Advanced Algebra Cumulative Review for Semester final

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Feb 20, 2024

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Advanced Algebra: C&C Final Exam Cumulative Review Problems 1
IOA 1.e: given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology (AA.DSR.2.5) 2
IOA 1.e Problem 1: When Andrew goes bowling, his scores are normally distributed with a mean of 160 and a standard deviation of 13. What percentage of the games that Andrew bowls does he score between 134 and 186? a. 68% b. 81.5% c. 95% d. 99% Problem 2: When Rashaad commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 34 minutes and standard deviation of 2.5 minutes. Determine the interval that represents the middle 95% of his commute times. 3
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IOA 1.e Problem 3: A store sells meat in small, medium and large size packages. The small packages have a mean weight of 1 pound and a standard deviation of 0.1 pound. If the weights for the small packages of meat are approximately normally distributed, about 95% of the data falls between which two values? a. 0.7 and 1.3 b. 0.8 and 1.2 c. 0.8 and 1.1 d. 0.9 and 1.1 Problem 4: When Camila goes bowling, her scores are normally distributed with a mean of 150 and a standard deviation of 12. What percentage of the games that Camila bowls does she score less than 137, to the nearest tenth ? 4
IOA 1.d: calculate and interpret z- scores as a measure of relative standing and as a method of standardizing units (AA.DSR.2.4) 5
IOA 1.d Problem 1: On a standardized exam, the scores are normally distributed with a mean of 300 and a standard deviation of 50. a) Find the z-score of a person who scored 235 on the exam. b) What does this z-score mean in the context of this scenario? Problem 2: At a restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 20 minutes and a standard deviation of 4 minutes. Which of the following BEST describes a wait time of 26 minutes? a) The wait time is 1.5 standard deviations above the mean. This means that this wait time is longer than 93.32% of the times . b) The wait time is 1.5 standard deviations above the mean. This means that this wait time is longer than 6.68% of the times . c) The wait time is 1.5 standard deviations below the mean. This means that this wait time is shorter than 6.68% of the times . d) The wait time is 1.5 standard deviations below the mean. This means that this wait time is shorter than 93.32% of the times . 6
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IOA 1.d Problem 3: A representative sample of trees in a forest shows the trees to have a mean circumference of 19.7 inches and a standard deviation of 1.3 inches. Assuming the circumferences are normally distributed, what is the z-score for a tree with a circumference of 22 inches? a. -6.85 b. -1.77 c. 1.77 d. 6.85 Interpret this z-score for this scenario. Problem 4: Nutritionists measured the sugar content for 32 drinks at a coffee shops. The drinks had a mean of 18 grams and a standard deviation of 5 grams. Which of the following best describes a cup of coffee with 14 grams of sugar? a. This drink has 0.8 g of sugar less than the mean of the 32 drinks. b. This drink is 0.8 standard deviations below the mean of the 32 drinks. c. About 80% of drinks have a lower sugar content than this drink. d. About 80% of drinks have a higher sugar content than this drink. 7
IOA 1.a: recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each; distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn (e.g., students might design and carry out a study with a recognition of error in the design of the study or evaluate a research study and critique the investigative measures and/or conclusions drawn about the data) (AA.DSR.2.1) 8
IOA 1.a Problem 1: Which question is best answered by an experiment? Select all that apply. a. Do nurses spend more time commuting than doctors? b. What effect does running every day have on cardiovascular health? c. Does taking a yoga class before the commute to work affect the heart rate of the drivers? d. How many nurses and how many doctors use running as a primary form of exercise? Problem 2: Which scenario represents randomization with respect to an observational study? 9
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IOA 1.a Problem 3: Which of the following is an observational study? Select all that apply. a. A company divides volunteers into two groups to test the popularity of a new version of a vitamin pill. b. A researcher visits classes and records the amount of times students answer questions. c. A government agency sends questionnaires to owners of small businesses in three states. d. A company records the time it takes its employees to park their car and enter the building. Problem 4: The United States Senate has 2 senators from each of the 50 states. Select all methods that would select a sample of 10 senators at random from the entire Senate. a. Take the 10 senators from the 5 largest states. b. Take the 10 senators who have served in the Senate the longest. c. Place the names of all 100 senators on individual slips of paper placed into a bowl. Thoroughly mix up the slips and then select 10 of them. d. Send letters to all 100 senators. Select the 10 that reply back the quickest. e. Number the senators from 1 to 100. Use a random number generator to make a list of 10 random integers between 1 and 100 inclusive 10
IOA 1.c: distinguish between population distributions, sample data distributions, and sampling distributions; use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language (AA.DSR.2.3) 11
