ࡱ> z|y^[ Ogbjbj 4ΐΐS` PPOOOcccc7c=KRF'F'F'=======@B=O)($F')()(= =777)( O=7)(=77:b<,6<.Kc5)< ~==0=<RIC.IC<ICO<F'"h'7''F'F'F'==2lF'F'F'=)()()()(ICF'F'F'F'F'F'F'F'F'P Y: Name:________________________________ Due: April 12, 2010 Algebra II - Pd ___ Easter Break Assignment Statistics Statistics is the study and interpretation of numerical data. If only one variable is being analyzed, it is called univariate statistics. There are a number of ways that data can be gathered for statistical studies. Data Gathering Methods: Surveys: used to collect information from a group of people. A survey is an efficient way of obtaining a wide range of information from a large number of people. Validity of surveys depends on the honesty of its participants. Census: type of survey in which an attempt is made to reach every member of an entire population. For example, the United States has a census every 10 years to count how many people are in the country. Sample: a subset of the entire population. Many times it is not cost-effective or productive to gather information from an entire population. When this is the case, a sample is used. It is always important to make sure that the survey sample is representative of the entire population. Simulation: Scientists and engineers use this to determine information about an event. For example, to determine the amount of weight a bridge can hold, engineers first simulate the effects statistically on a computer. Then, they use that data to draw conclusions about how the bridge should be constructed to bear the weight that it will be holding. Controlled Experiment: a study that is used to gather information about the effect of some kind of intervention. For example, medicine or an exercise program. The results obtained in from an experimental sample are compared to the results from a control sample. A control group is identical to the experimental group, except without one aspect whose effect is being tested. Observational Study: individuals are observed or certain outcomes are measured. Data is gathered without any intervention. There is no attempt made to affect the outcome. Bias and Random Sample: In statistics, it is important to determine if the information that is presented is somewhat biased, data used in the sample have come from sources that have a particular interest in the impact of the statistics. When this is the case, the sample is not representative of the general population, and it is said to be biased or skewed. In order that the sample reflects the properties of the entire group, the following three conditions must exist: The sample must be representative of the group being studied. The sample must be large enough to be effective. The selection should be random or determined in such a way as to eliminate any bias.  Problems: 1. Which would be the most unbiased group of people to ask about methods of Social Security reform? (1) adults at a senior citizen center (2) college students in a sociology class (3) members of a children's choir (4) shoppers at a mall 2. Reporters on a news show want to survey adults about their exercise habits. Where should they go to find an unbiased sample? (1) the boardwalk on the beach (2) the post office (3) an exercise gym (4) a rock-climbing expo 3. The city board of directors is considering raising the cost of parking tickets. Which group of citizens would be least biased on this issue? (1) fifty citizens who have never received a parking ticket (2) fifty citizens who have had at least three parking tickets (3) the first fifty people encountered on a city street (4) fifty citizens who do not have driver's licenses. 4. A survey completed at a large university asked 2,000 students to estimate the average number of hours they spend studying each week. Every tenth student entering the library was surveyed. The data showed that the number of hours that the students spending studying was 15.7 per week. Which characteristic of the study could create a biased result? (1) the size of the sample (2) the size of the population (3) the method analyzing the data (4) the method choosing the students who were surveyed Examples 5 - 10: Suggest a method (population survey, sample survey, simulation, observation, controlled experiment) that might be used to collect data for the study. 5. Number of texts sent a week by students in St. Francis Prep. 6. Amount of fertilizer to be used to produce a prize-winning watermelon. 