth grade student uses a calculator in math. Design an experiment to test your hypothesis. As students complete their work, collect the Diary entry. Using the Experimental Design Rubric, provide specific written feedback for each student in his or her diary. (See page 2 of the Associated File.) This is the formative assessment.

Part II:
9. Read the definition of Histogram in the Vocabulary for Statistical Sleuths. (See Unit Plan Associated File. Further information is available in Extensions.)

Another way to visualize a histogram is a collection of data that is arranged on a coordinate plane with a vertical and a horizontal axis. A histogram shows frequency for a variable within equal intervals.

The data on the horizontal axis usually represents the appropriate intervals for the kind of data it represents. (In other words, if the data represented the age of humans and there were intervals 150-160+, then there is a problem because no human has ever lived to that ripe old age.)

The vertical axis usually shows the frequency of the data.

A histogram looks like a bar graph; however, there are some differences between the two. In a histogram, both axes represent numerical values. However, in a bar graph, either axis can be any variable (examples-dog, Lisa, TV, etc.) and have no numerical value. On a histogram, there is no space between the bars, but in a bar graph there is some separation between the bars. The horizontal axis is usually represented with equal intervals (examples-# 0-9, 10-19, 20-29, etc.); however, in a bar graph there is only a single variable. A histogram can be misleading if there are too many or not enough bars used in the data set.

10. Remember the question from the diagnostic assessment (question #4): Marie conducted a survey of 390 eighth grade math students at Mystery Falls Middle School at the end of the school year. She asked them what the average number of hours each had spent on math homework per week. The results are located in the Histogram Activity Sheet. (See page 3 of the Associated File.)

11. Distribute copies of Histogram Activity Sheet to each student. (See page 3 of the Associated File.)

12. Identify each of the elements of this histogram with students.
Instruction:
(A) Point out that the vertical and the horizontal axis consist of numerical data.
(B) Point out that the data on the horizontal axis is displayed in equal intervals.
(C) Notice that the range of the data for the average number of hours spent on homework per week for eighth grade math students is from zero to fifteen hours which seems to be an appropriate number of hours for the group that we have collected data.
(D) The vertical axis represents the frequency for the data (Number of Students).
(E) Notice that there is no space between the bars on this histogram.

13. Write this entry in an area that all students can see, Detective Diary Entry #5: Students record the notes taken regarding the specific elements of a histogram.

14. Optional: Use the Frequency Table for Eighth Grade Students at Mystery Falls Middle School Teacher Resource to see the specific statistics for this histogram. (See page 6 of the Associated File.) You may want to show students how a histogram is created from a frequency table; however, this is an optional extension for the teacher to decide because the standard does not address the construction of a histogram. Also, this is question #4 from the diagnostic assessment for the Unit Plan: Statistical Sleuths.

15. Ask students to respond to questions A and B on the Histogram Activity Sheet. (See page 3 of the Associated File.) Collect the activity sheet when students complete their work. This is the formative assessment.

16. Use the Answer Key for the Histogram Activity Sheet to formatively assess (by going desk to desk or by collecting student work) students’ responses to the Histogram Activity Sheet. (See pages 3 and 4 of the Associated File.) Provide corrective formative feedback for students who did not read or interpret the histogram accurately. As a class, review the best answers for questions A and B. Create additional examples or use examples provided by your textbook of histograms for students who need more practice reading and interpreting this kind of data display.

17. Explain to students how these results (on the Histogram Activity Sheet) are an example of a biased sample. (See page 3 of the Associated File.) The statistics that were collected from Mystery Falls Middle School are an example of results from a biased sample because the only group that is represented is from a single location. In order to be a more representative sample of the average eighth grade student, the sample would have to be of people from other areas of the country or from people other than just the students at Mystery Falls Middle School.

18. Distribute Data Detective Diary #6 Reference Sheet. (See page 5 of the Associated File.) Ask students to read the hypothesis, and then read and interpret the table provided.

19. Write this entry in an area that all students can see, Detective Diary Entry #6 for students: Using the statistical results provided in the Data Detective Diary #6 Reference Sheet (see page 5 of the Associated File), answer the question: What conclusions can be drawn based on the inferences from the data? This serves as the formative assessment.

20. Collect students’ Detective Diaries.

21. After collecting the diaries, discuss as a class final conclusions that can be drawn from the data on the Data Detective Diary #6 Reference Sheet. (See page 5 of the Associated File.) Possible solutions for Detective Diary #6 include, but are not limited to the following:
(A) The most popular average number of hours spent on math homework by eighth grade students at Mystery Falls Middle School per week was the interval of four to ten hours.
(B) The range of average number of hours that eighth grade students at Mystery Falls Middle School spent on math homework per week was zero to eleven plus.
(C) The mode of the data was four to ten hours of math homework per week.
(D) A total of 130 students out of 390 students received a B average or higher when they did between four to ten hours of math homework per week (60 + 70 = 130).
(E) Only six students who did zero to three hours of homework per week in math received a B average or higher for the year and only forty students who did eleven plus hours of homework per week received a B average.
(F) Based on these statistics, the data from the table supports Mario's hypothesis.

22. Formatively assess students’ conclusions in Detective Diary #6. The rubric is not used to formatively assess this diary entry.

23. Return Detective Diaries before beginning tomorrow’s lesson, The Guise of a Graph Gumshoe.

#### Assessments

Students are taught the definition for three sampling techniques: random, systematic, and stratified, but not how to conduct the samples themselves. The student will not learn how to use these sampling techniques in this lesson.

Formative assessment: Students accurately complete the Histogram Activity Sheet. See the Answer Key for the Histogram Activity Sheet for possible responses. (See pages 3-4 of the Associated File.)

Formative assessment: Students complete Detective Diary Entry #4. Use Experimental Design Rubric to assess students' work. (See page 2 of the Associated File.)

Formative assessment: Students complete Detective Diary Entry #6. Use answers provided in step #21 of these Procedures to assess possible student answers.

#### Extensions

1. The Beacon Unit Plan associated with this lesson can be viewed by clicking on the link located at the top of this page or by using the following URL: http://www.beaconlearningcenter.com/search/details.asp?item=2958. Once you select the unit’s link, scroll to the bottom of the Unit Plan page to find the section Associated Files. This section contains links to the Unit Plan Overview, Diagnostic and Summative Assessments, and other Associated Files (if any).

2. This is the second lesson plan in the series.

3. Allow students to peer–assess each other’s diary entries if they are at a point where they can assess the content accurately enough to provide effective feedback.

This site offers and example of another lesson plan using similar concepts to the ones used in this lesson.
Rice (Gohan) Observations

This site offers comparisons and contrasts between histograms and bar graphs.
Histograms vs. Bar Graphs Discussion

This site offers a technical definition for a histogram.
Histogram

This site offers a 7th grade definition for histograms.
Histogram

This site offers use of different sampling techniques in a real-world context and offers a section titled, Sampling Techniques and Terminology.
Sampling Techniques and Terminology

This site offers additional definitions for random, stratified, and systematic sampling techniques.
Recommendations on Sampling

#### Attached Files

The Associated File for Sampling Snoops.     File Extension: pdf