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PROBABILITY AND STATISTICS SUMMER PACKET STUDENT NAME :…………………………. Due Date: Sept 9,2013 This packet is your FIRST GRADE FOR MARKING PERIOD 1 Student signature:…………………………………… Current Grade : …………….. Date Received :…………….. Parent signature: ……………. Date Received : ……………. Prob Stat summer packet 2013 Page 1 Topic: Summation Notation 5 Example 1: å3k 4 Example 2: k=1 åk 2 k=1 Solution Solution 3(1) + 3(2) + 3(3) +3(4) + 3(5)= 12 + 22 + 32 + 42 = 3 + 6 + 9 + 12 + 15 = 45 1 + 4 + 9 + 16 = 30 Practice Problems Directions: Evaluate each expression. 11 A. ( 2k 7) k 1 9 B. (8k 2 3k ) Answer: ______________ 2 2k 8) Answer: ______________ k 1 10 C. ( 4k Answer: ______________ k 1 Prob Stat summer packet 2013 Page 2 100 D. åk Answer: ______________ k=1 Topic: Slope of a Line (given two points) Slope Formula: m = y2 - y1 x2 - x1 Example 1: Find the slope of the line that passes through (1,2) and (3,4)= Example 2: Find the slope of the line that passes through (1,6) and (3,8)= Solution Solution Practice Problems Directions: For the problems below, find the slope of the line between each of the two given points. Write your answer in simplest form. A. (1, 5) and (7, 8) Answer: ______________ B. (-5, 9) and (5, 11) Answer: ______________ C. (12, 13) and (7, 13) Answer: ______________ Prob Stat summer packet 2013 Page 3 D. (-8, 2) and (-8, 7) Answer: ______________ Topic: Permutations n! Permutation Formula: Notation - nPr = ( n - r )! Example 1: Evaluate 3 P 7 Solution 7! = (7 – 3)! 7! = 7x6x5x4x3x2x1 4! 4x3x2x1 7 x 6 x 5 = 210 Example 2: How many different ways can first, second, and third place be awarded to 10 people? Solution n = 10 people r = 3 (1st , 2nd , 3rd place) 10! = 10! (10 – 3)! 7! = 10x9x8x7x6.. 7x6… 10x9x8 = 720 Directions: Evaluate the following expressions A. 12 P 3 Answer: _______________ B. 8 P 5 Answer: _______________ C. In how many ways can a president, a treasurer and a secretary be chosen from among 7 candidates? Prob Stat summer packet 2013 Page 4 Answer: _______________ D. How many ways can 1st, 2nd, 3rd, and 4th place be awarded to 10 runners? Answer: _______________ Topic: Combinations n! Combination Formula: Notation – nCr = ( n - r )!r! Example 1: Evaluate 3 C 7 Solution 7! = (7 – 3)!3! 4x3x2x1 7! = 4!3! 7x6x5x4! 4!x3x2x1 7 x 6 x 5 = 210 = 35 3x2x1 6 Example 2: Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose three people to go? Solution n=5 r=3 5! = 5! = 5 x 4 x 3! (5 – 3)!3! 2! 3! 2 x 1 x 3! 5 x 4 = 20 2x1 2 Directions: Evaluate the following expressions A. 12 C 3 Answer: _______________ B. 8 C 5 Answer: _______________ C. In how many ways can a president, a treasurer and a secretary be chosen from among 7 candidates? Prob Stat summer packet 2013 Page 5 Answer: _______________ D. Problem: From a group of 40 people, a jury of 12 people is selected. In how many different ways can a jury of 12 people be selected? Answer: _______________ Topic: Venn Diagrams Example 1: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? Example 2: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how are in either class? Solution Solution Two students are taking neither class. There are 38 students in at least one of the classes. Directions: Create a Venn diagram to display the information then solve. A. In a class of 87 students, 40 take Chorus, 53 take Band, and 16 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band? Prob Stat summer packet 2013 Page 6 B. In a school of 800 students, 185 students are taking Spanish, 160 students are on sports teams, and 72 students participate in both activities. How many students don’t participate in either activity? C. A veterinarian surveys 46 of his patrons. He discovers that 17 have hamsters, 21 have guinea pigs, and 19 have birds. Eight have hamsters and guinea pigs, and 5 people have a guinea pig and a bird. Seven have hamsters and a bird, and of these, 3 people have a guinea pig. How many patrons have none of these pets? Topic: Writing equations of lines (using slope & y-intercept) Slope Intercept Form: y = mx + b Example 1: Find the slope of the line Example 2: Find the slope of the line that passes