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10.2 Making Inferences from a Random Sample 7.RP.2C, 7.SP.1, 7.SP.2

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10.2 Making Inferences from a Random Sample 7.RP.2C, 7.SP.1, 7.SP.2 UNIT 5 Statistics MODULE 10 Random Samples and Populations 7.RP.2c, 7.SP.1, 7.SP.2 MODULEMODULE 1111 Analyzing and Comparing Data 7.SP.3, 7.SP.4 MATH IN CAREERS Entomologist An entomologist is a biologist who studies insects. These scientists analyze data and use mathematical models to understand and predict the behavior of insect populations. If you are interested in a career in entomology, you should study these mathematical subjects: • Algebra © Houghton Mifflin Harcourt Publishing Company • Image Credits: © Kitchin and Hurst/All Canada Photos/ and Hurst/All Canada © Kitchin • Image Credits: Company Mifflin Harcourt Publishing © Houghton Corbis • Trigonometry • Probability and Statistics • Calculus Research other careers that require the analysis of data and use of mathematical models. ACTIVITY At the end of the unit, check out how entomologists use math. Unit 5 305 Unit Project Preview A Sample? Simple! In the Unit Project at the end of this unit, you will choose a random group of people, conduct a survey, display and interpret the results of the survey, and calculate how many people in a much larger group would respond to your survey as your group did. To successfully complete the Unit Project you’ll need to master these skills: • Choose a random sample. • Conduct a survey. • Draw and interpret a box plot. • Make inferences from a random sample. 1. Kerry asked this question to 25 girls in her co-ed school: “Isn’t it about time that the school had a girls’ soccer team?” Do you think the school principal should use the results of Kerry’s survey to decide whether or not to have a girls’ soccer team? Explain. 2. What would have been a better question for Kerry to ask? Tracking Your Learning Progression This unit addresses important California Common Core Standards in the Critical Area of applying proportional relationships to random samples to solve problems. Domain 7.SP Statistics and Probability Cluster Use random sampling to draw inferences about a population. © Houghton Mifflin Harcourt Publishing Company • Image Credits: © Radius Images/Corbis • Image Credits: Company Mifflin Harcourt Publishing © Houghton The unit also supports additional standards. Domain 7.RP Ratios and Proportionality Cluster Analyze proportional relationships and use them to solve real-world and mathematical problems. 306 Unit 5 Preview Random Samples MODULE and Populations 10 LESSON 10.1 ESSENTIAL QUESTION ? Populations and How can you use random Samples samples and populations 7.SP.1 to solve real-world problems? LESSON 10.2 Making Inferences from a Random Sample 7.RP.2c, 7.SP.1, 7.SP.2 LESSON 10.3 Generating Random Samples 7.SP.2 Real-World Video Scientists place radio frequency tags on some animals within a population of that species. Then they track data, such as migration patterns, about my.hrw.com the animals. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mike Veitch Almay Images Almay Veitch ©Mike • Image Credits: Company Mifflin Harcourt Publishing © Houghton my.hrw.com my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your Scan with your smart Interactively explore Get immediate write-in student phone to jump directly key concepts to see feedback and help as edition, accessible on to the online edition, how math works. you work through any device. video tutor, and more. practice sets. 307 Are YOU Ready? Complete these exercises to review skills you will need Personal for this module. Math Trainer Online Practice Solve Proportions my.hrw.com and Help a 30 EXAMPLE __ = ___ 1 1.5 a× 1.5 = 1 × 30 Write the cross products. = 1.5a 30 Simplify. ____1.5a = ___30 Divide both sides by 1.5. 1.5 1.5 a = 20 Solve for x. x 45 x 1 2.5 x x 2 __ = __ _ = _ ___ = __ _ = _ 1. 16 40 2. 5 4 3. 10 50 4. 6 9 Find the Range EXAMPLE 29, 26, 21, 30, 32, 19 Order the data from least to greatest. 19, 21, 26, 29, 30, 32 range = 32 – 19 The range is the difference between the = 13 greatest and the least data items. Find the range of the data. 5. 52, 48, 57, 47, 49, 60, 59, 51 6. 5, 9, 13, 6, 4, 5, 8, 12, 12, 6 7. 97, 106, 99, 97, 115, 95, 108, 100 8. 27, 13, 35, 19, 71, 12, 66, 47, 39 Find the Mean EXAMPLE 21, 15, 26, 19, 25, 14 The mean is the sum of the 21 + 15 + 26 + 19 + 25 + 14 mean = _____________________ data items divided by the 6 120 number of items. = ___ 6 = 20 © Houghton Mifflin Harcourt Publishing Company Mifflin Harcourt Publishing © Houghton Find the mean of each set of data. 9. 3, 5, 7, 3, 6, 4, 8, 6, 9, 5 10. 