Author: Walter Vredeveld Learning Task: How Tall are Our Students? In this task, students will gather sample data, calculate statistical parameters and draw inferences about populations. Students will learn to take samples, understand different types of sampling (random, convenience, self-selected, and systematic) and sampling bias, extrapolate from a sample to a population and use statistical methodologies to make predictions regarding the full population. STANDARDS ADDRESSED IN THIS TASK: Advanced Algebra CCGPS Standards MCC9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more data sets. MCC9-12.S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. MCC9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Math III GPS Standards MM3D1. Students will create probability histograms of discrete random variables, using both experimental and theoretical probabilities. MM3D2. Students will solve problems involving probabilities by interpreting a normal distribution as a probability histogram for a continuous random variable. a. Determine intervals about the mean that include a given percent of data. c. Estimate how many items in a population fall within a specified interval. Advanced Mathematical Decision Making (AMDM) GPS Standards DATA ANAYLSIS AND PROBABILITY - Students will explore representations of data and models of data as tools in the decision making process. MAMDMD2. Students will build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays. MAMDMD3. Students will apply statistical methods to design, conduct, and analyze statistical studies. Mathematical Process Standards GPS MM#P1.Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. 1|Page d. Monitor and reflect on the process of mathematical problem solving. MM#P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. STANDARDS FOR MATHEMATICAL PRACTICE 1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE In order for students to be successful, the following skills and concepts need to be maintained: Students will need to be familiar with histograms, normal distributions and with statistical measures of central tendency (mean) and dispersal (standard deviation) Students must understand the use of z-scores to make predictions about a population from smaller samples. Students must understand sampling techniques and sampling types, including random, convenience, self-selected, and systematic and sampling bias Students must be able to read and create tabular data to be analyzed. ESSENTIAL QUESTIONS How do I conduct a sampling operation to gather data about a population? How do I develop statistical parameters of the data that was gathered? How do I make predictions about the total population, based on the sample? MATERIALS: Scientific calculator. Tape Measure or other measurement device (“height pipes” can be created from thin 1 wall 12 inch PVC pipe). GROUPING Teams of two to four students. 2|Page TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION I. Purpose To give students the experience of gathering sample data, calculating statistical parameters and drawing inferences about populations. II. Data Sampling (Height Measurement) a. Each team will utilize a measuring tape or “height pipe” and the “Height Sampling Data” worksheet. They will measure other students, recording Grade, Gender and Height (to the nearest inch), and record their observations on the worksheet. (This works best when a wide range of students are available, e.g. during lunch) b. Teams should NOT duplicate students measured, i.e. ask the subject if they have already been measured by another team. If they answer yes, then find another student to measure. III. Develop Statistical Parameters of the Sample a. Draw a Histogram Chart for your sample data. Height Sample Histogram 16 14 Number of Students 12 10 8 6 4 2 0 45 50 55 60 65 70 Height in Inches 75 80 85 b. Calculate the mean, standard deviation and % of the total population (total number of students in the population*) for your sample. (Use the Sample Standard Deviation S on your calculator). Sample Size (n) % Total Population* * Need to obtain total population numbers for comparison. 3|Page Mean ( x ) Standard Deviation (S) c. Draw a Normal Distribution Chart for your sample data. Normal Distribution - Student Height 25 # of Students 20 15 10 5 0 21 22 -3ϭ IV. -2ϭ 23 x -1ϭ Height - Inches 24 +1ϭ 25 +2ϭ +3ϭ In-depth Sample Analysis a. “Disaggregate” your data into gender and grade and compare your sample size to the total school population* breakdown. Grade Female Sample Mean Total School* Male % Sample 9th 10th 11th 12th TOTAL * Need to obtain total population numbers for comparison. 4|Page Mean Total School* % Total School* b. Draw a Histogram of your Means by Gender and Grade. Height 74 72 70 68 66 64 62 60 58 V. 9th 9th 10th 10th 11th 11th 12th 12th Female Male Female Male Female Male Female Male Draw Inferences about the Entire Population Using your normal distribution and the z-score table determine what percent of students are predicted to be: Less than 5 feet VI. Between 5 feet and 6 feet − = Greater than 6 feet Representative Sample? a. How would you describe your sampling methods? __ Self-Selected __ Convenience __ Systematic __ Random Describe your method: b. Do you think that your sample is representative of the total population? Why? 5|Page Height Sampling Data Worksheet # Grade Gender 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 6|Page Height (inches) # Grade Gender 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Height (inches) Z-Score Table 7|Page Answers will vary depending on the sample data. This answer key is representative of how the response should be structured. Data Sample - Answer Key (For Sample below) II. Data Sampling (Height Measurement) 8|Page III. Develop Statistical Parameters of the Sample a. Draw a Histogram Chart for your sample data. Height Sample Histogram 8 Number of Students 7 6 5 4 3 2 1 0 60 61 62 63 64 65 66 68 69 71 72 73 74 75 Height in Inches b. Calculate the mean, standard deviation