Data Representation Lesson Practice

Answer Key I. Data Representation 1. b 2. e 3. b 4. c 5. 15/58 II. Measure of Center and Spread 6. a 7. 2.5 8. a 9. a 10. c 11. d 12. b 13. c 14. b - c 15. b 16. c II. Probability 17. a. 1/3, b. 2/3, c. 1, d. 0, e. 2/3 18. d 19. e 20. 7/10 21. c 22. 3/5 23. d 24. b 25. 60 26. 112/225 27. d III. Statistical Concepts 28. d 29. d - e 30. 74 31. e 32. d 33. b 34. b – c - d Contents I. Data Representation II. Measure of Center and Range III. Probability IV. Statistical Concepts V. Answer Key I. Data Representation There are different ways that data can be represented. It largely depends on the type of variable you use. Some of the most common representations include tables, dot plots, histograms, pie charts and bar graphs. Two main types of variables for data include categorical and quantitative. Eye color is an example of categorical because eye color could be blue, brown, green or hazel. Weight would be an example of quantitative or numerical because someone can weigh 128 pounds. Examples Data - Eye Color A student did a survey on eye color and found that blue, blue, brown, hazel, green, brown, brown, brown, hazel, green, hazel and brown. We will be using this data for a categorical frequency table, a bar chart and a pie chart. Categorical Frequency Table is used to organize categorical data in a table form. Eye Color Brown Hazel Blue Green Frequency 5 3 2 2 Example #1 Using the table above, what is the sample size? Frequency is the number of times that type of eye color occurs. There are 5 people who have brown eyes and 2 have blue eyes. If I add the values (5+3+2+2), this gives us a sample size of 12. Bar Chart or Graph represents categorical data by using vertical or horizontal bars whose heights represent the frequencies of the data. Eye Color of Different Students 6 4 2 0 Brown Hazel Blue Green Pie Chart or Graph is a circle that is divided into sections or wedges according to the percentages or frequencies in each category of the distributions. The pie chart with same data would look like this: Eye Color of Different Students Green Brown Blue Hazel One way that proportions can be used is to find the central angle and the percentage. 𝑝𝑎𝑟𝑡 𝑡𝑜𝑡𝑎𝑙 = 𝑎𝑛𝑔𝑙𝑒 360 or 𝑝𝑎𝑟𝑡 𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 100 or 𝑎𝑛𝑔𝑙𝑒 360 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 100 Example #2 Using the data above for eye color, what is the central angle for brown eyes? What is the percentage? For brown eyes, remember there were 5 students out of total of 12 students. Solving for angle 𝑝𝑎𝑟𝑡 5 𝑥 = 𝑡𝑜𝑡𝑎𝑙 12 360 360 ∙ 5 = 12𝑥 1800 = 12𝑥 150 = 𝑥 The central angle is 150⁰ Solving for percent 𝑝 5 = 12 100 500 = 12𝑝 41.6666 … = 𝑝 The percent is 41.7%, rounded to the nearest tenth of a percent. Age of Dogs A student did a survey given to dog owners at her school and asked them the age of the dogs. She found 5, 7, 6, 6, 5, 9, 3, 4, 1, 8, 5, 7, 7, 6, 3, 5, 6, 5, 4 and 2. We will be using this data for a quantitative frequency table, a histogram and a dot plot. Quantitative Frequency Table is the organization of numerical data in table form, using classes and frequencies. The classes can be a single value or a range of values. Age 1 2 3 4 5 6 7 8 9 Frequency 1 1 2 2 5 4 3 1 1 Example #3 Using the frequency table above, what is the total number of dogs? By adding the frequencies (1+1+2+2+5+4+3+1+1), we get a total of 20 dogs. The frequency table with numerical data can be adjusted to combine ages or a range of ages. Age 1-3 4-6 7-9 Frequency 4 11 5 The Histogram is a graph that displays the data by using connected vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes. Using a frequency table, a histogram can be created. Like a bar chart, a histogram uses frequency but differ in the type of variable it uses. Visually, this is shown by the bars touching. A Dotplot is a statistical graph in which each data value is plotted as a point (dot) above the horizontal axis. The dotplot is different from a scatterplot. The scatterplot shows the relationship between two variables where the dotplot only shows the distribution of one variable. An example of the same data as a dotplot: On the dotplot, each point represent a value or a dog. The x-axis describes the value associated with the data point. Practice Problems 1. The table below shows revenue, circulation, and number of pages for the Math Center Times Newspaper. 