8.1, 9.4, 11.3, 6.7, 6.2, 7.5 308 Unit 5 Reading Start-Up Vocabulary Review Words Visualize Vocabulary ✔ box plot (diagrama de caja) Use the ✔ words to complete the right column of the chart. data (datos) dot plot (diagrama de puntos) Box Plots to Display Data interquartile range Definition Review Word (rango entre cuartiles) ✔ lower quartile A display that uses values (cuartil inferior) from a data set to show how ✔ median (mediana) the values are spread out. spread (dispersión) The middle value of a survey (estudio) data set. ✔ upper quartile The median of the lower (cuartil superior) half of the data. Preview Words The median of the upper biased sample half of the data. (muestra sesgada) population (población) Understand Vocabulary random sample (muestra aleatoria) Complete each sentence, using the preview words. sample (muestra) 1. An entire group of objects, individuals, or events is a . 2. A is part of the population chosen to represent the entire group. 3. A sample that does not accurately represent the population is a . Active Reading Tri-Fold Before beginning the module, create a © Houghton Mifflin Harcourt Publishing Company Mifflin Harcourt Publishing © Houghton tri-fold to help you learn the concepts and vocabulary in this module. Fold the paper into three sections. Label the columns “What I Know,” “What I Need to Know,” and “What I Learned.” Complete the first two columns before you read. After studying the module, complete the third column. Module 10 309 GETTING READY FOR Random Samples and Populations Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. 7.SP.1 Understand that statistics can What It Means to You be used to gain information You will learn how a random sample can be representative of a about a population by population. examining a sample of the population; generalizations EXAMPLE 7.SP.1 about a population from a Avery wants to survey residents who live in an apartment building. sample are valid only if the She writes down all of the apartment numbers on slips of paper, sample is representative of and draws slips from a box without looking to decide who to that population. Understand survey. Will this produce a random sample? that random sampling tends to produce representative samples The population is all of the residents or people who live in the and support valid inferences. apartment building. The sample is a valid random sample because every apartment number has the same chance of being selected. 7.SP.2 Use data from a random sample What It Means to You to draw inferences about a You will use data collected from a random sample to make population with an unknown inferences about a population. characteristic of interest. Generate multiple samples (or EXAMPLE 7.SP.2 simulated samples) of the same Alexi surveys a random sample of 80 students at his school and finds size to gauge the variation in that 22 of them usually walk to school. There are 1,760 students at the estimates or predictions. school. Predict the number of students who usually walk to school. number in sample who walk number in population who walk Key Vocabulary _____________________ = _____________________ size of sample size of population population (población) 22 x __ = ____ The entire group of objects or 80 1,760 individuals considered for a 22 = ___ · survey. x 80 1,760 sample (muestra) 38,720 = _____ = A part of the population. x 80 484 Approximately 484 students usually walk to school. © Houghton Mifflin Harcourt Publishing Company Mifflin Harcourt Publishing © Houghton Visit my.hrw.com to see all CA Common Core Standards explained. my.hrw.com 310 Unit 5 LESSON 7.SP.1 Populations and … Understand that random sampling tends to produce 10.1 representative samples and Samples support valid inferences. ESSENTIAL QUESTION ? How can you use a sample to gain information about a population? EXPLORE ACTIVITY 7.SP.1 Random and Non-Random Sampling When information is being gathered about a group, the entire group of objects, individuals, or events is called the population. Because gathering information about each member of a large group can be difficult or impossible, researchers often study a part of the population, called a sample. The size of a sample and the way the sample is First Number chosen can have an effect on whether the sample Cube is representative of the population or not. 8 9 13 18 24 15 1 A vegetable garden has 36 tomato plants arranged in a 6-by-6 array. The gardener 34 42 46 20 13 41 2 wants to know the average number of tomatoes on the plants. Each white cell 29 21 14 45 27 43 3 in the table represents a plant. The number in the cell tells how many tomatoes are on 22 45 46 41 22 33 4 that particular plant.
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