TEACHER NOTES Building Concepts: Sample Proportions Lesson Overview In this TI-Nspire lesson, students investigate the effect of sample size Learning Goals on variability by comparing the distribution of sample proportions with 1. Identify sampling variability as the population proportion. the variation from sample to A statistic computed from a random sample can be used as an sample in the values of a estimate of that same characteristic of the population from sample statistic; which the sample was selected. 2. understand that the shape, center, and spread of simulated sampling distributions of sample proportions for a given sample size will be fairly predictable; 3. understand that the sampling variability among samples is related to size of the samples; as the sample size increases, the variability decreases; 4. recognize that using a sample statistic from an unknown population to understand some characteristic of the population is based on knowing how statistics from samples drawn from known populations behave and that simulation can be a tool to approximate this behavior. ©2016 Texas Instruments Incorporated 1 education.ti.com TEACHER NOTES Building Concepts: Sample Proportions Prerequisite Knowledge Vocabulary Sample Proportions is the nineteenth lesson in a series of lessons • random sample: that explore the concepts of statistics and probability. This lesson representative of the population builds on the concepts of the previous lessons. Prior to working on from which it was drawn this lesson students should have completed Probability and • sampling variability: the Simulation, Law of Large Numbers and Why Random Sampling? variation from sample to sample Students should understand: in the values of a sample • that random sampling is likely to produce a sample that is statistic representative of the population; • sampling distribution of a • how to use simulation to collect data. statistic: the collection of sample statistics from all possible samples of a given size from a specific population • simulated sampling distributions: modeling a collection of sample statistics from a specific population Lesson Pacing This lesson should take 50–90 minutes to complete with students, though you may choose to extend, as needed. Lesson Materials • Compatible TI Technologies: TI-Nspire CX Handhelds, TI-Nspire Apps for iPad®, TI-Nspire Software • Sample Proportions_Student.pdf • Sample Proportions_Student.doc • Sample Proportions.tns • Sample Proportions_Teacher Notes • To download the TI-Nspire activity (TNS file) and Student Activity sheet, go to http://education.ti.com/go/buildingconcepts. ©2016 Texas Instruments Incorporated 2 education.ti.com TEACHER NOTES Building Concepts: Sample Proportions Class Instruction Key The following question types are included throughout the lesson to assist you in guiding students in their exploration of the concept: Class Discussion: Use these questions to help students communicate their understanding of the lesson. Encourage students to refer to the TNS activity as they explain their reasoning. Have students listen to your instructions. Look for student answers to reflect an understanding of the concept. Listen for opportunities to address understanding or misconceptions in student answers. Student Activity: Have students break into small groups and work together to find answers to the student activity questions. Observe students as they work and guide them in addressing the learning goals of each lesson. Have students record their answers on their student activity sheet. Once students have finished, have groups discuss and/or present their findings. The student activity sheet can also be completed as a larger group activity, depending on the technology available in the classroom. Deeper Dive: These questions are provided for additional student practice and to facilitate a deeper understanding and exploration of the content. Encourage students to explain what they are doing and to share their reasoning. Mathematical Background A central question in statistics is how to use information from a sample to begin to understand something about the population from which the sample was drawn. Collecting data from every member of the entire population can be time consuming and often impossible. The best way to collect data in such a situation is to use a random sample of the population: A statistic computed from a random sample, such as the sample proportion, can be used as an estimate of that same characteristic of the population from which the sample was selected. In prior lessons, students investigated variability within a single sample. In this lesson, students begin to differentiate between the variability within a single sample and the variability inherent in a statistic computed from each sample when samples of the same size are repeatedly selected from the same population. Understanding variability from this perspective enables students to think about how far a proportion of “successes” in a sample is likely to vary from the proportion of “successes” in the population. The variability in samples can be studied using simulations. The collection of sample statistics from all possible samples of a given size from a population is called a sampling distribution. The complete sampling distribution of all possible values of a sample statistic for samples of a given size is typically difficult to generate but a subset of that distribution based on simulated sample statistics can be used to approximate the theoretical distribution. Students should note that sampling distributions of sample statistics computed from random samples of the same size from a given population tend to have certain predictable attributes. For example, while sample proportions vary from sample to sample, they cluster around the population proportion. In the case of sample means, the sample means cluster around the mean of the population. In both cases, the distributions of the sample statistic for a given sample size have a fairly predictable shape and spread. ©2016 Texas Instruments Incorporated 3 education.ti.com TEACHER NOTES Building Concepts: Sample Proportions Part 1, Page 1.3 Focus: Students develop an understanding of sampling variability by generating random TI-Nspire samples from a population where the Technology Tips proportion of successes is known and observing the variability from sample to b accesses sample. page options. On page 1.3, students can select the bag to e cycles through generate a random sample (size 30) from a proportion, sample population where the proportion of size, show successes is 0.5 and display the result on proportion/show the dot plot. Selecting the bag again will count, and draw. generate a new random sample. After 10 · selects single samples, selecting the bag generates highlighted 10 samples at a time. segments and Proportion changes the population proportion of “successes.” displays length. Sample Size changes the size of the sample. / . resets the page to the original Show Prop displays in the table and graph the proportion of successes after screen. each count. Clear Data clears sample data but maintains population proportion. Reset resets the page. Class Discussion Teacher Tip: In the following questions, students use a simulated distribution of successes from random samples of a given size (30) to make conjectures about the general behavior of such a distribution—the shape, center, and spread (as measured by the spread of successes) are relatively predictable after about 50 samples. Be sure students understand what the rows and columns represent in the table. Students may find it useful to use the scratchpad to do some of the calculations. Have students… Look for/Listen for… Car colors vary from year to year and brand to brand, but the most popular color for a car is white. About 25% of all cars sold in the United States are white. Suppose you randomly sampled 30 cars in a grocery store parking lot and counted the number of white cars. ©2016 Texas Instruments Incorporated 4 education.ti.com TEACHER NOTES Building Concepts: Sample Proportions Class Discussion (continued) • Describe a way to choose a random sample Answers will vary. You could number the cars of 30 cars in the parking lot. and randomly choose 30 of the numbers or you could number the rows and the cars in the rows, randomly choose five rows and then randomly choose six cars from each row. (Note that you might want students to use page 2.2 in Activity 18 to generate such a random sample, where the blocks would correspond to rows and the cells to the cars in a row.) • About how many white cars would you Answer: About 7 or 8. expect to see in your sample? • Would you be surprised to see 10 white cars Answers may vary. Students might think 10 white in your sample? 20? Why or why not? cars is possible, and others might think it is not; some students will begin to question whether the sample was random if you see that 20 of the 30 cars are white. • On page 1.3, set the proportion to match the Answer: Set the population proportion to 0.25. A information about the number of white cars. dot at 11 indicates the random sample of 30 cars Select the bag on page 1.3 to draw a random had 11 white cars. sample. Explain what the dot on the number line represents. If you were able to select a different sample of the Answers will vary. Students should comment same size, which of the following do you think is that: a. is likely to occur since 8 cars is very close likely to be true about the number of white cars in to 25% of 30 cars, b. is unlikely since the number the sample? Explain your reasoning. of white cars will vary from sample to sample, c. is somewhat unlikely since 10 is a little more than a. Eight cars in the sample will be white. 30% of the sample of size 30, and that d. is b. The number of white cars will be the same as almost certain since it would be very unusual to the number in the first sample.