1960-2000 Year Annual Advertising Average Weekly Revenue (billions of Circulation (in dollars) thousands) 1960 7 90 1970 17 98 1980 23 102 1990 37 145 2000 29 113 Which of the following can be concluded from the data above? Average Number of Pages per Newspaper 11 25 36 52 61 a. Annual advertising revenue, average weekly circulation, average number of pages were all the greatest in 1990 b. In each ten-year period, when the average circulation decreased, annual advertising revenue also decreased c. In each ten-year period, when the average circulation decreased, the average number of pages decreased 2. Daniel Tiger is interested in the relationship between the weather conditions and whether the trolley is on time or delayed. For 6 months, he records the information below. Weather Condition Sunny Cloudy Rainy Snowy Total On Time 130 15 5 0 150 Delayed 5 6 15 4 30 Total 135 21 20 4 180 Based on the table above, which one of the following statements is true? a. b. c. d. e. More than half of the delays happened on rainy days The fewest number of delays happened on sunny days The greatest number of delays happened on cloudy days When the weather was cloudy, it was more likely for the trolley to be delayed than on time When the weather was rainy, it was more likely for the trolley to be delayed than on time 3. The graph below shows eye color of 50 different students. Eye Color of Different Students Green Brown Blue Hazel According to the pie chart, approximately how many students have hazel eyes? a. b. c. d. e. 5 13 25 30 45 4. The dot plot below shows the ages of different dogs in a pet store. What percentage is less than the age of 5? a. b. c. d. 20% 25% 30% 45% 5. Use the table below to answer the following question. Income State University Community College Under $35,000 30 45 $35,000 to $49,999 85 62 $50,000 and over 40 28 Researchers surveyed recent graduates of two different postsecondary schools about their incomes. The table above displays the data from the survey. What fraction of the graduates in the sample had an income of under $35,000? Give your answers as a simplified fraction. II. Measures of Center and Range For data, there are four main measures of central tendency. They include mean, median, mode, and midrange. You also might see range. Range is a measure of variation along with variance and standard deviation. A measure of variation tells you something about the spread of the data or how much the data varies. Definitions and Examples Mean is the average. Sum of the values divided by the number of items. Example #1 The mean of 8, 11, 17, 18, 20, 30 is _____. 8 + 11 + 17 + 18 + 20 + 30 6 ≈ 17.3 Answer: 17.3 Weighted mean or average is the sum of the values times their weights divided by the sum of the weights. Weighted mean can be used with dot plots and frequency tables or charts. Example #2 What is the mean using the frequency table below? Value 20 30 40 Answer: 34 Frequency 2 5 8 20 ∙ 2 + 30 ∙ 5 + 40 ∙ 8 510 = = 34 2+5+8 15 Median is the middle number. If the list is not in order, you will need to arrange the values in ascending order (least to greatest). They must be in order. If odd number of items, it is the middle number. If even number of items, it is the average of the middle two numbers. Example #3 The median of 5, 6, 7, 8, 10, 21, 23, 26 is _______. The middle two numbers are 8 and 10. The average of the two values is 9. Answer: 9 Example #4 The median of 4, 6, 7, 8, 10, 15, 21 is _______. The middle number is 8. Answer: 8 Mode is the most frequent number that occurs. It is possible to have no mode or more than one mode, but you will rarely see this. Example #5 The mode of 1,1,2,2,3,3,3,3,4,4,5 is _______. 3 occurs the most (four times) Answer: 3 Midrange is the average of the high and low. Example #6 The midrange of 4, 6, 7, 8, 10, 15, 21 is _______. 𝑀𝑖𝑑𝑟𝑎𝑛𝑔𝑒 = ℎ𝑖𝑔ℎ + 𝑙𝑜𝑤 21 + 4 = = 12.5 2 2 Answer: 12.5 Range is the difference between highest value and lowest value. It is a way to measure the spread. Example #7 The range of 4, 6, 7, 8, 10, 15, 21 is _______. 