AP/UConn Statistics Summer Packet Welcome to Advanced Placement Statistics. The AP/UConn Statistics course is built around four main topics: exploring data, planning a study, probability as it relates to distributions of data, and inferential reasoning. Among leaders of industry, business, government, and education, almost everyone agrees that some knowledge of statistics is necessary to be an informed citizen or a productive worker. Numbers are regularly used and misused to justify opinions on public policy. Quantitative information is the basis for decision-making in virtually every job within business and industry. Included in this packet is a brief description of basic statistics with corresponding exercises. There are also questions in the wrap-up section which require an assimilation of statistical ideas. Lastly, there is a multiple choice section. These questions review concepts that all AP/Uconn Statistics students should know before entering the class. You are expected to complete this packet before the start of school. You are also expected to show the work needed to get the correct answers. The packet will be graded as a homework assignment, with the grade based on completion only. I will answer questions concerning the packet during the first few days of school. Good luck! I am looking forward to an exciting year working with students who are prepared for class and are willing to challenge themselves. Feel free to email me at ncowles@ctreg14.org if there are any questions over the summer.
I. Measuring Central Tendency Some measures of central tendency are the mean, median, and mode. Consider the following set of data. 15, 11, 19, 15, 14, 13, 17, 11, 12, 17, 15, 14, 15, 14 To begin, order the fourteen numbers: 11, 11, 12, 13, 14, 14, 14, 15, 15, 15, 15, 17, 17, 19 To find the mean (xx ), divide the sum of the numbers by 14. For this set of data, xx = 14.4. The median is the middle number. If there are an even number in the list, it is the average of the two middle numbers. For this set of data, the median is 14.5. The mode is the number that occurs the most frequently. For this set of data, the mode is 15. II. Measuring Spread The first quartile (Q1) is the median of the first half of the data. The third quartile (Q3) is the median of the second half of the data. For the data above, Q1 is 13 and Q3 is 15. The range is the difference of the highest and lowest data points. For the data above, the range is 8. The 5 number summary is the minimum, Q1, median, Q3, maximum. The 5 number summary for the data above is 11, 13, 14.5, 15, 19. Exercises 1. Find the mean, median, mode, range, and 5 number summary for the following test scores. 32, 72, 81, 95, 98, 58, 77, 75, 83, 97, 45, 89, 93, 57, 82, 97, 52, 75
2. The weights (in pounds) of eleven children are as follows: 39, 52, 40, 45, 46, 55, 48, 40, 43, 47, 44. Find the mean, median, mode, range, and 5 number summary for the data. III. Organizing Data At a car dealership, the number of new cars sold in a week by each salesperson was as follows: 5, 8, 2, 0, 2, 4, 7, 4, 1, 1, 2, 2, 0, 1, 2, 0, 1, 3, 3, 2 a. This is a frequency distribution b. This is a line plot for the for the data. data. c. This is a bar graph for the data.
Exercises 3. Twenty students in a class were asked how many cars their family owned. The results were as follows: 2, 2, 3, 2, 1, 2, 2, 3, 2, 0, 1, 0, 1, 1, 2, 2, 3, 2, 3, 5 Construct a frequency distribution and a line plot for this data. 4. Each of the members of a recent high school graduating class was asked to name his/her favorite among these subjects: English, world language, history, mathematics, science. The results are shown in the table. Construct a bar graph that shows these results. English 22 World Language 40 History 40 Mathematics 53 Science 33
IV. Constructing Stem-and-Leaf Plots and Histograms Consider the following set of data. 63, 52, 84, 83, 51, 32, 58, 35, 45, 41, 65, 75, 59, 67, 25, 46 The graph shown is a stem-and-leaf plot (sometimes called a stemplot) of the data. The graph shown is a histogram of the data. Exercises 5. Construct a stem-and-leaf plot for the data 15, 59, 66, 42, 48, 23, 70, 81, 35, 51, 68, 29, 77, 92, 85, 16, 37, 59, 61, 76, 40, 25, 86, 11
6. Construct a histogram for the same data. V. Venn diagrams and probabilities Consider the following problem: A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all three kinds of pets, what is the probability that a randomly chosen patron has none of these pets? Start by drawing a Venn diagram with 3 circles (dogs, cats, fish). Start in the center of the Venn diagram. If no one has all 3 pets, there are 0 patrons in the intersection of all three circles. Next fill in the patrons who have two types of pets.
There are 7 patrons who have dogs so far (4 + 0 + 3), so there are 7 who have only a dog. Similarly, there are 5 who have only a cat (10-4 - 0-1) and 1 who has only a fish (5-3 - 0-1). If you add all of the individual numbers together, we have accounted for 21 patrons. This means that there are 5 patrons who lie outside the Venn diagram. That means that the probability that a randomly chosen patron has none of these pets is 5 which is 0.1923 or 19.23%. 26 Exercise 7. There are three language classes that the 200 Upper School students at The Bear Creek School may take: Spanish, French, and Latin. 121 students take Spanish, 40 students take French, and 28 students take Latin. 3 students take both Latin and Spanish, 4 students take both French and Latin, 2 students take both Spanish and French, and no students take all three classes. Draw a Venn diagram to represent this information. a. What is the probability that a student selected at random will take Spanish only? b. What is the probability that a student selected at random takes none of these languages? c. What is the probability that a student selected at random takes Latin but not French?
VI. Applications 8. Find a set of 20 numbers that will satisfy the following conditions: The median is 24. The range is 42. To the nearest whole number the mean is 24. No more than three numbers are the same. Explain your strategy. 9. Calculate each of the following probabilities and arrange them in order from least to greatest. I. The probability that a fair die will produce an even number. II. A random digit from 1 to 9 (inclusive) is chosen, with all digits being equally likely the probability that when it s squared it will end with the digit 1. III. The probability that a letter chosen from the alphabet will be a vowel. IV. A random number between 1 and 20 (inclusive) is chosen the probability that its square root will not be an integer.
