Algebra 2 Honors: Probability and Statistics Semester 2, Unit 7: Activity 36

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1 Resources: SpringBoard- Algebra 2 Online Resources: Algebra 2 Springboard Text Unit 7 Vocabulary: Density curve Z-score Normal distribution Normal curve Sample Survey Response Bias Simple random sample Experiment Explanatory variable Response variable Completely randomized design Randomized block design Matched pairs design Single-blind study Double-blind study Observational study Confounding variable Statistic Algebra 2 Honors: Probability and Statistics Semester 2, Unit 7: Activity 36 Unit Overview In this unit, students study normal data distributions and solve problems using tables and technology. An examination of bias provides students with a reason to develop simple random samples from a population of interest. Students create simulations with and without technology to test conjectures about data. Margin of error is applied to population proportions and an informal understanding of statistical significance is developed. Student Focus Main Ideas for success in lessons 36-1, 36-2, 36-3, & 36-4: Revisit single-variable statistics concepts (data distributions and representations, shape, center, and spread). Further develop an understanding of normal distributions using z-scores, tables, and technology. Lesson 36-1: Math Tip: when a graphical representation shows that data has a tail in one direction, the data is described as skewed in the direction of the tail, either left or right. Skewed left Math Tip: data can be described as unimodal if it has one maximum in a graphical representation. Data with two maxima can be described as bimodal. Unimodal Bimodal Page 1 of 16

2 Margin of error Simulation Sample proportion Sampling distribution Critical value Statistically significant Math Vocabulary: Density curves have special characteristics. They are always drawn above the x-axis and the area between the curve and the x-axis is always 1. Math Vocabulary: the numbers that are the results of the difference between each x-value and the mean divided by the standard deviation of the data calculate the z-score. These scores standardize the data of different types so that comparisons to the mean can be made. The positive z-score indicates the data point is above the mean, while the negative z-score indicates that the data point is below the mean. Lesson 36-2: Math Vocabulary: Normal distribution: symmetrical, unimodal, bell-shaped. Described by the mean and standard deviation. Math Tip: rule: in a normal distribution approximately 68% of the data lies within one standard deviation of the mean, 95% of the data lies within two standard deviations of the mean, and 99.7% of the data lies within three standard deviations of the mean. Page 2 of 16

3 Lesson 36-3: In a small-breed dog show, the competitor weight is distributed normally, with a mean of 32.0 pounds and a standard deviation of 3.5 pounds. Sparky the Wonderpup is a competitor that weighs 30.3 pounds. Which graph shows the proportion of dogs that weigh less than Sparky? Lesson 36-4: Page 3 of 16

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5 Resources: SpringBoard- Algebra 2 Online Resources: Algebra 2 Springboard Text Unit 7 Vocabulary: Density curve Z-score Normal distribution Normal curve Sample Survey Response Bias Simple random sample Experiment Explanatory variable Response variable Completely randomized design Randomized block design Matched pairs design Single-blind study Double-blind study Observational study Confounding variable Statistic Algebra 2 Honors: Probability and Statistics Semester 2, Unit 7: Activity 37 Unit Overview In this unit, students study normal data distributions and solve problems using tables and technology. An examination of bias provides students with a reason to develop simple random samples from a population of interest. Students create simulations with and without technology to test conjectures about data. Margin of error is applied to population proportions and an informal understanding of statistical significance is developed. Student Focus Main Ideas for success in lessons 37-1, 37-2, & 37-3: Investigate the process of sampling a population and identifying possible bias in samples. Explore experimental and observational studies. Lesson 37-1: Math Vocabulary: A survey is a study in which subjects are asked a question or series of questions. Math Vocabulary: An answer provided by a subject to a survey question is called a response. Math Vocabulary: A sample is part of a population of interest. Data are collected from the individuals in the sample. Math Vocabulary: A sample shows bias if the composition of the sample favors certain outcomes. Math Vocabulary: A simple random sample (SRS) is a sample in which all members of a population have the same probability of being chosen for the sample. Emilia wants to send a survey to a representative sample of students at her school. After assigning each student on the school roster a number, which method(s) would create a simple random sample? Check all that apply. Page 5 of 16