𝑅𝑎𝑛𝑔𝑒 = ℎ𝑖𝑔ℎ − 𝑙𝑜𝑤 = 21 − 4 = 17 Answer: 17 Variance and standard deviation are difficult to calculate. They are related to one another. The square root of variance equals the standard deviation. They both measure how much the data varies from the mean. You will not be asked to calculate these statistical measures, but you may be asked a conceptual question. If a question wants you to show the variance or the standard deviation, your answer would include one or the other. Practice Problems 6. The mean of the numbers below is 37.4. How does the median compare to the mean? 34, 28, 39, 54, 32 a. It is about 3 less than the mean b. It is equal to the mean c. It is about 3 more than the mean d. It is about 1 less than the mean e. It is about 1 more than the mean 7. What is the median of the numbers in the list below? 2.2, 4.3, 3.4, 2.5, 1.0, 1.2, 1.8, 3.2, 3.0, 2.5, 1.5 8. The median and range of 50 measurements are 25 and 82, respectively. If 8 is added to each of the measurements, which of the following statements regarding the median and range of the modified measurements must be true? a. The median will increase, but the range will stay the same b. The median will stay the same, but the range will increase c. The median and range will both increase d. The median will stay the same, but the range will decrease e. The median and the range will stay the same 9. Given the list of seven values: 23, 40, 65, 62, 51, 78, and 53. If the number 54 is added to the list as the 8th number, which of the following will be true? a. The mean will increase and the median will increase b. The mean will decrease and the range will decrease c. The mean will increase and the median will decrease d. The mean will decrease and the median will increase e. The mean will decrease and the median will stay the same 10. Seth ran the 50-yard dash 10 times during the fall. His times, in seconds are listed below. What is the range of Seth’s times? 12.3, 14.5, 11.3, 15.2, 16.3, 12.3, 11.2, 13.2, 11.1, 15.6 a. 3.3 b. 4.3 c. 5.2 d. 6.1 e. 6.3 11. Use the graph below to answer the following question: Sales for 7-day weeks 80 70 60 50 40 30 20 10 0 Week 1 Week 2 Week 3 Week 4 Sales Which is the closest to the average sales per day over the four-week period? a. 225 b. 80 c. 56 d. 8 e. 10 12. The ages, in years, of 7 children are 7, 6, 8, 11, 5, 9 and a. If the average of the 7 ages is 8, what is the value of a? a. 9 b. 10 c. 11 d. 12 e. 13 13. Given the following frequency distribution, what is the median of the sample? Value Frequency 0 3 1 4 2 11 3 12 4 3 5 8 a. 2.8 b. 3.5 c. 3 d. 4.5 e. 5 14. The following is a list of the weights of thirteen soccer players. 138, 145, 152, 128, 164, 132, 125, 133, 126, 140, 151, 159, 162 If a fourteenth weight is added to the list and the range remains the same, which of the following could be the weight of the fourteenth player? Choose all correct answers. □a. 121 □b. 129 □c. 163 □d. 165 □e. 170 15. Which of the following will show the variance in age of the high school basketball team? a. the range of the ages b. the standard deviation of the ages c. the mean of the ages d. the median of the ages e. the midrange of the ages 16. The dot plot below shows the ages of different dogs in a pet store. What average age of the 20 dogs? (Try using a weighted average) a. b. c. d. e. 5 5.15 5.2 6 8 III. Probability Probability measures likelihood. Some basic rules of probability include: 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 (𝑤ℎ𝑎𝑡 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡) 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Probability is always a number between 0 and 1. It can be 0 (no chance) or 1 (a sure thing). Some other rules include the addition rule and the multiplication rule. When it says “or”, use addition. When it says “and,” use multiplication. For examples, we will use a standard 52 deck of cards. It is not necessary for you to memorize the cards. Example #1 What is the probability of getting an ace? 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑔𝑒𝑡𝑡𝑖𝑛𝑔 𝑎𝑛 𝑎𝑐𝑒 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑐𝑒𝑠 4 1 = = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 52 13 Remember to simplify your fraction. Addition rule 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑜𝑟 𝐵 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 + 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵 − 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐴 𝑎𝑛𝑑 𝐵 If A and B cannot happen at the same time, then probability of A and B is equal to 0. Example #2 What is the probability of getting an ace or a spade? 