SAMPLING Judo Math Inc.

Transcription

1 SAMPLING 2013 Judo Math Inc.

2 7 th grade Statistics Discipline: Yellow Belt Training Order of Mastery: 2. Random Sampling (7SP1) 3. Representative Sampling (7SP1) 4. Making generalizations about a population (7SP2) 5. Making predictions based on data (7SP2) Welcome to the Yellow Belt Statistics is the field of math that involved collecting and analyzing data often in large quantities. Chances are that last year you did a lot of analyzing of numeric data. Whether it was ages of kids in your class, or number of pets your friends have, you gathered data and probably even made some graphs! This year as a seventh grader, you re going to have to kick this up a notch. We are going to be discussing who to survey and how to survey and lots more! Here in the yellow belt you will be receiving a lot of initial practice and then wrapping things up with a mini project where you become a statistician yourself and present your findings to a larger audience! Good luck grasshopper. Standards Covered: 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions Judo Math Inc.

3 1. Random Sampling Have you ever watched the election for president on TV and wondered how they predict the winner before the election takes place? Or how they can predict how many of a certain bird live in North America without being able to see them all? These predictions are made using a statistics tool called random sampling. Here are some important vocabulary words for this unit: Vocabulary Population: the entire group being studied. Sample: part of the population being surveyed Random Sample: choosing subjects from a population through unpredictable means. All subjects have an equal chance of being selected out of the population being researched. Because it is nearly impossible to ask everyone in a population a question, researchers often use a method called randome sampling to select a small amout of people from the population. They then use data gathered about the small group to make generalizations about the whole populations. Statistical measures like mean, median, and mode are then used to analyze that data. FOR EXAMPLE: You want to find out how many texts middle school students in your school send in one day. By surveying a random sample of the group, you can make a generalization about the entire group Can you think of a case where making a generalization like this might not work? Can you think of some ways to get a truly random group of students? 1

4 Benefits of Random Sampling it is the best way to ensure that results are unbiased it consistently provides results that are valid it makes it easy for researchers to draw conclusions about large populations. Risks of random sampling there is no way to guarantee that the results that come are 100% accurate The sample may not be representative of the larger population: sampling error every survey comes with measures of uncertainty TRULY RANDOM?: If three students are to be selected from the class for a special project, a fair way to make the selection is to put all the student names in a box, mix them up, and draw out three names at random. For each of the scenarios, identify the population and the sample. Then reflect on if you think that the sample is random enough to generate data to make predictions about the whole population. 1. To gauge students preference for a new school mascot, the Student Council President surveys her soccer team. Population: Sample: Good Random Sample?: 2. To determine the number of students who carry backpacks in school, Tina collects data on the first 100 students who enter the building. Population: Sample: Good Random Sample?: 2

5 3. To determine the number of people who have iphones in the USA, Sam asks his entire class. Population: Sample: Good Random Sample?: 4. You are buying pizza for a party at your middle school, in which 350 students will attend. a) How would you use sampling to make a generalization about students preferred pizza toppings b) Why not survey all 350 students? c) Describe the sample you would survey, and explain why you chose that sample. 5. Describe a sample of each of the following populations. a) Stop lights in the United States b) American politicians c) Professional athletes d) Hospital employees 3

6 6. In a poll of Mr. Briggs s math class, 67% of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion and suggest a way to gather better data to determine what subject is most popular. You become the statistician Investigate the texting habits of middle school students in the United States by conducting a survey in class today. a) Write the survey question, survey the students in your class, and record the data. b) Calculate the average number of texts sent (use mean, median, mode, or range). c) Use the data from your sample to make a generalization about the population. Make sure to identify the population and sample. 4

7 2. Representative Sample Let s build on our knowledge of random samples by adding a few more vocabulary words: Bias Sample: A sample in which all individuals, or instances, were not equally likely to have been selected. A non-random sample. Unbiased sample: A random sample where all individuals were equally as likely to be selected. Representative Sample: A subset of a population that accurately reflects the members of the entire population. A representative sample should be an unbiased indication of what the population is like. Random Sampling methods (unbiased) Simple Random Sample: Draw names out of a hat, or use a computer to randomly generate people. Stratified Random Sample: Breaks the population into groups (like boys and groups) and then picks a random group of each. Systematic Random Sample: Ask every 3 rd or 6 th or x th person that you come up on from the given population. NOT random sampling methods (biased) Convenience Sample: You ask whoever is easiest to get to (your friends, people who live near you) Voluntary Response Sample: The survey is optional and you just use the data of the people who respond. 1. Researchers want to estimate the number of orange trees in the United States that have been infected with a disease. Identify which type of sample is used in each case. a. The researchers collect data from apple trees on a farm in California. b. The researchers send a mail survey to orange farmers asking them to record the number of their trees that are infected. 5

8 2. For each scenario, identify the type of sample and discuss whether it will generate valid inferences. a. At band practice, the principal distributes a survey about whether to cut funding for music programs or athletic programs. b. The principal invites students to a meeting after school to express their opinions about the funding cuts. 3. You are collecting data about the allowance received by middle school students in the United States each month. a. Describe a convenience sample. b. Describe a voluntary response sample. c. Discuss why the samples you described may not generate valid inferences. 4. You are studying the favorite snacks of movie-goers. Label each biased sample as a voluntary response sample or a convenience sample and explain why you made your selection that way. The first 20 people in line for the matinee Adults at the movie theater People who agree to fill out your survey 6

9 5. Your principal is ordering sweatshirts for the all the students at your school. In order to determine what size t-shirts to order, the principal plans on surveying a sample of students about what size t-shirt they wear. The principal surveys 30 members of the football team about their t-shirt sizes. a) Is this a representative sample? Explain. b) What should the principal do to ensure he has a representative sample? 6. Which is an example of an unbiased sampling method that could be used to predict the color of leaves in September? a) 100 fallen leaves collected from the ground b) 100 leaves on tree branches c) 50 fallen leaves and 50 leaves on branches d) 50 fallen oak leaves and 50 oak leaves on branches Explain why you made your selection. 7

10 7. A market researcher wants to know how year old women spend their money. Which group would be a representative sample? a) year old women at the mall b) men and women commuting to work c) Women of all ages commuting to work d) year old women commuting to work Explain why you made your selection. 8. In a complete paragraph, describe why the following sampling method is biased, and suggest an unbiased method. You want to know San Diego s favorite fast-food restaurant. You randomly ask 30 people their preference as they leave one of the fast-food restaurants in town. 9. An ice-cream company wants to find out if its ice cream is the favorite in the state. Which group would be a representative sample? a) Customers who visit their store b) Employees of ice cream stores in the state c) People at the state fair d) Adults entering a gym in the capital city Explain why you made your selection. 8

11 10. Which is an unbiased sampling method for predicting the type of payment most frequently used at a grocery store? Record the type of payment used by a) Students at the local high school b) Every 10 th customer entering the store c) Every 10 th customer in the cash-only line d) Visa card holders Explain why you made your selection. 11. Choose a population and something you would like to learn about it. Identify examples of biased sampling methods and unbiased sampling methods. 1. Population: 2. Desired information: 3. Examples of biased sampling methods: 4. Examples of unbiased sampling methods: 9

12 3. Using Simulations to make predictions (multiple random samples) The variability in samples can be studied by means of simulation. For example, students are to take a random sample of 50 seventh graders from a large population of seventh graders to estimate the proportion that has football as their favorite sport. Suppose, for the moment, that the true proportion is 60%, or How much variation can be expected among the sample proportions? The scenario of selecting samples from this population can be simulated by constructing a population that has 60% red chips and 40% blue chips, taking a sample of 50 chips from that population, recording the number of red chips, replacing the sample in the population, and repeating the sampling process. (This can be done by hand or with the aid of technology, or by a combination of the two.) Record the proportion of red chips in each sample and plot the results. The dot plots in the margin shows results for 200 such random samples of size 50 each. Results of simulations Proportions of red chips in 200 random samples of size 50 from a population in which 60% of the chips are red. Proportions of red chips in 200 random samples of size 50 from a population in which 50% of the chips are red. Proportions of red chips in 200 random samples of size 50 from a population in which 40% of the chips are red around 0.60, but it is not too rare to see a sample proportion down around 0.45 or up around Thus, we might expect a variation of close to 15 percentage points in either direction. Interestingly, about that same amount of variation persists for true proportions of 50% and 40%, as shown in the dot plots. For example, 15 of the random samples had exactly 25.00% red chips; only 2 of the random samples had exactly 62.50% red chips, and so on Reflect on this case, where might you use this type of sampling and why might the same variation (15%) be present in all of these samples? 10

13 Now you practice 1. Which describes the generation of multiple random samples? a) Instead of choosing 50 people for a sample, choose 100 people. b) Survey every 5th person who enters the store until you have 100 people, then survey every 10th person who enters the store until you have 100 people. c) Survey only people who volunteer. d) Pick 30 phone numbers out of a hat to be surveyed, then repeat the survey a week later on the same sample. 11

14 2. The Valentine s Day Contest: A hotel holds a Valentine's Day contest where guests are invited to estimate the percentage of red marbles in a huge clear jar containing both red marbles and white marbles. There are 11,000 total marbles in the jar: 3696 are red, 7304 are white. The actual percentage of red marbles in the entire jar (33.6%= ) is known to some members of the hotel staff. Any guest who makes an estimate that is within 9 percentage points of the true percentage of red marbles in the jar wins a prize, so any estimate from 24.6% to 42.6% will be considered a winner. To help with the estimating, a guest is allowed to take a random sample of 16 marbles from the jar in order to come up with an estimate. (Note: when this occurs, the marbles are then returned to the jar after counting.) One of the hotel employees who does not know that the true percentage of red marbles in the jar is 33.6% is asked to record the results of the first 100 random samples. A table and dotplot of the results appears below. Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50% % % % % % % % % % 1 Total:

15 a. Assuming that each of the 100 guests who took a random sample used their random sample's red marble percentage to estimate the whole jar's red marble percentage, how many of these guests would be "winners"? b. How many of the 100 guests obtained a sample that was more than half red marbles? c. Should we be concerned that none of the samples had a red marble percentage of exactly 33.6% even though that value is the true red marble percentage for the whole jar? Explain briefly why a guest can't obtain a sample red marble percentage of 33.6% for a random sample of size 16. d. Recall that the hotel employee who made the table and dotplot above didn't know that the real percentage of red marbles in the entire jar was 33.6%. If another person thought that half of the marbles in the jar were red, explain briefly how the hotel employee could use the dotplot and table results to challenge this person's claim. Specifically, what aspects of the table and dotplot would encourage the employee to challenge the claim? e. Design a simulation that takes a large number of samples of size 16 from a population in which 65% of the members of the population have a particular characteristic. For each sample of size 16, compute the percentage of red items in the sample. Record these percentages, and then summarize all of your sample percentages using a table and dotplot similar to those shown above. In what ways is your dotplot similar to the dotplot used in this task? In what ways does it differ? 13

16 3. Estimate the mean word length in a book by randomly sampling words from the book 4. Suppose your school election is coming up. What would be some methods you could use to predict the winner before it even happens? Give 2 different ideas below, explaining each idea in at least two sentences. Also reflect on how far off the estimate or prediction might be. 14

17 5. The average amount of allowance 10th graders receive according to one random sample was $20 a month. Another random sample was generated from the same population and produced an average monthly allowance of $30. A third random sample from the same population had a mean monthly allowance of $32. What inferences might you make? 15

18 4. Making generalizations about a population (7SP2) Generalization is a very common human process. We all draw conclusions about reality from a limited amount of experience. This saves us effort, but it can mislead us, because our experiences may be so limited or selective that the conclusions drawn from them are quite wrong. These are key words to use when making inferences Tends to increase/decrease Twice as many as More/less than 1. Here is a table of student responses when students were asked their favorite color: Favorite Color Number of Students Blue 7 Green 6 Yellow 6 example of a WRONG generalization: Most students like blue Why is this generalization incorrect? Give a few examples of correct generalizations. Try using the key words from above to do this. 16

19 2. Here is a graph depicting the percent of people who voted in the last election and their level of education. % Voted 100% 90% 84% 86% 80% 70% 70% 75% 60% 50% 40% 40% 30% 20% 10% 0% Less than high school High school degree Some college College degree Graduate degree Kyle makes the generalization that people with graduate degrees vote more often than people with only college degrees. 1. Explain why Kyle s inference may be invalid 2. Make at least 3 inferences from the graph above that are valid. 17

20 3. To predict middle school students ice cream preferences, a random sample of 1,300 students was surveyed about their preferences between ice cream flavors. a. Make two inferences about the population. b. Can you think of any inferences that someone could make that would be incorrect? 4. The following table shows students favorite color preferences in your school: Yellow Black Green Orange Blue Total Which of these would you say is the MOST VALID INFERENCE: Black is the least preferred color. More students prefer blue than prefer black and green combined Blue and green are the preferred colors Orange is preferred to black. Explain how and why you made your selection: 18

21 5. Two samples of 700 people were conducted on people s favorite ice create the following table: Chocolate Chip Cookie Dough Rainbow Sherbert TOTAL Sample Sample a) Create a double bar graph showing this data: b) What are some inferences you can make from this data? c) Your friend infers that chocolate chip is preferred to rainbow sherbert. Why might this inference not be valid? 19

22 Mini Project (random and representative samples): Now you construct a simulated study where you take at least 5 random samples. Record your results in a table and then plot the results on either a bar graph or dot plot and then make some generalizations prediction. If you are having a struggle coming up with ideas for this, talk to people around you and see if you can put your heads together to create an epic simulation study! Be prepared to present your findings to the class. Also feel free to use technology for any part of this if it is available to you. Step 1: Create your question The question should be able to be categorized. Look through this packet for examples of questions and then try to come up with your own original question! Step 2: Create your table to record data: 20

23 Step 3: Graph the results Step 4: Make generalizations Now prepare either a poster or short powerpoint presentation on your findings to present to the class! 21