7. Number of hours of sleep obtained by a whale living in the ocean. 8. Number of hours spent watching television by members of your COR. 9. Effect of an advertising campaign on "name recognition" of a new product. 10. Amount of time for the red, green, and yellow lights to reaming active in a busy intersection. Measures of Central Tendency: Measures of central tendency are summary statistics that indicate where the center of a collection of data lies. Three common measures of central tendency are mean, median, and mode. Mean: The mean refers to the arithmetic average of the data and its symbolized as  EMBED Equation.DSMT4 , read as "x bar." The mean is the most common measure of central tendency. The mean is the sum of all of the data values divided by the number of data values. Median: The median is the middle number of a data set arranged in numerical order. Therefore, whenever we are looking for the median, we must write our data in numerical order. To find out which position is the median, add 1 to the total number of data values and divide by 2. The value in that position, counting from the first or last term, is the median. If there is no middle term, find the average of the middle two terms to arrive at your median. Mode: The mode is the most commonly repeated data value in a set. Some collections of data, often referred to as distributions, will have no mode and others will have multiple modes. Standard Deviation The standard deviation shows how much variation there is from the "average" (mean). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values. Finding Mean, Median and Standard Deviation using the calculator. Press STAT 1.Edit Enter the data into List 1. If there is a frequency list or any second list, enter that data into List 2. Press STAT , move the arrow to CALC, and press 1. 1-Var Stats , then Enter If there is a frequency list, Press STAT CALC 1. 1-Var Stats 2nd 1, 2nd 2 Enter A list of statistical information will be displayed:  EMBED Equation  mean of the data  EMBED Equation  sum of the values  EMBED Equation  sum of the squares of the values  EMBED Equation  sample standard deviation  EMBED Equation  EMBED Equation  population standard deviation (ALWAYS USE POPULATION FOR STANDARD  EMBED Equation  number of items in sample DEVIATION UNLESS THE PROBLEMS SAYS  EMBED Equation  lowest value THE WORD SAMPLE)  EMBED Equation  first quartile  EMBED Equation  median  EMBED Equation  third quartile  EMBED Equation  maximum value (Many of these items, you will not use. Just copy the numbers needed to answer the question. Also, mode is not given you still have to find that by hand)    Problems: 11. What is mean of 37, 54, 72, 89, 74, 83, 90, and 93? 12. The table below lists the scores for a multiple-choice test is Ms. A's class. Find the mean, median, and mode for the data to the nearest hundredth. Test Grade10095908580757060Frequency12276532 13. What is the mean of the data in the accompanying table? Scores25201110Frequency3254 (1) 11 (2) 14.5 (3) 15 (4) 16 14. The term snowstorms of note applies to all snowfalls over 6 inches. The snowfall amounts for snowstorms of note in Utica, New York, over a four-year period are as follows: 7.1, 9.2, 8.0, 6.1, 14.4, 8.5, 6.1, 6.8, 7.7, 21.5, 6.7, 9.0, 8.4, 7.0, 11.5, 14.1, 9.5, 8.6 What are the mean and population standard deviation for these data, to the nearest hundredth? (1) mean = 9.46; standard deviation = 3.74 (3) mean = 9.45; standard deviation = 3.74 (2) mean = 9.46; standard deviation = 3.85 (4) mean = 9.45; standard deviation = 3.85 15. Andrew wanted a raise in his allowance for doing yard work. His father said he should find out the "average" payment others in the neighborhood received. Andrew surveyed the other families on the block and discovered that they paid the following prices for having their lawns cut and raked: $18.00, $22.00, $17.50, $15.50, $25.00, $17.50, $20.00, $15.00, $26.00, $28.00. a. Which "average" should Andrew use to promote his increase in his allowance? b. Which "average" might his father use to rebut Andrew's argument? 