4 52 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝑎𝑐𝑒 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑠𝑝𝑎𝑑𝑒 = 13 52 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝑎𝑐𝑒 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒 = 1 52 4 13 1 16 4 + − = = 52 52 52 52 13 Example #3 What is the probability of getting a two or a seven? 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑡𝑤𝑜 = 4 52 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑠𝑒𝑣𝑒𝑛 = 4 52 4 4 8 2 + = = 52 52 52 13 Multiplication rule 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑎𝑛𝑑 𝐵 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 × 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵 Depending on the situation, the probability of B might be different. Example #4 If you replace the card after picking it, what is the probability of getting two kings? 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑘𝑖𝑛𝑔 = 4 52 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑘𝑖𝑛𝑔 = 1 1 1 4 4 ∙ = ∙ = 52 52 13 13 169 4 52 Example #5 If you don’t replace the card after picking it, what is the probability of getting two kings? 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑘𝑖𝑛𝑔 = 4 52 3 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑘𝑖𝑛𝑔 = 51 since you are not replacing the card 4 3 1 1 1 ∙ = ∙ = 52 51 13 17 221 Sometimes it might be convenient to use the complementary rule. 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑛𝑜𝑡 𝐴 = 1 − 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 Example #6 What is the probability of not getting a 10? Two ways to get the answer. First way is to count up the not 10’s. 48 12 = 52 13 Second way is to use the complementary rule. 1− 4 1 13 1 12 =1− = − = 52 13 13 13 13 Practice Problems 17. Fill-in-the-blank. A jar contains 4 red marbles and 2 black marbles. If you take a marble without looking, find the probability of getting: (Complete each sentence.) a. a black marble is ____ b. a red marble is ____ c. a red or black marble is ____ d. a blue marble is _____ e. not a black marble is ____ 18. If 3 black marbles are added to the jar above, what would be the probability of picking a red marble? a. 7/9 b. 1/3 c. 2/3 d. 4/9 e. 1/6 19. Katie has a bag of 120 marbles. Some of the marbles are blue and the rest are silver. If the probability of picking a silver is 3/5, how many silver marbles are in the bag? a. 36 b. 48 c. 50 d. 60 e. 72 20. There are 20 turtles in an aquarium. 4 are box, 6 are red-eared slider, and 10 are painted. What is the probability that a blindfolded person reaching in will pick a box turtle or a painted turtle? Express your answer as a fraction. 21. There are 60 math books in the bookshelf. Fourteen of the books are in math history, ten are in calculus, eight are in statistics and probability, four are in geometry, and the rest are in algebra. What is the probability that a book chosen at random will not be a math history book? a. 7/30 b. 1/3 c. 23/30 d. 1/6 e. 5/6 22. Ten cards are marked with one of the following numbers on it: 1, 2, 3, 5, 6, 11, 17, 18, 19, and 20. If a card is picked at random, what is the probability that it will be a prime number on it? Express your answer as a fraction. 23. A bag contains a number of solid-colored Skittles, of which 6 are red, 7 are green, 8 are yellow, and the rest are orange. If a person were to take one Skittle out of the bag at random, the probability of getting a red Skittle is 2/9. How many orange Skittles are in the bag? a. 3 b. 4 c. 5 d. 6 e. 7 24. There are 7 books and only one of them is white. There are 9 notebooks and only one of them is blue. What is the probability of randomly selecting a white book and a blue notebook? a. 2/63 b. 1/63 c. 1/8 d. 1/16 e. 9/16 25. There are 600 students in a group. If the probability of picking a boy from the group is 0.45, how many more girls are in the group than boys? 26. There are 140 blue jelly beans and 160 red jelly beans in a bowl. One jelly bean is picked from the bowl, the color is noted, and then the jelly bean is returned to the bowl. Then another jelly bean is selected from the bowl and the color is noted. What is the probability that exactly one red jelly bean is picked in these two tries? Express your answer as a fraction. 