6 Margin of error Simulation Sample proportion Sampling distribution Critical value Statistically significant Lesson 37-2: Math Vocabulary: An experiment applies a treatment (a condition administered) to experimental units to observe an effect. Math Vocabulary: The explanatory variable is what is thought to be the cause of different outcomes in the experiment. In the simple experiments, the explanatory variable is simply the presence or absence of the treatment. Math Vocabulary: The effect of the explanatory variable is called the response variable. Math Vocabulary: A completely randomized design implies that all experimental units have the same probability of being selected for application of the treatment. Math Vocabulary: A randomized block design involves first grouping experimental units according to a common characteristic and then using random assignment within each group. Math Vocabulary: A matched pairs design involves creating blocks that are pairs. In each pair, one unit is randomly assigned the treatment. Sometimes, both treatments may be applied, and the order of application is randomly assigned. Academic Vocabulary: A placebo is a treatment applied to an experimental subject that appears to be the experimental treatment, but in fact is a treatment known to have no effect. A scientist is testing a growth booster for roses. She wants to compare the booster to a placebo, testing them on both miniature and standard-sized flowers. She plans on applying the booster to the miniature roses and the placebo to the standard sized flowers. What is the best explanation for why this is a poor experimental design? Lesson 37-3: Math Vocabulary: In an observational study, a researcher observes and records measurements of variables of interest but does not impose a treatment. Math Tip: The results of an observational study can only imply an association. The results of an experiment, by imposing a condition, can imply causation. Math Vocabulary: A third unmeasured variable that may be associated with both of the measured variables is called a confounding variable. This variable is confounded with one of the other two, and therefore is a potential explanation of the association. Page 6 of 16

7 Page 7 of 16 Research shows an association between the number of roads built in Europe and the number of children born in the United States. What is a confounding factor that could explain this relationship?

8 Resources: SpringBoard- Algebra 2 Online Resources: Algebra 2 Springboard Text Unit 7 Vocabulary: Density curve Z-score Normal distribution Normal curve Sample Survey Response Bias Simple random sample Experiment Explanatory variable Response variable Completely randomized design Randomized block design Matched pairs design Single-blind study Double-blind study Observational study Confounding variable Statistic Algebra 2 Honors: Probability and Statistics Semester 2, Unit 7: Activity 39 Unit Overview In this unit, students study normal data distributions and solve problems using tables and technology. An examination of bias provides students with a reason to develop simple random samples from a population of interest. Students create simulations with and without technology to test conjectures about data. Margin of error is applied to population proportions and an informal understanding of statistical significance is developed. Student Focus Main Ideas for success in lessons 39-1 & 39-2: Investigate how to use a margin of error in an estimate of a population proportion. Use simulation models for random samples. Lesson 39-1: Math Vocabulary: The margin of error indicates how close the actual proportion is to the estimate of the proportion found in a survey of a random sample. Math Vocabulary: Survey results converted into a proportion; for each result is called the sample proportion. A local newspaper publishes a survey that says two-thirds of townspeople prefer pizza chain A over pizza chain B with 95% confidence that the margin of error is ±4%. What is the lowest possible percentage of people who prefer pizza chain B, rounded to the nearest percent? Page 8 of 16

9 Margin of error Simulation Sample proportion Sampling distribution Critical value Statistically significant Lesson 39-2: Math Vocabulary: The distribution of proportions of those who indicate they are satisfied for all possible samples of size n from the population is called the sampling distribution of the population for that statistic. Math Vocabulary: A critical value for an approximately normal distribution is the z-score that corresponds to a level of confidence. Page 9 of 16

10 Name class date Algebra 2 Unit 7 Practice Lesson 36-1 Use this information for Items 1 5. The test scores for 15 students are 72, 74, 74, 79, 82, 82, 82, 82, 86, 86, 86, 87, 87, 89, and a. Find the mean, median, and mode of the data. b. What is the standard deviation? 2. Use appropriate tools strategically. Make a dot plot of the data and describe the shape of the distribution. Lesson 36-2 Use this information for Items The mean maximum jumping height of U.S. high school high jumpers is 5 ft, 11.5 in., with a standard deviation of 2.25 in. The data are normally distributed. 6. Make sense of problems. a. What percent of U.S. high school high jumpers have a maximum jump height between 21 and 11 standard deviations from the mean? b. What percent of high jumpers have a maximum jump height that is more than 2 standard deviations from the mean? c. What percent of high jumpers have a maximum jump height less than 3 standard deviations from the mean? 7. a. What jump height is 2 standard deviations above the mean? 3. Reason abstractly. How can you tell from the mean, median, and mode that the data are not exactly symmetric? Based on the data summaries, how are the data skewed? 4. Which test score would you expect to be associated with a z-score of 12? A. 77 B C D Based on the z-score, which is less extreme, a test score of 79 or of 86? Why? b. What jump height is 3 standard deviations below the mean? 8. Given a representative sample of 200 U.S. high school high jumpers, how many would you expect to be able to clear the bar when it is set at 6 ft 4 in.? A. 6 jumpers B. 5 jumpers C. 3 jumpers D. 0 jumpers 9. Given a representative sample of 200 high jumpers, how many would you expect to be able to clear the bar when it is set at 5 ft 4.75 in.? 10. Attend to precision. Statistically speaking, how many jumpers should a sample include so that you could expect to have at least 1 jumper able to clear the bar at 6 ft 4.25 in.? 2015 College Board. All rights reserved. 1 SpringBoard Algebra 2, Unit 7 Practice Page 10 of 16

11 Name class date Lesson 36-3 Use this information for Items The heights of Major League Baseball players in the year 2012 were approximately normally distributed, with m in. and s in. 11. Which of the following z-scores is the best estimate for a player height of 75 in.? A B C D Lesson 36-4 Use this information for Items The heights of Major League Baseball players in the year 2012 were approximately normally distributed, with m in. and s in. 16. How many of the 1400 players are expected to have heights between 69.5 in. and 76.5 in.? A. 600 B. 800 C D A certain player has a height with a z-score of 1.2. Which of the following is the best estimate of his height? A in. B. 75 in. C. 80 in. D in. 13. Construct viable arguments. Brian claims that his favorite team includes 3 players, each over 6 ft 10 in. tall. Assuming that there are 1400 players in Major League Baseball, is his claim reasonable? Why or why not? 14. Critique the reasoning of others. Belle claims that since there are 1400 players in Major League Baseball and 77 3 in. represents a z-score of 1.0, 4 then there are 1178 players who are taller than 77 3 in. Is she correct? Why or why not? Use appropriate tools strategically. Use technology to determine the number of players with z-scores for height between and Use appropriate tools strategically. Use technology to determine the expected percentage of players with heights in each of the following ranges. a. 70 in. and 76 in. b. 72 in. and 73 in. c. 77 in. and 79 in. d in. and 80 in. 19. How many more players are expected to have heights above 73 in. than between 71 in. and 74 in.? A. 200 B. 300 C. 400 D How tall is a player whose height is greater than 90% of Major League Baseball players, to the nearest tenth of an inch? 20. How many more players are expected to have heights above 76.5 in. than below 70 in.? College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 11 of 16

12 Name class date Lesson 37-1 Jamie wants to evaluate the negative effects of computer gaming on student grades. 21. Jamie decides to limit her study to the 27 students in her math class. Which statement is most likely to be true? A. Jamie will need a representative sample. B. Jamie will have difficulty identifying the population in her study. C. Jamie will need to use a random number generator or table. D. Jamie will be able to evaluate the entire study population. 22. Critique the reasoning of others. Jamie has decided to use the ninth-grade class as the population for her study. After some consideration, she decides to interview ninth-graders who shop in the local used-game store as a sample of the ninth-grade population. Is Jamie s sample free from bias? Why or why not? 25. Why is a simple random sample most likely to be free from bias? A. The composition of the sampling method favors certain outcomes. B. The sampling method will contain common characteristics in each sample. C. The sample data collected will always be smaller than with other methods. D. Members of the sample population have the same probability of being chosen for the sample. Lesson 37-2 Use this information for Items A new nutritional supplement claims to increase long-distance runner endurance by 25%. A study to evaluate the claim is planned using a representative sample of 50 male and 50 female long-distance runners. 26. Construct viable arguments. A study is designed to give the supplement to 100 runners before they race in Florida. The study will compare the results to the same runners when they raced in Colorado before taking the supplement. Is this a reasonable study method? Why or why not? 23. Make sense of problems. After conducting her evaluation on the students from the game store, Jamie presents her findings and her instructor tells her that she should have been more careful with her sample choice. What might Jamie have done to make sure her sample was more representative of the ninth grade? 24. Sayber is in charge of social activities at his office party. He decides to poll his friends when they get together for Monday Night Football to see if they agree with his plan to play football trivia at the party. He concludes from his study that football trivia is supported by over 90% of people in his office. Are Sayber s results likely to be accurate? Explain your answer. 27. Which of the following is an example of using the block design to determine the possible difference in effects on male and female runners? A. Have each male and each female run around the block before and after receiving the supplement. B. Block the females from getting the supplement to compare the results to the males who received it. C. Randomly select a block of 25 males and 25 females to receive a placebo. D. Block the males and females from discussing the results of the study with each other. 28. If the developer of the supplement commissioned the study, how would a double-blind study improve the accuracy of the results? College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 12 of 16

