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1 "#\$%"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" Common Core Math 3 Unit 1 - Statistics A P E X HIGH SCHOOL 1501 LAURA DUNCAN R OAD A P E X, NC 27502

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8 Normal Distributions Warm Up Questions 1. At the doctor s office, there are three children in the waiting room. The children are ages 3, 4, and 5. Another 4 year old child enters the room. a) What will happen to the mean of the children s ages when the new child enters? b) What will happen to the standard deviation of the children s ages? 2. Which of the following data sets will have the largest standard deviation? Which will have the smallest standard deviation? A. 3, 6, 9, 12, 15 B. 5, 7, 9, 11, 13 C. 1, 5, 9, 13, Is it possible for a data set to have a standard deviation of 0? If so, give an example. If not, tell why not. 4. For the following data set, find the mean, median, and standard deviation: 4, 6, 6, 8, 10, 12, 16, Two normal distributions have the same mean. Distribution A has a standard deviation that is half of the standard deviation of distribution B. Sketch both graphs on the same axis. Compare and contrast the graph of distribution A with that of distribution B. 6. Give an example of a data set that is not normally distributed. Draw a curve that represents your data set. 7. Hillary scores 78 on her Chemistry test and her friend Alissa scores 83 on her Biology test. Alissa believes that she did better than Hillary, but Hillary says that the Chemistry test was harder. The mean score on the Chemistry test was 74 with a standard deviation of 2. The mean score on the Biology test was 79 with a standard deviation of 4. a. Which girl did better on her respective test? b. If Sean s z-score on the Chemistry test was -1.5, find his test score.

9 8. Heights of kindergarten children are normally distributed. With a mean of 40 in and a standard deviation of 6 in. a. What percent of kindergarten children are between 40 in and 46 in tall? b. What percent of kindergarten children are less than 40 in tall? c. What percent of kindergarten children are between 34 in and 46 in tall? d. If 1000 kindergarteners are selected at random, how many will be less than 46 in tall? e. If 1000 kindergarteners are selected at random, how many will be greater than 52 in tall? 9. The results of an AFM test are normally distributed with a mean of 74 and a standard deviation of 8. Find the percent of individuals that scored as indicated below. i. Greater than 78 ii. Between 60 and 72 iii. Less than 78 iv. Between 78 and 86 v. Greater than A weight loss clinic guarantees that its new customers will lose at least 5 lb by the end of their first month of participation or their money will be refunded. If the loss of weight of customers at the end of their first month is normally distributed with a mean of 6.7 lb and a standard deviation of 0.81 lb, find the percent of customers that will be able to claim a refund. Sampling and Experimental Design 11. Let s gather data about how our school population feels about introducing smart lunch on campus. Think of three ways we could gather information to make an informed decision on whether students feel that this would be beneficial or not. 12. A local newspaper ran a survey by asking, Do you support the development of a weapon that could kill millions of innocent people? Is the survey question biased and why? 13. A journalist goes to a campground to ask people how they feel about air pollution. Determine whether this method of collecting data is biased and why.

10 14. A report by the California Citrus Commission stated that cholesterol levels can be lowered by drinking at least one glass of a citrus product each day. Is this report biased and why? 15. Decide which method of data collection you would use to collect data for each study (observational study, experiment, simulation, or survey): a) A study of the salaries of college professors in a particular state b) A study where a political pollster wishes to know if his candidate is leading in the polls. c) A study where you would like to determine the chance of getting three girls in a family of three children. d) A study of the effects of a fertilizer on a soybean crop e) A study of the effect of koalas on Florida s ecosystem 16. What kind of bias is Calvin introducing in his study? Explain. 17. Explain what bias there is in a study done entirely online. Populations & