Unit 8: Statistics. SOL Review & SOL Test * Test: Unit 8 Statistics

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1 Name: Block: Unit 8: Statistics Day 1 Sequences Day 2 Series Day 3 Permutations & Combinations Day 4 Normal Distribution & Empirical Formula Day 5 Normal Distribution * Day 6 Standard Normal Distribution & Z Scores * SOL Review & SOL Test * Day 7 Day 8 Day 9 Test Review Test Review Test: Unit 8 Statistics *Days 5 and 6 will also be SOL Review days; Test Review and Test will take place after the SOL test

2 Tentative Schedule of Upcoming Classes Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 A Friday 4/22 Notes: Sequences B Monday 4/25 A Tuesday 4/26 Notes: Series B Wednesday 4/27 A Thursday 4/28 Notes: Permutations & Combinations B Friday 4/29 Skills Check #2 A Monday 5/2 Notes: Normal Distribution & B Tuesday 5/3 Empirical Rule A Wednesday 5/4 Notes: Normal Distribution * B Thursday 5/5 A Friday 5/6 Notes: Normal Distribution B Monday 5/9 Z Scores * SOL Review & SOL Test Day 7 Day 8 Day 9 A Thursday 5/19 B Friday 5/20 A Monday 5/23 B Tuesday 5/24 A Wednesday 5/25 B Thursday 5/26 Absent? Test Review: Days 1-3 Test Review: Days 4-6 Unit 8 Test See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need. Notes are always posted online on the calendar. (If links are not cooperative, try changing to list mode) Handouts and homework keys are posted under assignments You may also Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions! Need Help? Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10. Ms. Huelsman is in L402 on Wednesday mornings. Need to make up a test/quiz? Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman s classroom

3 SEQUENCES & SERIES NOTES Day 1: Sequences Objective: In these notes we will recognize and write rules for number patterns, analyze arithmetic & geometric sequences, and compare & contrast sequences. Find the pattern to the following sequence of numbers. 1, 6, 11, 16, 1, 2, 4, 8, 2, 9, 28, 65, 1, 4, 9, 16, 3, 5, 7, 9, -1, -8, -27, -64, -4, -8, -12, -16, 5, 8, 11, 14, 17, 1, 5, 25, 125, What is a sequence? Finite ; infinite

4 A sequence can be specified by an equation, or rule. Using the rule you can find the terms of the sequence. Find the first six terms of the following sequences. an = 2n + 5 an = n A recursive sequence gives the beginning terms or terms of a sequence and then a recursive equation that tells how an is related to one or more preceding terms. (Recursive you will need to find the previous terms in order to get the next one) Find the first six terms of the following sequences. a 1 = 1 an = 3 an - 1 a 1 = 5 an = an 1 + 4

5 There are two SPECIAL types of sequences: Arithmetic and Geometric An Arithmetic Sequence has a common difference, denoted as d. (Add or subtract the same amount). Determine if the following are arithmetic sequences: -4, 1, 6, 11, 16, 8, 5, 2, -1, -4, 3, 5, 9, 15, 23, 17, 14, 11, 8, 5, Formula: an = a1 + (n 1)d where a 1 is the first term d is the common difference n is the n th term of the sequence Write a Rule for the nth term of the sequence. Then find a , -10, -13, -16, 2. a 1 = - 4 and d = -2 Write a Rule given a Term and Common Difference 1. a 27 = 263 d = a 19 = 48 d = 3

6 A Geometric Sequence has a constant ratio, denoted by r. (Multiplying by a constant) Determine if the following are geometric sequences: 625, 125, 25, 5, 1, 1, 1, 2, 6, 24, 120, 5-4, 8, -16, 32, -64, 81, 27, 9, 3, 1, Formula: an = a1 r n-1 where a 1 is the first term r is the common ratio n is the n th term of the sequence Write the Rule for nth term. Then find a 7 3, 12, 48, 192, Rule a 7 = a 1 = 4, r = 3 Write a rule for the nth term of the sequence. Determine whether the given sequence is arithmetic or geometric. Then find a 10. 4, 9, 14, 19, 2, 6, 18, 54, 162,