13 Name class date 29. Reason abstractly. According to the results of the double-blind study, 62% of the runners receiving the supplement demonstrated a 20% increase in endurance, compared to 45% of runners demonstrating a 25% increase while on the placebo. Did the study support the developer s claim? 30. During the race that included the supplement and placebo, there were so many runners that the race was conducted over a period of two days, with the women running on the first day in the rain and the men on the second day under sunny skies. Given this situation, describe a possible benefit of conducting a matched-pairs design. Lesson 37-3 Use this information for Items A software development company monitors water consumption from the water cooler and determines that employees who drink more than 4 cups of water each day are more productive. 31. Was this an observational study or an experiment? Why? 32. Identify the population and question of interest. A. The population is the set of employees using the cooler, and the question of interest is if four cups of water are too many. B. The population is the set of company employees, and the question is if drinking more water is healthier. C. The population is the set of productive employees, and the question is how much water they drink. D. The population is the set of company employees, and the question of interest is the effect of drinking water on productivity. 33. Reason abstractly. The company president decides to institute a company policy to remove the water coolers and give every employee a large bottle containing a minimum of 4 cups of water to drink each day. After three weeks under the new policy, the president notes that overall production has gone down significantly. Describe a confounding variable that could have affected the results. Use this information for Items A certain company allows employees to work from home a maximum of 4 days per month. The company has recently installed software on the company-owned computers that employees use for work at home or in the office. The software monitors keyboard usage to track how much employees are working. 34. After three weeks it is determined that employees work more at home than in the office. Identify the population and question of interest. 35. Construct viable arguments. After reviewing the initial results of the working from home study, the company president decides to require that all employees work from home full time. Was this a reasonable response to the results from the study? Identify a confounding variable that may have affected the results. Lesson Tony is debating whether to open a golfing supply store or a camping supply store in his hometown. He wants to take into account the potential profitability of each type of store based on average purchase amount as recorded by sales from similar stores in a nearby town. Over a one-hour period, he asks customers leaving other golf and camping stores how much they spent in the store. Which two statistics would be most appropriate for Tony to calculate from the data based on the goal of his study? A. mean and median B. mode and range C. standard deviation and extrema D. IQR and variance College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 13 of 16