16. For the data 14, 18, 21, 19, 27, 23, 17, which statement is true? (1) mean = median (2) mean < median (3) mean > median (4) median = mode 17. From 1984 to 1995, the winning scores for a golf tournament were 276, 279, 279, 277, 278, 278, 280, 282, 285, 272, 279, and 278. Find the Sample Standard Deviation for this data. 18. One hundred senior girls were interviewed about their price limit for the "perfect" prom dress. Their responses were summarized in the table below. Determine the mean, median, and mode for the set of data. Maximum Price (prom dress)$100$150$200$250$300# of Girls2322261415 19. The scores on a mathematics test are 42, 51, 58, 64, 70, 76, 76, 82, 84, 88, 88, 90, 94, 94, 94, and 97. For this set of data, find the standard deviation to the nearest tenth. 20. Using the scores in the accompanying table, find the standard deviation to the nearest hundredth. Find the mean to the nearest tenth. Scores6065707580Frequency26485 xifi87389491393695221. Find, to the nearest tenth, the standard deviation of this set of data. Find the mean. BINOMIAL EXPANSION How to find a one term in a binomial expansion such as find the 5th term of (3x 3)12 Lets call the term we are looking for r (or the rth term)  To find the rth term of (a + b)n we use: n C r an - (r - 1) br -1 for (ax + b)n  Ex: Find the 5th term of (3x 4)12 n = 12, r = 5, a = 3x and b = (-4) n C r-1 an - (r - 1) br -1 write the formula 12 C 5-1 (3x)12 - (5 - 1) (-4)5 -1 substitute n, r, x and b 12 C 4 (3x)12 - (4) (-4)4 simplify 12 C 4 (3x)8(-4)4 simplify Now you use your calculator to solve to get 12 C 4 : type 12, then  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Math.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\RightArrow.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\RightArrow.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\RightArrow.png" \* MERGEFORMAT  to PRB then hit  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\3.png" \* MERGEFORMAT  for nCr, then type 4 your screen will say 12 nCr 4 hit  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Enter.png" \* MERGEFORMAT and it equals 495. Then you need to find 3 to the 8th power = 6561 (get by typing  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\3.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Caret.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\8.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Enter.png" \* MERGEFORMAT ) You then find (-4) to the fourth power = 256 (get by typing  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\LeftParen.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Negative.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\4.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\RightParen.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Caret.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\4.png" \* MERGEFORMAT  INCLUDEPICTURE "E:\\..\\..\\..\\AppData\\Local\\Temp\\Enter.png" \* MERGEFORMAT ) Now substitute those into the formula. (495)(6561x8)(256) Multiply all of the numbers and variable to get your answer 831,409,920x8  Ex 2 :Find the 3rd term of (x + 3y)9 n = 9, r = 3, a = x and b = 3y nCr-1 an-(r-1) br-1 write the formula 9C3-1 (x)9-(3-1) (3y)3-1 substitute n, r, x and b (9C2)(x)9-2(3y)2 simplify (36)(x)7(3y)2 use calculator to find 9C2 (36)(x7)(9y2) = 324x7y2 multiply coefficients for final answer  Find the rth root of each of the following binomials. 22) Find the 3rd term of (x + y)6 23) Find the 6th term of (x 5)8 24) Find the 4th term of (2x + 7y)8 25) Find the 4th term of (1 x)7 26) Find the 10th term of (x + 3)12 27) Find the 3rd term of (3x 2)10 PROBABILITY: BERNOULLI EXPERIMENTS nCr sr f n-r n = number of trials r = exactly # of successes used for "exactly" s = probability of a success r successes f = probability of a failure n ( r = number of failures  Ex 1: A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. What is the probability, the thousandth, of getting exactly 6 questions correct on this test? Solution: n = 10 nCr sr f n-r r = 6 n r = 4   0.005505024   The probability of getting exactly 6 questions correct is about 0.006  Ex 2: When rolling a die 100 times, what is the probability of rolling a 4 exactly 25 times? Solution: n = 100 nCr sr f n-r r = 25 n r = 75   QUOTE   0.0098258819  The probability of rolling 4 exactly 25 times is approximately 0.010  EXERCISES: 28. If the probability that it will rain on any given day this week is 60%, find the probability that it will rain exactly 3 out of 7 days this week. 29. The Coolidge familys favorite television channels are 3, 6, 7, 10, 11, and 13. If the Coolidge family selects a favorite channel at random to view each night, what is the probability that they choose exactly three even numbered channels in five nights? Express your answer as a fraction or as a decimal rounded to four decimal places. 