10. Two pain relievers, A and B, are being compared for relief of post surgical pain. Twenty different strengths (doses in milligrams) of each drug were tested. Eight hundred post surgical patients were randomly divided into 40 different groups. Twenty groups were given drug A. Each group was given a different strength. Similarly, the other twenty groups were given different strengths of drug B. Strengths used ranged from 210 to 400 milligrams. Thirty minutes after receiving the drug, each patient was asked to describe his or her pain relief on a scale of 0 (no decrease in pain) to 100 (pain totally gone). The strength of the drug given in milligrams and the average pain rating for each group are shown in the scatterplot below. Drug A is indicated with A s and drug B with B s. a. Based on the scatterplot, describe the effect of drug A and how it is related to strength in milligrams. b. Based on the scatterplot, describe the effect of drug B and how it is related to strength in milligrams.
c. Which drug would you give and at what strength, if the goal is to get pain relief of at least 50 at the lowest possible strength? Justify your answer based on the scatterplot. 11. An advertising agency in a large city is conducting a survey of adults to investigate whether there is an association between highest level of educational achievement and primary source for news. The company takes a random sample of 2,500 adults in the city. The results are shown in the table below. Primary Source for News HIGHEST LEVEL OF EDUCATIONAL ACHIEVEMENT Not High School Graduate High School Graduate But Not College Graduate College Graduate Total Newspapers 49 205 188 442 Local television 90 170 75 335 Cable television 113 496 147 756 Internet 41 401 245 687 None 77 165 38 280 Total 370 1,437 693 2,500 a. If an adult is to be selected at random from this sample, what is the probability that the selected adult is a college graduate or obtains news primarily from the internet? b. If an adult who is a college graduate is to be selected at random from this sample, what is the probability that the selected adult obtains news primarily from the internet? c. When selecting an adult at random from the sample of 2,500 adults, are the events is a college graduate and obtains news primarily from the internet independent? Justify your answer.
12. Approximately 3.5 percent of all children born in a certain region are from multiple births (that is, twins, triplets, etc.). Of the children born in the region who are from multiple births, 22 percent are left-handed. Of the children born in the region who are from single births, 11 percent are left-handed. a. What is the probability that a randomly selected child born in the region is left-handed? b. What is the probability that a randomly selected child born in the region is a child from a multiple birth, given that the child selected is left-handed? VII. Multiple Choice. 1. A state university has 25,000 students: 0.1 percent of these students are majoring in mathematics. How many math majors are there at this university? a. 2.5 b. 25 c. 250 d. 2500 2. In the US, 1 person out of every 250 has worked in the fast food industry, and 6 out of every 15,000 have management experience in fast food. What percentage of workers with fast food experience has also served as managers? a. 16% b. 10% c. 6% d. 1% 3. You hold a bag filled with 7 green, 5 red, and 8 blue marbles. What is the probability of selecting a blue marble at random? a. 2 5 b. c. 1 10 8 13 d. 12 10
4. You hold a bag filled with 7 green, 5 red, and 8 blue marbles. How would you calculate the probability of selecting first a green marble, then a red one, if you do not replace the first marble drawn before selecting the second? a. b. 7 20 5 19 7 13 20 20 c. 20 20 7 5 d. 7 20 5 20 5. Pepsi has long been famous for their blind taste test. What exactly does the word blind mean in this setting? a. That only blind people were allowed to perform the taste test. b. That the person tasting the drinks was blindfolded. c. That the person tasting the drinks didn t know which brand they were tasting. d. That the person giving the drinks didn t know which brand they were giving. 6. The probability of flipping a coin and getting heads is 0.5. What does this mean? a. Every time you flip a coin, you ll get exactly 0.5 heads. b. Over time, the percent of heads becomes close to 50%. The more times the coin is flipped, the closer to 50% it should become. c. For every two coins you toss, you ll get one head. d. You have to toss a coin more than 10 times to see these results, but once you do, you ll get an equal number of heads and tails. 7. Which pair of variables has a strong positive correlation? a. Outside temperatures and cold remedy sales. b. Number of miles you drive and amount of gas left in your gas tank. c. Hours you worked at a fast food restaurant and the amount of your paycheck. d. The price of coffee in China and the number of visitors per day at Disney World.
8. A team is given 1:19 odds that they will win the Super Bowl. What is the probability that this team wins? a. 1/19 b. 1/20 c. 19/1 d. can't be determined 9. Which of the following is NOT a plausible probability? a. 0 b. 0.0001 c. 0.50 d. 1.01 10. When two fair dice are rolled, what is the probability of getting a sum of 7 given that the first die rolled is an odd number? a. 1/6 b. 1/9 c. 1/2 d. 1/12 11. If a fair coin is tossed five times and comes up heads all five times, then the probability of a tail on the sixth toss is a. 1/32 b. 3/32 c. 1/2 d. slightly less than 1/2 12. A spelling quiz is taken by 12 girls and 20 boys. The mean score for the girls is 8.25 and the mean score for the boys is 7.3. Of the following, which is closest to the mean score for all of the students? a. 7.66 b. 7.71 c. 7.76 d. 7.78 13. In which of the following situations would it be most difficult to use a census? a. To determine what proportion of licensed bicycles on a university campus have lights b. To determine what proportion of students in a high school support wearing uniforms c. To determine what proportion of single-family dwellings in a small town have two-car garages d. To determine what proportion of fish in Lake Michigan are bass