27. Mark flips a fair coin five times. It comes up heads, heads, heads, tails, and heads. What is the probability that coin will come up heads on the next flip? a. 1/5 b. 1/3 c. 4/5 d. 1/2 e. ¾ IV Statistical Concepts In this section, we will look at other concepts that you may come across relating to statistics. Definitions Sampling is the act of selecting your sample. The best way to select your sample is randomly, where all possibilities are included. If the method selects from a particular group, there might be bias. If the method is convenient, there is definitely bias. The most valid method would be random and anyone from the population can be selected. Linear regression tries to find a relationship between two variables. The scatterplots are referring to the points that fall on the graph. The stronger the relationship (or correlation) the more it will look like a straight line. A strong positive linear relationship means the points almost form a straight line that is increasing from left to right. A strong negative relationship means the points almost form a straight line that is decreasing from left to right. When thinking of positive or negative relationship, think slope. Statistical question is a question that needs statistics to answer it. It is not asking something specific. It is asking something about a group. Practice Problems 28. Tampa Bay is considering legislation to force the National Football League to change the team name from Buccaneers to something less controversial. Which of the following methods of selecting the sample will yield the most valid information about the feelings of all the residents of the Bay area? a. Choose a random sample of the football players b. Interview a random sample at football game c. Poll those “friends from their Facebook page d. Poll a random sample from the residents who live in Tampa e. Interview all the coaches 29. A principal wants to select a random sampling of faculty from his staff. Which of the following is an example of random sampling? Choose all that apply. □a. selecting the last 15 teachers who enter the meeting □b. selecting the first 15 teachers who show up to work on Monday □c. e-mailing a survey and taking the first 15 who fill it out □d. choosing 15 names of all the teachers from out of a hat □e. choosing every 5th name of a list of teachers 30. The ages of a group of children are summarized in the frequency table below. How many children were in the group? Age Frequency 2 5 3 12 4 13 5 17 6 18 7 9 31. Which of the following scatterplots most strongly suggests a negative linear relationship between x and y? 4 3 3 2 2 1 1 0 0 0 a. 1 2 3 4 d. 3 3 2 2 1 1 b. 0 -0.5 0 -3 -2 1 2 1 2 3 4 0 1 2 3 4 0 0 -4 0 -1 0 3 4 e. -1 -1.5 -2 -2.5 -3 c. The number of students 32. WGU decided to experiment with changing the price of a special class for GRE test prep. From the scatterplot below, which of the following conclusions can be drawn? Cost of the course a. There is no correlation between cost and the number of students b. There is a strong positive correlation between cost and the number of students c. There is a weak positive correlation between cost and the number of students d. There is a strong negative correlation between cost and the number of students e. There is a weak negative correlation between cost and the number of students Math score of your basic skills exam Math score of your basic skills exam 33. Which scatterplot shows the hopeful results between hours studied and scores on your basic skills exam? Number of hours studied Number of hours studied d. Math score of your basic skills exam Math score of your basic skills exam a. Number of hours studied Number of hours studied e. Math score of your basic skills exam b. Number of hours studied c. 34. Which of the following questions are statistical questions? Indicate all such questions. □a. How many course mentors are in the math center? □b. How tall is the average course mentor in the math center? □c. Is there a correlation between the number of appointments with the math center and their math score on the Praxis Core? □d. How many students use the math center on a weekly basis? □e. When was the math center established?