30. Ginger and Mary Anne are planning a vacation trip to the island of Capri, where the probability of rain on any day is 0.3. What is the probability that during their five days on the island, they have no rain on exactly three of the five days? 31. During a recent survey, students at Franconia College were asked if they drink coffee in the morning. The results showed that two-thirds of the students drink coffee in the morning and the remainder do not. What is the probability that of six students selected at random, exactly two of them drink coffee in the morning? Express your answer as a fraction or as a decimal rounded to four decimal places. 32. Which fraction represents the probability of obtaining exactly eight heads in ten tosses of a fair coin? (1) __45__ (2) __64__ (3) __90__ (4) __180__ 1024 1024 1024 1024 33. The probability that Kyla will score above a 90 on a mathematics test is 4/5. What is the probability that she will score above a 90 on three of the four tests this quarter?  QUOTE    QUOTE    QUOTE    QUOTE   34. The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 0.7. Which expression can be used to find the probability that it will rain on exactly three of the seven days they are there? (1)  EMBED Equation  (2)  EMBED Equation  (3)  EMBED Equation  (4)  EMBED Equation  AT LEAST/AT MOST Some questions will ask for at least or at most a certain amount of favorable outcomes: There are five questions on a test, the probability of getting a question correct is 0.25. a) What is the probability of getting at least 3 questions correct? b) What is the probability of getting at most 3 questions correct?  a) At least 3 correct means you can get exactly 3 correct, exactly 4 correct, or exactly 5 correct. Use Bernoullis Formula to find the Probability of exactly 3, then exactly 4 and exactly 5 and we add the 3 probabilities together P (at least 3 correct) = P(exactly 3) + P(exactly 4) + P(exactly 5)  QUOTE   +  QUOTE   +  QUOTE   = 0.087890625 + 0.0146484375 + 0.00097656525 = 0.10351515625  b) At most 3 correct means you can get exactly 0, exactly 1, exactly 2, or exactly 3 correct. P(at most 3 correct) = P(exactly 0) + P(exactly 1) + P(exactly 2) + P(exactly 3)  QUOTE   +  QUOTE   +  QUOTE   +  QUOTE   = 0.237346875 + 0.3955078125 + 0.263671875 + 0.087890625  = 0.9844171875 EXERCISES: 35. Tim Parker, a star baseball player, hits one home run for every ten times he is at bat. If Parker goes to bat five times during tonights game, what is the probability that he will hit at least four home runs? 36. The probability that a planted watermelon seed will sprout is  EMBED Equation . If Peyton plants seven seeds from a slice of watermelon, find, to the nearest ten-thousandth, the probability that at least five will sprout. 37. On mornings when school is in session in January, Sara notices that her school bus is late one-third of the time. What is the probability that during a 5-day school week in January her bus will be late at least three times? 38. Dr. Glendon, the school physician in charge of giving sports physicals, has compiled his information and has determined that the probability a student will be on a team is 0.39. Yesterday, Dr. Glendon examined five students chosen at random. a) Find, to the nearest hundredth, the probability that at least four of the five students will be on a team. b) Find, to the nearest hundredth, the probability that exactly one of the five students will not be on a team. 39. On any given day, the probability that the entire Watson family eats dinner together is 2/5. Find the probability that, during any 7-day period, the Watsons eat dinner together exactly 3 times. Name:________________________________ Due: Monday April 12, 2010 Algebra II Pd ___ Statistics Easter Break Assignment Complete the Statistics packet that you were given in class. Do all of the work on looseleaf, then write all of your answers on this sheet and attach it to the looseleaf. Missing this sheet will result in a 5 point penalty. If the work is not attached you will get a zero on the assignment. Work must be stapled with a staple or you will lose 2 points. DO NOT ASK ME OR ANY OTHER TEACHER TO BORROW THEIR STAPLER! This must be handed into E012 before 8:30 am on Monday, April 12, 2010. ABSOLUTELY NO LATE WORK WILL BE ACCEPTED. It cannot be handed in after 8:30, during class, during a free, on Tuesday or any other time. Do not interrupt my