7 Day 2: Series What is a series? Find the Sum of a Series: 7 5ii ii=2 5 (3kk 2 7) kk=3 8 2 kk kk=4 Find the Sum of a Series using the calculator! 7 8ii ii=4 8 3 kk kk=5 5 (kk 2 + 1) kk=3 22 (4kk 5) kk=5

8 Sum of a Finite Arithmetic Series: n Sn = ( a1 + an) 2 The sum of the first n terms of an arithmetic series is: What are the following sums? Use the formula above to determine the answer. Then verify using your calculator. 1) 7 i= 1 3i a1 = an = n = 2) 9 i= i a1 = an = n = 3) i a1 = an = n = i= 1 Sum of a Finite Geometric Series: S 1 r 1 r n n = a1 The sum of the first n terms of a geometric series is: What are the following sums? Use the formula above to determine the answer. Then verify using your calculator. 4) 4 2 i a1 = n = r = i= 1 5) (3) i a1 = n = r = i= 1 6) 4 5( 3) i a1 = n = r = i= 1

9 Sum of an Infinite Geometric Series: The sum of an infinite geometric series is: a1 S = 1 r if r < 1 7) 8) (.3) i a1 = r = i= 1 i 1 2 i= 1 3 a1 = r = 9) i 1 i= 1 4 a1 = r = 10) Challenge: when we are given a list: a1 = r= In Class Practice: Review of Sequences (Day 1) 1) Write the rule for the nth term for the following sequences. Using your rule, find a10. 5, 14, 23, 32, 41,. 152, 76, 38, 19,. 2) Write a rule for the nth term of the arithmetic sequence that has a16 = 52 and d = 5. (hint: you don t have a1!) 3) Write a rule for the nth term of the geometric sequence that has a1 = - 2 and r = 5.

10 Day 3: Permutations & Combinations Fundamental Counting Principle (Multiplication Principle): If there are m ways to do one thing and n ways to do another, then there are m n ways of doing both in sequence. (Multiply m by n) This rule can be extended to any number of events happening in sequence. Number of Letters in alphabet Number of single digits Example 1: Student IDs are 5 characters long, consisting of 3 letters and 2 digits. How many student IDs are possible? Example 2: In a local restaurant, there are 3 choices for appetizer, 7 choices for entrée, 4 items for dessert, and 5 choices for drinks. In how many ways can a customer order a full meal? Example 3: Calculate the number of possible License Plates using the AAA ### format, which represents 3 letters of the alphabet followed by 3 digits (numbers). Example 4: Calculate the number of possible license plates above if the letters of the alphabet cannot repeat. PERMUTATIONS: (Order matters!) An ORDERING of n objects is a PERMUTATION of those objects. For example, there are 6 permutations of the letters A, B, and C. ABC ACB BAC BCA CAB CBA Fundamental Counting Principle: = But we don t have to use ALL of the objects in our set. We can choose a SUBSET.

11 Permutations: USE WHEN: Order matters! The number of permutations of n objects taken r at a time is represented by: n! n P r =, where 0 r n Alternate Notation: P ( n, r) ( n r)! (Remember: n is the objects in our WHOLE set, r is the number in our ordered SUBSET.) n! is read as n factorial and n!= n(n 1)(n 2) (1) Ex. 4! = (4)(3)(2)(1) = 24 0! = 1 (by definition) Example 1: Eight women are competing in the final round of the Olympic ice-skating competition. In how many different ways can three of the women finish first, second, and third winning the gold, silver, and bronze medals, respectively? n = r = n P r = On Calculator: (number in whole set) (number in subset) Using fundamental Counting Principle: 8 MATH, PRB,2 3 ENTER In how many different ways can all the women finish the competition? Assume no ties. n = r = n P r = (number in whole set) (number in subset) Using fundamental Counting Principle: Example 2: The board of directors for a company has 12 members. One member is the president, another is the vice-president, another is the secretary, and another is the treasurer. How many ways can these positions be assigned? Using idea of permutation: Using Fundamental Counting Principle:

