Algebra 1 Summer Institute 2014 The Sampling Proclivity Summary Goals Understand what is average by simple random sampling Recognize the variability in subjective and random sampling techniques and understand how to measure it Recognize how bias can occur in the sampling process Draw conclusions about the effect of sample size on statistical information Explore the sampling distributions of stratified random samples and clustered samples Technology Source How we select a sample is extremely important. Improper or biased sample selection can produce misleading conclusions. Sample selection is biased if it systematically favors certain outcomes. Materials Poster paper Markers Participant Handouts LCD Projector Facilitator Laptop Excel 1. Random Rectangles 2. Sampling Rectangles 3. Distribution of Sample Average 4. Sample Size 5. Data Record 6. Sampling Method Navigating through Data Analysis in Grades 9-12 Estimated Time 90 minutes Mathematics Standards Common Core State Standards for Mathematics MAFS.6.SP.1: Develop understanding of statistical variability 1.1: Recognize that a statistics question as one that anticipates variability in the data related to the question and accounts for it in the answers. MAFS.6.SP.2: Summarize and describe distributions 2.5: Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or measurement variable 1.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme points (outliers). Standards for Mathematical Practice 1 Algebra 1 Summer Institute 2014 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically Instructional Plan In data analysis, we use graphs, tables, and numerical summaries to study the variation present in our data. Often, we want to extend our interpretation to a larger group beyond the particular group studied. Such generalizations are only valid, however, if the data we examine are representative of that larger group. If not, our interpretation may misrepresent the larger group! The entire group that we want information about is called the population. We can gain information about this group by examining a portion of the population, called a sample. To gain useful information, the sample must be representative of the population. A representative sample is one in which the relevant characteristics of the sample members are generally the same as the characteristics of the population. There are several good reasons that we use samples to study populations; chief among them are feasibility and cost. For instance, in a nationwide political survey of the population of all voters in the United States, it would be difficult, if not impossible, to poll every voter. It would also be quite expensive. Statistical theory shows that a survey of a 1,000 carefully selected voters suffices to represent the opinions of the millions of people in the population of voters. Another problem in answering questions about a population arises when we want to inspect or test products. For example, testing an air bag to see if it works properly averages that we have to destroy it. We certainly can't test every air bag, but testing a carefully selected sample of air bags will tell us what we need to know about all the air bags in the population. How we select a sample is extremely important. Improper or biased sample selection can produce misleading conclusions. Sample selection is biased if it systematically favors certain outcomes. If we select only Democrats to participate in a political survey, the outcome will reflect Democrats' opinions, but not other political parties'. If we personally select a sample of students we know and like for a school survey, we have just eliminated the differing opinions of those whom we do not know and like. We need to select our sample in an unbiased fashion. Random sampling is a way to remove bias in sample selection. For example, to pick a random sample of 20 people out of a population of a 1,000, you might put all 1,000 names in a hat, then draw 20 of them. Random sampling attempts to reduce bias in 2 Algebra 1 Summer Institute 2014 sample selection, since every member of the population has an equal chance of being selected. There are different random sampling methods and some are simple random sampling, stratified sampling, and cluster sampling. Simple random sampling is the basic sampling technique where we select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has an equal chance of being included in the sample. Every possible sample of a given size has the same chance of selection; i.e. each member of the population is equally likely to be chosen at any stage in the sampling process. There may often be factors which divide up the population into sub-populations (groups / strata) and we may expect the measurement of interest to vary among the different sub-populations. This has to be accounted for when we select a sample from the population in order that we obtain a sample that is representative of the population. This is achieved by stratified sampling. A stratified sample is obtained by taking samples from each stratum or sub-group of a population. (Slides 2, 3, 4) The “Random Rectangles” handout shows the population of rectangles that participants will use for the following activities. Each small square in the figure represents an area of 1 square unit. NOTE: in this activity we refer to the measures of center as “average” since the term mean will be formally introduced in later chapters. 1. Distribute the handouts to all participants of “Sampling Rectangles” activity pages and “Random Rectangles” which will be used with all the activities in this session. The activity directs students to select 5 rectangles that they think are “typical” of the group of rectangles on the page. Give them a limited time – maybe 15 seconds – to choose their 5 typical rectangles and to record the areas. 