class to attempt to hand this in because it will not be accepted. If you are going to be absent on Monday, April 12, 2010 make sure to send your work in with someone else because it will not be accepted late. 1) __________________________________ 2) __________________________________ 3) __________________________________ 4) __________________________________ 5) __________________________________ 6) __________________________________ 7) __________________________________ 8) __________________________________ 9) __________________________________ 10) __________________________________ 11) __________________________________ 12) __________________________________ 13) __________________________________ 14) __________________________________ 15) __________________________________ 16) __________________________________ 17) __________________________________ 18) __________________________________ 19) __________________________________ 20) __________________________________ 21) __________________________________ 22) __________________________________ 23) __________________________________ 24) __________________________________ 25) __________________________________ 26) __________________________________ 27) __________________________________ 28) __________________________________ 29) __________________________________ 30) __________________________________ 31) __________________________________ 32) __________________________________ 33) ___________________________________ 33) __________________________________ 34) __________________________________ 35) __________________________________ 36) __________________________________ 37) __________________________________ 38) __________________________________ 39) __________________________________ Model Problem 1: Car manufactures want to explore the desirability of pre-installed infant seats in their new car designs. Which of the following groups would be most likely to provide an unbiased sample for a survey? (1) adult shoppers at a supermarket with young children in shopping carts (2) couples signing up to win a honeymoon package at a bridal expo (3) adults visiting a car show (4) a group of teenagers playing ball at the park Answer: (3) is the least biased because (1) contains people who already need car seats; (2) involves couples who might be planning on having kids; (4) involves teenagers for whom parenthood is not relevant. Model Problem 2: A new medicine intended for use by adults is being tested on five men whose ages are 22, 24, 25, 27, and 30. Does the sample provide a valid test? Answer: No. The sample is too small, the sample includes only men, and the sample does not include adults over the age of 30. Model Problem 3: Nine members of the basketball team played during all or part of the last game. The number of points scored by each of the players was: 21, 15, 12, 9, 8, 7, 5, 2, 2. What is the mean? Answer: mean = 9 Model Problem 4: An English teacher recorded the number of spelling errors in the 40 essays written by students. The table below shows the number of spelling errors and the frequency of the number of errors, that is, the number of essays that contained that number of misspellings. Find the mean number of spelling errors for these essays. Errors012345678910Frequency13226975212 Answer: mean = 5.1 Model Problem 5: Find the standard deviation, to the nearest tenth, of the following set of data: 11, 17, 31, 18, 25, 12, 19, 13, 15. Answer:  EMBED Equation x = 6.1 Model Problem 6: Geoff stocked up on canned goods at the grocery store. The frequency table to the right shows the price of the canned goods. Find the median price of the canned goods. Price (in cents)5673788286Frequency49755Answer: The median is 78, but your answer is $0.78, since the data is given in cents. Model Problem 7: Find the mode of the following data: 67, 54, 91, 67, 83, 46, 72, 54, 91, 81, 75, 67, 54, 88. Answer: Since 67 and 54 both appear three times, both of these serve as modes. This data set is said to be bimodal. Model Problem 8: In May 2008, the government predicted that gasoline prices would peak at a national average of $3.73 a gallon in June. However, this prediction turned out to be incorrect. The average weekly prices of gasoline in Rochester from May 7 to July 16, 2008 were: $3.85, $3.92, $4.09, $4.12, $4.18, $4.20, $4.20, $4.20, $4.23, $4.24. Find the mean, median, and mode for these data. 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