and % of the total population (total number of students in the population*) for your sample. (Use the Sample Standard Deviation S on your calculator). Sample Size (n) % Total Population* Mean ( x ) Standard Deviation (S) 47 3.4% 69.0 4.0 * Need to obtain total population numbers for comparison. Total School population of 1,374 9|Page c. Draw a Normal Distribution Chart for your sample data. Normal Distribution - Student Height 90 80 # of Students 70 60 50 40 30 20 10 0 IV. 1 57 -3ϭ 2 61 -2ϭ 3 65 -1ϭ 4 69 x Height - Inches 5 73 +1ϭ 77 +2ϭ 6 81 +3ϭ 7 In-depth Sample Analysis a. “Disaggregate” your data into gender and grade and compare your sample size to the total school population* breakdown. Grade 9th 10th 11th 12th TOTAL Female Sample Mean 5 3 5 2 15 65.0 65.7 63.8 65.0 64.7 Total School* 189 151 141 160 641 Male % Sample Mean 2.6% 2.0% 3.5% 1.3% 3 3 17 9 32 70.0 73.7 69.9 69.8 70.3 2.3% * Need to obtain total population numbers for comparison. Total School population of 1,374 10 | P a g e Total School* 230 184 174 145 733 % 1.3% 1.6% 9.8% 6.2% 4.4% Total School* 419 335 315 305 1374 b. Draw a Histogram of your Means by Gender and Grade. Height by Gender/Grade 76 74 72 70 68 66 64 62 60 9th 65 70 Female Male 10th 65.7 73.7 Female V. 11th 63.8 69.9 12th 65 69.8 Male Draw Inferences about the Entire Population Using your normal distribution and the z-score table determine what percent of students are predicted to be: = 60 −69 4 = 2.25 100% minus % less than 5 feet + % greater than 6 feet = = − 72 − 69 = 0.75 4 Less than 5 feet (60 inches) Between 5 feet and 6 feet Greater than 6 feet (72 inches) 1.2% 76.1% 22.7% 11 | P a g e VI. Representative Sample? b. How would you describe your sampling methods? __ Self-Selected __ Convenience __ Systematic __ Random Describe your method: The samples were obtained by measuring student heights during 1st lunch (one lunch out of three). Students were asked if we could measure their height and we collected the data. c. Do you think that your sample is representative of the total population? Why? NO. Students were only measured during one lunch. Any possible sample data from the other two lunches was missed. Looking at the data, juniors and seniors are over-represented and freshmen and sophomores are under-represented. Also more male than female measurements were taken. The students taking the measurements were juniors and seniors. They were more easily able to approach fellow juniors and seniors to be sampled. Males may have been more willing to volunteer than females. The lunch period sampled MAY have contained more males and/or more upper classmen. This was definitely a CONVENIENCE sample. 12 | P a g e Answers will vary depending on the sample data. This answer key is representative of how the response should be structured. Aggregate Data - Answer Key (For ALL Sample Data Collected) III. Develop Statistical Parameters of the Sample a. Draw a Histogram Chart for your sample data. Aggregate of All Samples 35 # of Students 30 25 20 15 10 5 0 52 55 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 Height (inches) b. Calculate the mean, standard deviation and % of the total population (total number of students in the population*) for your sample. Team # Sample Size (n) % Total Population Mean ( x ) Std Deviation S 1 40 2.9 66.0 3.8 2 3 40 31 2.9 2.2 67.9 65.0 3.6 3.0 4 5 6 7 8 38 33 40 29 47 2.8 2.4 2.9 2.0 3.4 65.4 65.8 68.4 67.4 69.0 4.1 4.0 5.4 4.2 4.0 TOTAL 297 21.6 67.0 4.3 * Need to obtain total population numbers for comparison. Total School population of 1,374 13 | P a g e c. Draw a Normal Distribution Chart for your sample data. Aggregate Normal Distribution - Student Height 90 80 # of Students 70 60 50 40 30 20 10 0 1 54.1 2 -3ϭ IV. 58.4 -2ϭ 3 62.7 4 67.0 5 71.3 75.6 -1ϭ x +1ϭ +2ϭ Height - Inches 6 79.9 7 +3ϭ In-depth Sample Analysis a. “Disaggregate” your data into gender and grade and compare your sample size to the total school population* breakdown. Grade 9th 10th 11th 12th TOTAL Female Sample Mean 19 23 49 36 127 63.2 62.9 64.3 63.6 63.7 Total School* 189 151 141 160 641 Male % Sample Mean 10.0% 15.2% 34.8% 22.5% 22 25 59 64 169 69.0 68.8 69.4 69.9 69.4 19.8% * Need to obtain total population numbers for comparison. Total School population of 1,374 14 | P a g e Total School* 230 184 174 145 733 % 9.6% 13.6% 33.9% 44.1% 23.1% Total School* 419 335 315 305 1374 b. Draw a Histogram of your Means by Gender and Grade. Aggregate Height by Gender Grade 72 70 68 66 64 62 60 58 9th 63.2 69.0 Female Male 10th 62.9 68.8 Female V. 11th 64.3 69.4 12th 63.6 69.9 Male Draw Inferences about the Entire Population Using your normal distribution and the z-score table determine what percent of students are predicted to be: = 60 −67 4.3 = 1.63 100% minus % less than 5 feet + % greater than 6 feet − = = 72 − 67 = 1.16 4.3 Less than 5 feet (60 inches) Between 5 feet and 6 feet Greater than 6 feet (72 inches) 5.3% 82.5% 12.2% 15 | P a g e VI. Representative Sample? c. How would you describe your sampling methods? __ Self-Selected __ Convenience __ Systematic __ Random Describe your method: The samples were obtained by measuring student heights during 1st lunch (one lunch out of three). Students were asked if we could measure their height and we collected the data. d. Do you think that your sample is representative of the total population? Why? NO. Students were only measured during one lunch. Any possible sample data from the other two lunches was missed. Looking at the data, juniors and seniors are over-represented and freshmen and sophomores are under-represented. Also more male than female measurements were taken. The students taking the measurements were juniors and seniors. They were more easily able to approach fellow juniors and seniors to be sampled. Males may have been more willing to volunteer than females. The lunch period sampled MAY have contained more males and/or more upper classmen. This was definitely a CONVENIENCE sample. 16 | P a g e

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