through (1,2) and has a that passes through (1,6) and (3,8) = slope of 5. Solution Solution 8-6 2 Find the slope first: = = 2 3 -1 1 Plug in the slope (m) and point (x,y), then solve for b: Use the m = 2 and any point and plug 2 = 5(1) + b it in to the slope-int formula to solve 2=5+b for b: -3 = b 6 = 1(2) + b Now take m = 5 & b = -3 and write your equation in slope-int form: y = 5x – 3 6=2+b 4=b Now take m = 2 & b = 4 and write your equation in slope-int form: y = 2x + 4 Directions: Write the equation of the line using the given information. A. Passes through (2,4); slope of 3 Prob Stat summer packet 2013 Answer: _______________ Page 7 1 5 Answer: _______________ C. Passes through (0, 0); m = -4 Answer: _______________ D. Passes through (5, 1) & (3, 0) Answer: _______________ B. Passes through (-5, 8); m = E. Passes though (-2, 3) & (-2, -1) Topic: Standard Deviation Standard Deviation Formula: s= Answer: _______________ å( x - m) 2 N Example: What is the population standard deviation for the numbers: 75, 83, 96, 100, 121 and 125? Solution Firstly find the mean: Mean = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100 Next find the variance. To calculate the Variance, take each difference, square it, and then average the result: (75 - 100)2 + (83 - 100)2 + (96 - 100)2 + (100 - 100)2 + (121 - 100)2 + (125 - 100)2 = (-25)2 + (-17)2 + (-4)2 + (0)2 + (21)2 + (25)2 = 625 + 289 + 16 + 0 + 441 + 625 = 1,996 So the Variance = 1,996 ÷ 6 = 332.66... The Standard Deviation is just the square root of the Variance = √(332.66...) Prob Stat summer packet 2013 Page 8 = 18.2 correct to 1 decimal places Directions: Calculate the standard deviation for each problem A. Ten friends scored the following marks in their end-of-year math exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82% Answer: ______________ B. A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314 Answer: ______________ Topic: Stem and Leaf Plots Prob Stat summer packet 2013 Page 9 Example: "32" is split into "3" (stem) and "2" (leaf). Directions: Find the mean, median, mode, and range for each. A. Mean: _____________ B. Mean: _____________ Median: _____________ Median: ____________ Mode: _______________ Mode: _______________ Range: _______________ Range: ______________ Prob Stat summer packet 2013 Page 10 Topics: Frequency Tables A Frequency Histogram is a special histogram that uses vertical columns to show frequencies (how many times each score occurs): Here I have added up how often 1 occurs (2 times), how often 2 occurs (5 times), etc, and shown them as a histogram. Directions: Find the mean, median, mode, range, and the quartiles. Prob Stat summer packet 2013 Page 11 Mean = __________ Median = _________ Mode = ___________ Range = ___________ Q1 = ______________ Q2 =_____________ Q3 = _____________ Topics: Box-and-Whisker Plots (Quartiles) Example: 5, 8, 4, 4, 6, 3, 8 Put them in order: 3, 4, 4, 5, 6, 8, 8 Cut the list into quarters: And the result is: • Quartile 1 (Q1) = 4 • Quartile 2 (Q2), which is also the Median, = 5 Quartile 3 (Q3) = 8 Directions: Find the quartiles (Q1, Q2, Q3) for each example: A. 8, 11, 20, 10, 2, 17, 15, 5, 16, 15, 25, 6 Prob Stat summer packet 2013 Page 12 B. 13, 18, 6, 20, 25, 11, 9, 18, 3, 30, 16, 9, 8, 23, 26, 17 C. 4, 5, 6, 8, 9, 11, 13, 16, 16, 18, 20, 21, 25, 30, 31, 33, 36, 37, 40, 41 D. Probability & Statistics Summer Packet Student Name: _________________________ Summer Resource Contact: Due Date: 9/10/2013 [email protected] [email protected] This packet is the FIRST TEST GRADE of Marking Period 1 Student Responsibilities I will be able to improve my math performance by: Completing the summer math packet Prob Stat summer packet 2013 Page 13 Reviewing math skills throughout the summer ____________________________ _________________ Student Signature Current Grade _______________ Date Received Parent Responsibilities Parents will be able to ensure student success in math by Supporting the math goal of ACHS Monitoring student completion of the summer packet ____________________________ _________________ Parent Signature Date Received Prob Stat summer packet 2013 _________________ Date Reviewed Page 14