14 Name class date 37. Model with mathematics. Tony finds that one out of every three shoppers made a purchase in the camping store. How can he simulate whether or not a customer who visits the camping store will make a purchase? 38. Model with mathematics. Tony determines that the camping supply store purchases tend to be either small purchases between $2 and $4 or larger purchases between $25 and $29, with an approximately equal number of each. Describe a simulation method that Tony could use to generate additional sample data. 43. Which of the following probabilities would best indicate that Brian should not reject the null hypothesis in his experiment? A B C D Express regularity in repeated reasoning. An experiment is conducted to test the likelihood that a claimed result is valid. According to the experiment, the probability that the claimed result could have occurred by chance is Did the experiment reject the claim? Defend your answer. 39. A certain thunderstorm has an average of six lightning strikes less than or equal to 1 second long every minute. Describe a simulation that could model the probability of a lightning strike during any given second. Lesson 38-2 Use this information for Items During the Leonid meteor shower, observers can see as many as 600 meteors per hour. Brian s friend claims to have seen 1 meteor per second for 5 seconds in a row. 40. a. What is the probability of a meteor falling in a given second given the maximum rate of 600 meteors per hour? Explain your answer. b. Describe a simulation that Brian could use to determine how likely it is that his friend is exaggerating. 41. The null hypothesis is the statement used in a simulation or experiment that assumes the status quo, meaning that there is no change from stated belief. What would be the null hypothesis in Brian s simulation experiment? What would be the significance of rejecting the null hypothesis in this situation? 42. Reason quantitatively. Brian states that he believes his friend is not exaggerating. He conducts his simulation to determine that the probability of rolling the same value five times in a row is What should be Brian s conclusion? Lesson A study indicates, with a 95% confidence level, that 34% of respondents were in favor of a new law, with a margin of error of 4%. Which of the following statements is an accurate interpretation of the study? A. 95% of every 34 respondents are in favor of at least 4% of the new law. B. 95 out of every 100 researchers are certain that between 32% and 36% of respondents are in favor of the law. C. The researcher is 95% confident that between 30% and 38% of respondents are in favor of the law. D. Researchers are between 91% and 99% confident that at least 34% of respondents are in favor of the law. 46. Make sense of problems. A given study indicates, with a 95% confidence level, that 52% of students scored at least a C (70%) on a recent test, with a margin of error of 3%. Write a sentence to interpret these results College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 14 of 16

15 Name class date Use this information for Items A certain study indicates, with a 90% confidence level, that 72% of young children who eat candy become more active, with a margin of error of 3%. 47. Construct viable arguments. If 150 children were given candy in an experiment, and 112 became more active, did the experiment return the expected results? Explain your answer. 48. If 100 children were given candy and only 52% became more active, would this invalidate the original study? Why or why not? 49. If 50 children were given candy, how many would you expect to become more active, according to the data from the study? Lesson Which z-score corresponds to a 95% confidence interval? A B C D Use this information for Items A university wants to determine the proportion of students that have grade-point averages above 2.9. The university conducts a random survey of 250 students and found that 210 have a grade-point average of 2.9 or above. 51. Identify the following. a. the question of interest b. the sample proportion c. the population 52. Attend to precision. What is the standard deviation of the sample proportion? 53. Use appropriate tools strategically. Use technology to compute the margin of error for a 95% confidence level. 54. Interpret the results in sentence form. Lesson 40-1 Use this information for Items A student study group is attempting to determine if students retain more information when they highlight the important information in the texts they study. They study 500 students over a period of several months. The students are randomly separated into two groups of 250. One group deliberately highlights any important information as they come across it in a text, and the other group does no highlighting at all. The mean test scores of each group for each of four tests taken during the study are presented in a table. Highlight No Highlight Identify the following. a. the question of interest b. the treatment imposed c. Is this an observation or an experiment? 56. Which of the following is the most appropriate test statistic? A. the number of students choosing to highlight B. the number of students with good test scores C. the mean test scores of each group D. the standard deviation of the test scores of each group 57. Make sense of problems. Interpret the meaning of a positive test statistic if the test scores are evaluated by arithmetic mean. 58. Attend to precision. Compute the mean and median test scores for each group over the four tests. 59. If the results were determined not to be statistically significant, what would the study group conclude based on the experiment? College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 15 of 16

16 Name class date Lesson Determine the ratio of means of the two groups, Highlight and No Highlight. 61. Interpret the ratio from Item 60 in the context of test scores earned by highlighters versus nonhighlighters. 62. Reason quantitatively. Suppose the study group determines that the probability that a ratio of means between 1.02 and 1.04 occurring by chance is 23%. Would it be reasonable to conclude that the results are statistically significant? Explain your answer. 64. If the ratio were 1.43 rather than 1.03, which of the following would you expect to find? A. The 1.43 ratio is more statistically significant than the 1.03 ratio. B. The 1.43 ratio is less statistically significant than the 1.03 ratio. C. The 1.43 ratio is no more or less significant than the 1.03 ratio. D. The statistical significance cannot be used in this study. 63. Reason quantitatively. If the ratio of means between Highlight and No Highlight were 1.43, what would that indicate in terms of the study? College Board. All rights reserved. SpringBoard Algebra 2, Unit 7 Practice Page 16 of 16