IOA 1 Problem 1: A farmer wants to determine whether a nutritional supplement will have an effect on cows’ milk production. For one week, he gives the supplement to half of his cows and then measures their milk output. Then, he compares the milk output of the cows that took the supplement with the milk output of the cows that did not get the supplement. Classify the situation as a sample survey, an observational survey, or an experiment. Explain your reasoning. If it is an experiment, identify the treatments. Problem 2: 12
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IOA 1.c Problem 3: Decide whether a sample survey, an observational study, or an experiment is the best method for each scenario. Describe how to obtain a random sample. You want to compare the average income of public school teachers with and without advanced degrees in a certain city. Problem 4: Identify the population, the sample, and the characteristic of interest for the situation. A learning center claims that students can improve college entrance exam scores by taking their prep course. In order to test the claim, an independent organization records the exam scores of 145 randomly selected students before taking the prep course and their exam scores after taking the prep course 13
IOA 1.f: model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population and decide if the specified model accurately reflects real outcomes (AA.DSR.2.6) 14
IOA 1.f Problem 1: Myriah hates doing the dishes, but her parents insist that she must help out and do them sometimes. Since Myriah wants to leave it to chance whether or not she will have to do the dishes, Myriah’s mom proposes that they roll two dice. If the sum of the dice is 6 or less, Myriah will do the dishes. If the sum is 7 or more, one of her parents will do the dishes. How often can Myriah expect to have to do the dishes? Problem 2: Which statement is true regarding the sampling distribution of the sample mean? a. The sampling distribution shows the sampling variability of repeated samples. b. The sampling distribution mean is used as an estimate of the population’s mean. c. The sampling distribution of the sample mean will be a nearly normal distribution when the sample size is large. d. All of the above are true. 15
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IOA 1.f Problem 3: Steven takes a series of samples of size n = 40 from a population of size N = 300. He examines the averages of the proportions in each of the samples. Steven could expect: a. The averages of the proportions to be the same in each sample. b. The averages of the proportions to vary from sample to sample. c. The mean of the averages of the proportions from the samples to equal the standard deviation of the population. d. The mean of the averages of the proportions to equal to the mean of the standard deviations from the samples. Problem 4: A group of community organizers wanted to estimate the proportion of families in the community who had someone in their household with asthma. Team 1 took a random sample of 20 families, team 2 took a random sample of 30 families and team 3 took a random sample of 40 families. Below are their sampling distributions shown as histograms. Which histogram likely matches each team? Explain. 16
IOA 1.g: given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability (AA.DSR.2.7) 17
IOA 1.g Problem 1: A survey was given to a random sample of the residents of a town to determine whether they support a new plan to raise taxes in order to increase education spending. The percentage of people who said the supported the plan to raise taxes was 59%. The margin of error for the survey was 4%. Write a confidence interval for the percentage of the population that favors the plan. Problem 2: A scientific report says that the proportion of the population that is left-handed is 0.10 with a margin of error of 0.02. Which of the following is a plausible estimate for the proportion of the population that is left- handed? a. 0.07 b. 0.087 c. 0.128 d. 0.130 18
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IOA 1.g Problem 3: In an election poll, Candidate A is leading with 53% of the vote while Candidate B has 47% of the vote. The poll has a margin of error of 4.5%. Which of the following statements is TRUE? a. Candidate A can be predicted the winner because the mean percentage for the poll is greater than 50%. b. Candidate B cannot be predicted the winner because the percentage of voters is between 42.5% and 51.5%. c. Both candidates can be predicted the winner because it is possible for either candidate to get more than 50% of the vote. d. Neither candidate can be predicted the winner because it is possible for either candidate to get more than 50% of the vote. Problem 4: A study was commissioned to find the mean weight of the residents in certain town. The study found the mean weight to be 191 pounds with a margin of error of 7 pounds. Which of the following is a reasonable value for the true mean weight of the residents of the town? a. 181.7 b. 182.2 c. 195.7 d. 198.6 19
IOA 1.h: summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used; communicate statistical information using written and oral reports (AA.DSR.2.8) 20
IOA 1.h Problem 1: You want to perform an experiment to determine the support for a new stadium at the local high school. You choose a random sample of 40 senior citizens and interview them to represent the population for your experiment. Explain how this sampling method is biased. Problem 2: A researcher wants to know whether elementary school students enrolled in music classes do better on standardized math tests. He gathers data from 10 different elementary schools that have records for all of their students. Is the given method of data collection a sample survey, an observational study, or an experiment? Explain your reasoning. Identify the population, the sample, and the characteristic of interest 21