2. Next, participants find the average area of the rectangles that they have chosen as typical. (Slide 5) 3. Participants choose five more rectangles, this time by generating random numbers between 1 and 100 and selecting the rectangles that correspond to those numbers. (The random numbers could be generated by Excel by using the command “=RANDBETWEEN(1,100)”). They record their data for this “random” sample of rectangles, again calculating the average for the five rectangles’ areas. (Slide 6) 4. Now the participants are ready to pool their results. They could make a dot plot of the two distributions – one for the average of the areas of the randomly chosen rectangles and the other for the average of the areas of the subjectively chosen rectangles. Participants could use the “Distribution of Sample Average Areas” to make the dot plot. Display the sampling distributions. Ask participants to record a measure of center (decide together if the average is going to be used) and a 3 Algebra 1 Summer Institute 2014 measure of spread (range) for both distributions in order to compare the two methods of choosing a sample of rectangles. The activity Sampling Rectangles can help participants begin to see the difference between a sample chosen by using one’s judgment and one chosen at random. Randomization People often think that they can make good judgments about what is typical. This activity focuses on what randomization brings to the sampling process. This activity highlights three facts: i) ii) iii) Subjectively chosen samples may be biased True random samples are in fact carefully chosen to ensure that every set of a specified size consisting of individuals in the population under consideration is equally likely to be selected. Though an individual event is not predictable, there is an underlying pattern in long-term random behavior. 5. The population in this activity is the given set of rectangles. Because the entire population is known (which is not usually the case), the population average – the parameter – of all the rectangles’ areas can also be known. It can readily be determined to be 7.42 square units. Draw a vertical line on the display of the distributions through 7.42 on each plot to illustrate how the distributions from the two sampling techniques vary with respect to the actual population average. How do the two sample averages compare to 7.42? If using the median instead of the average, the population median is 6. (Slide 7) Sample size In this activity the 100 rectangles will still be used. Here participants investigate sample sizes of 10. 6. After participants have worked individually to find the sample average area of one sample of 10 rectangles, using random numbers, they can pool their data and work as a class to plot the simulated sampling distribution of the sample average areas for the new sample size. Compare the simulated sample average distributions of size 5 and 10 by using the dot plots. They could also use graphing software that allows results to be displayed. Discuss with the whole group the following questions: a. Is the distribution of the sample size 10 more like a bell shape? b. What distribution shows a center closer to the population average for all 100 rectangles of 7.42? c. What distribution shows a wider spread? 4 Algebra 1 Summer Institute 2014 d. In general, what can a sample of 5 or 10 individuals tell about a particular population? e. How many individuals should be sampled? Formal statistical inference explores how confident one can be in reasoning about a population from a sample. Participants’ work can communicate to them the important message that as the sample size increases, the variability of the sampling distribution for a sample statistics decreases. (Slide 8) Other Sampling Methods Stratified Random Sampling In this method the population is divided into two or more groups called strata, according to some criterion, such as geographic location or grade level, and subsamples are randomly selected from each group. Task A of the activity Sampling Methods illustrates these ideas. Here the “width” of a rectangle is understood as the rectangle’s horizontal dimension. Table 1 and Table 2 show the numbers of the rectangles divided in to groups: those with widths less than 3, and those with width greater than or equal to 3. These two strata are not of equal size. There are 59 rectangles with widths less than 3 and 41 with widths greater than or equal to 3. 7. Participants choose 5 rectangles (using the random number generator) from each strata – that is 5 small rectangles and 5 large rectangles for a sample of 10. To calculate the sample average of the rectangles’ areas for the combined strata sample, they must use the population proportion, as follows: (Slide 9) 59 41 ∙ (𝑚𝑒𝑎𝑛 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑓𝑟𝑜𝑚 𝑇𝑎𝑏𝑙𝑒 1 ) + 100 100 ∙ (𝑚𝑒𝑎𝑛 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑓𝑟𝑜𝑚 𝑇𝑎𝑏𝑙𝑒 2) 8. Pooling the information from all participants, create a simulated sampling distribution and find the sample average. Describe the shape, center and spread of the distribution. Cluster Sampling Cluster sampling is often used when studies look across large populations whose members may be widely dispersed and a reliable list of all the individuals or objects in the population is not available. 5 Algebra 1 Summer Institute 2014 In Task B of the activity Sampling Methods, participants explore the technique of cluster sampling by considering the 100 rectangles again, this time in clusters that they will sample to make another simulated sampling distribution of the sample average areas. Table 3 has the rectangles divided in clusters of 5 by proximity. 9. Using a random number generator, participants select 2 of the twenty clusters. Find the average area of the sample of 10 rectangles. Compile the class data and show all sample averages in a simulated sample distribution. (Slide 10) 10. Have a class discussion comparing the sampling methods used in this session. a. Does the sample size have an effect on the results of the investigation? b. What observations can be made about the sampling methods explored: simple random sampling, stratified random sampling, and cluster sampling? (Slide 11) 6