12 Combinations: USE WHEN: Order DOES NOT matter! A combination is an arrangement without regard to order. The notation ( n r) C, represents the number of combinations of n distinct objects taken r at a time. The formula is: n! n C r = C( n, r) = r!( n r)! Combination is not the same as Permutation! In counting combinations of letters (see previous example), ABC is the same as ACB, BAC, BCA, CAB, and CBA. Only one combo all 3 letters in each! Example 1: The board of directors for a company has 12 members. A committee of 3 members will be chosen to examine cost savings. How many possible committees could be formed? 12 n = r = C On Calculator: MATH, PRB,3 3 ENTER Combinations are much easier using the calculator, but we can still use Fundamental Counting Principle we just have to take into account different arrangements How many different ways can 3 people be arranged? Think, ABC is the same group as BCA, etc. Example 2: From a class of 20 students, how many groups of 4 students can the teacher form? Remember, if you are using Fundamental Counting Principle, this is the same group, regardless of order: Billy, JoAnn, Karen, and Jack JoAnn, Jack, Karen, Billy Billy, Karen, Jack, JoAnn Etc How many different ways are there to arrange 4 students? This is what you must divide by if using Fundamental Counting Principle! Example 3: The manager of an accounting department wants to form a three person advisory committee from the 16 employees in the department. In how many ways can the manager form this committee?

13 Day 4: Normal Distributions Empirical Rule The graph of a normal distribution a normal curve. Normal Distributions: Rule Every normal curve has the following characteristics: The mean, median, and mode are equal. They are bell-shaped and symmetrical about the mean. The curve never touches the x-axis, but it comes closer to the x-axis as the curve gets farther from the mean. The total area under the curve is equal to 1, or 100%. Empirical Rule: For a data set with a symmetric distribution, approximately 68% of observations fall within one standard deviation of the mean ( µ ± σ ), 95% of observations fall within two standard deviations of the mean ( µ ± 2 σ ), and 99.7% of observations fall within three standard deviations of the mean ( µ ± 3 σ ). Be sure to recognize symmetry and equal areas involved in the Rule: 68: 95: 99.7:

14 Example: A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a mean of 12.4 ounces and with a standard deviation of 0.2 ounces. The company has asked you to determine the following probabilities to aid in consumer relations concerning the weight of the bags purchased. a. If you purchase a bag filled by this dispenser what is the likelihood it has less than 12 ounces? b. If you purchase a bag filled by this dispenser what is the likelihood it has more than 12 ounces? c. If you purchase a bag filled by this dispenser what is the likelihood it has less than 12.6 ounces? d. If you purchase a bag filled by this dispenser what is the likelihood it has between 12 and 12.6 ounces? e. What weight of the bag is represented by the 84 th percentile? 84 th Percentile means f. A weight in what range would represent the bottom 16% of the weights? Explain the difference between percentage and percentile:

15 Day 5: Standard Normal Distributions What if we want to find a percent that is not covered by the rule? Objective: In these notes we will describe data using statistical measures and analyze normal and standard normal distributions. Recall the example from last class: A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a mean of 12.4 ounces and with a standard deviation of 0.2 ounces. 1. Go to 2 nd VARS (DISTR) 2. Choose # 2:normalcdf( 3. Type in : normalcdf(lower bound, upper bound, µ, σ) 4. Or, if you normalized the data first, type in : normalcdf(lower bound, upper bound) a. If you purchase a bag filled by this dispenser what is the likelihood it has less than 12.3 ounces? min: max: mean: st dev: b. If the factory produces 5,000 bags of potato chips in one hour, how many bags would you expect to weigh less than 12.3 ounces? c. If you purchase a bag filled by this dispenser what is the likelihood it has more than ounces? min: max: mean: st dev: d. If you purchase a bag filled by this dispenser what is the likelihood it has between 12 and 12.5 ounces? min: max: mean: st dev:

16 EXAMPLE 2: Consider the time required for a college student to complete a standardized exam. Suppose that for the population of students at a particular university is normally distributed with a mean of 53 minutes and standard deviation 10 minutes. Use your calculator. a. Find the proportion of students who take between 35 and 55 minutes to complete the test. b. If 60 minutes is allowed for the exam, what proportion of students at this university would be unable to finish in the allotted time? (Think about which minutes represent a student not finishing if they are only given 60 minutes total) c. Given that the college had 870 students take the standardized exam, how many students do you expect to finish the test in 60 minutes? d. How much time should be allowed for the exam if we wanted 80% of the students taking the test to be able to finish in the allotted time? 1. Go to 2 nd VARS (DISTR) 2. Choose # 3:invNorm( 3. Type in : Area: µ: σ: Enter How much time should be allotted for the exam if we wanted 90% of the students taking the test to be able to finish in the allotted time? Explain when we use the calculator command normalcdf vs. invnorm :

17 Day 6: Normal Distributions Standard Normal Curve & Z scores The standard normal curve standardizes scores by changing x-values into z-values. This change the mean to zero, and each standard deviation represents a z-score of 1. Z-Score the z-value for a particular x-value the number of standard deviations the x-value lies above or below the mean µ z = x µ σ where x is an element of the data set, μ is the mean of the data set, and σ is the standard deviation of the data set Example 1: Mean = 60 and SD = 5 What number is 1 standard deviation above the mean? What number is 2 standard deviations below the mean? What is the z-score of the number 75 in this normal distribution? How about 50? What number has a z-score of z = 2.2 in this normal distribution? How about -1.6? What is the z-score of the number 67 in this normal distribution?

18 EXAMPLE 2: Test Scores 1. The class average on a math test was 84.5 and the standard deviation was 4.4 Find the z-score for a test score of A national achievement test is administered annually to 9 th graders. The test has a mean score of 100 and a standard deviation of 15. If John s z-score is 1.20, what was his score on the test? 3. Jill scores 680 on the mathematics part of the SAT. The distribution of SAT scores in a reference population is normally distributed with mean 500 and standard deviation 100. Kelly takes the ACT mathematics test and scores 27. ACT scores are normally distributed with mean 18 and standard deviation 6. Find the standardized scores for both test takers. Assuming that both tests measure the same kind of ability, who has performed better? Explain. 3 characteristics of a data set that is normally distributed: Summary of Normal Distribution normal curve representing the empirical rule percentage vs. percentile normalcdf() vs. invnorm() calculator command

19 Day 7: Unit Review of Days 1 3 Sequences, Series, Permutations, Combinations Part 1: Review of Sequences & Series ARITHMETIC SEQUENCES & SERIES GEOMETRIC SEQUENCES & SERIES aa nn = aa 1 + (nn 1)dd aa nn = aa 1 rr (nn 1) SS nn = nn 2 (aa 1 + aa nn ) SS nn = aa 1(1 rr) nn (1 rr) SS = aa 1 (1 rr) if r <1 If you are looking for the nth term, regardless of being given a sequence or a series, you will use the formula for an! 1. Write a rule for the nth term (that means an) in the series: (Hint: is it arithmetic or geometric?) 2. Write a rule for the nth term (that means an) in the sequence: -12, -9, -6, -3, (Hint: is it arithmetic or geometric?) 3. Find the sum of the first 12 terms of the following: (Hint: is it arithmetic or geometric?) 4. Write a rule for the nth term (that means an) of the sequence: 11, 11, (Hint: is it arithmetic or geometric?) 88, Find the sum of the series: (Hint: is it arithmetic or geometric?) 6. Find the common ratio of the series: Find the sum of the infinite series:

20 Essential Questions: Sequences & Series: 1. Explain the difference between a sequence and series. 2. a. What must be true for an infinite geometric series to have a sum? Why? b. What is the formula for the sum of an infinite geometric series? 3. Create a sequence that is NEITHER geometric nor arithmetic. List the first 7 terms. QUICK QUESTIONS: SEQUENCES AND SERIES QUESTION ANSWER A ANSWER B 1 Arithmetic or Geometric? 28, 24, 20, 16, Arithmetic Geometric 2 Arithmetic or Geometric? 2, 4, 8, 16, 32, Arithmetic Geometric 3 Arithmetic or Geometric? 6 1 2(3) i Arithmetic Geometric i = 1 4 Arithmetic or Geometric? i Arithmetic Geometric 5 Arithmetic or Geometric? i = 3 10 i Arithmetic Geometric i = 1 6 What is r? 125, 25, 5, 1, 5 7 What formula would you use? 1, 1 2, 1 4, 1 8, an = a1(r)n-1 a n = a 1 1 r What formula would you use? -10, -8, -6, - 8 an = a1 + d(n-1) a 4,.. n = n 2 a 1 + a n What formula would you use? 9 (3)i 1 a n = n 2 a 1 + a 1 r ( n) a n = a n 1 1 r 10 What formula would you use? i = i a n = n 2 a 1 + a n 11 What formula would you use? a n = a 1 1 r i = 1 i = 1 i ( ) ( ) an = a1 + d(n-1) Does not Exist 12 What is the common difference: 7, 5, 3, What is the common ratio: 4, 6, 9, 13.5,