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IOA 1.h Problem 3: Suppose that two cell phone companies, E-Phone and Unlimited, claim that two of their comparable cell phone models have a mean battery life of 10 hours. If the mean battery life is 10 hours, does that indicate that all E-Phone’s phones and all of Unlimited ‘s phones have a 10 hour battery life? Problem 4: Wildlife biologists recorded the weights of grizzly bears in two different populations. The normal curves represent the weights of the bears in Population A and the weights of bears in Population B. Explain what the difference in the standard deviation means in terms of this population situation. 22
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IOA 4.a: graph and analyze key characteristics of quadratic functions using formal notation (i.e., interval and set notation); relate key characteristics to the real-world situation the model represents; 23
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IOA 4.a Problem 1: A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. What is the maximum height reached by the rocket, and how long does it take the rocket to reach that height? y = -16x 2 + 161x + 69 Problem 2: A company sells widgets. The amount of profit, p(x), made by the company, is related to the selling price of each widget, x, by the given equation: p(x) = -x 2 + 81x - 539 . a) Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar . b) What price should the widgets be sold for for the company to make the maximum profit? 24
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IOA 4.a Problem 3: A tennis ball is dropped from a certain height. Its height, h , in feet is given by h = -16t 2 + 19 where t represents the time in seconds after launch. What is the ball’s initial height? Problem 4: Mayerlin kicks a football. Its height, h, in feet is given by h = -16t 2 + 80t where t represents the time in seconds after kick. How long is the football in the air? a. 2.5 seconds b. 5 seconds c. 80 seconds d. 100 seconds 25
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IOA 4.e: write and solve quadratic equations (e.g., solve by inspection, taking square roots, factoring, completing the square, applying the quadratic formula) with real coefficients and use the solution to explain a real-world situation, 26
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IOA 4.e Problem 1: A garden designer designed a square decorative pool. The pool is surrounded by a walkway. On two opposite sides of the pool, the walkway is 8 feet. On the other two opposite sides, the walkway is 10 feet. Write an equation for the total area of the pool. What does a solution to the equation mean in this situation? Problem 2: The revenue from a youth league baseball game depends on the price per ticket, . Here is a graph that represents the revenue function, R . Select all the true statements. A. R (5) is a little more than 600. B. R(600) is a little less than 5. C. The maximum possible ticket price is$15. D. The maximum possible revenue is about$1,125. E. If tickets cost$10, the predicted revenue is$1,000. F. If tickets cost$20, the predicted revenue is$ 1,000. 27
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IOA 4.e Problem 3: A frog jumps in the air. The height, in inches of the frog is modeled by the function h(t) = 60t - 75 t 2 , where t is the time after it jumped, measured in seconds. Solve 60t - 75 t 2 = 0 . What do the solutions tell us about the jumping frog? Problem 4: A tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeled by the equation f(t) = 4 + 12t - 16t 2 , where t is measured in seconds since the ball was thrown. 1. Find the solutions to the equation. 2. What do the solutions tell us about the tennis ball? 28
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IOA 4.g: create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations (e.g., create and solve quadratic equations involving projectile motion) (AA.FGR.5.7) 29
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IOA 4.g Problem 1: An object is launched at 19.6 meters per second (m/s) from a 58.8 meter tall platform. The equation for the object’s height s at time t seconds after launch is s(t) = -4.9t 2 + 19.6t + 58.8 , where s is in meters. When does the object strike the ground? Problem 2: An object is launched directly upward at 64 feet per second (f/s) from a platform 80 feet high. What will be the object’s maximum height? When will it attain this height? 30
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IOA 4. Problem 3: Jason jumped off a cliff into the ocean in Jamaica while vacationing with his family. His height as a function of a time could be modeled by the function h(t) = -16t 2 + 16t + 480 , where t is the time in seconds and h is the height in feet. Jason hit the water after how many seconds? Problem 4: Jason jumped off a cliff into the ocean in Jamaica while vacationing with his family. His height as a function of a time could be modeled by the function h(t) = -16t 2 + 16t + 480 , where t is the time in seconds and h is the height in feet. What was the highest point that Jason reached? 31
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IOA 3.a: rewrite radical expressions as expressions with rational exponents; extend the properties of integer exponents to rational exponents (AA.FGR.4.1) 32
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IOA 3.a Problem 1: Which of the following is equivalent to the expression Problem 2: Which of the following is equivalent to the expression 33
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IOA 3 Problem 3: a) Rewrite the expression below in radical form: -27 b) Rewrite the expression below using rational exponents: Problem 4: Simplify 4 3 34
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Average rate of change 39
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