21 Part 2: Review of Combinations, Permutations & Fundamental Counting Principle I. Essential Questions 1. Explain the difference between a combination and permutation. Include which calculator command you would use for each. 2. Choose any sports team to use as a data set, a.) write a question that would be answered with a COMBINATION b.) write a question that would be answered with a PERMUTATION II. Practice with Counting Show any work needed to solve each problem. Include any calculator commands used. 1. You want to create an ID code for all your customers based on three characters. The first character must be a letter of the alphabet, and the second and third must each be a digit between 1 and 9, inclusive (which means 1 and 9 are included). How many such codes are there? 2. A license plate is to consist of two letters followed by three digits. Determine how many different license plates are possible if repetition of characters is allowed. 3. Ryan is building his Pinewood Derby Race car. To make the car he needs to pick one body style, one color of paint, and one type of wheels. If there are 5 body styles to choose from, 10 colors of paint to choose from, and 3 types of wheels to choose from, how many possible different cars could Ryan come up with?

22 Part 3: Sequences and Series Application Question A movie theater has 80 rows. If the 8 th row has 32 seats, and the 20 th row has 68 seats, how many seats are in the 80 th row? This is a tricky question! We don t have a1 and we don t have d! So, let s find d: From row 8 to row 20, how many rows did you gain? (20 8) From row 8, with 32 seats, to row 20, with 68 seats, how many seats were added? (68 32) Using this information you can find d, the common difference for every row, how many seats are added? d = Now that we ve found d, we can find a1 by plugging what we know into the formula aa nn = aa 1 + (nn 1)dd You can use either a8 = 32 or a20 = 68. You will plug in the number of seats for an and the row number for n. Solve for a1: a1 = Finally, we can write a general rule to find the number of seats in any row: aa nn = aa 1 + (nn 1)dd As always, you should plug in for a1 and d (both values you just found!) Now you can find how many seats are in the 80 th row! (What do you need to plug in for n?) Find the total number of seats in the theater (S80) Hint: is this an arithmetic or geometric series?

23 Day 8: Unit Review of Days 4-6 Statistics (Standard Normal Distribution, Empirical Formula) The length of wear on Spinning Tires is normally distributed with a mean of 60,000 miles and a standard deviation of 5,000 miles. 1. Shade the region under the curve that represents the fraction of tires that last between 50,000 miles and 70,000 miles. Use the empirical rule to determine what fraction (percentage) of tires that represents. 2. If there are 100,000 tires sampled, how many would last between 50,000 and 70,000 miles? 3. In one instance, a tire lasted 62,000 miles. At what percentile is this tire s length of wear? 4. What length of wear would be the 84 th percentile? Hint: use empirical rule 5. What length of wear would be the 90 th percentile? Hint: you can t use empirical rule

24 The length of time it takes to groom a dog at Shaggy s Pet Shoppe is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. 1. Shade the region under the curve that represents the percent of dog grooming times greater than 65 minutes. What is that percent? 3. In one instance, a dog took an hour to groom. At what percentile is this length of time? 4. What length of time would represent the 84 th percentile? Hint: use empirical rule 5. What length of time would be the 90 th percentile? Hint: you can t use empirical rule 6. A university gives an admission qualifying exam. The results are normally distributed with a mean of 500 and a standard deviation of 100. The admissions department would like to accept only students who score in the 80 th percentile or better. Determine what score is associated with the 80 th percentile and which students would qualify for admission?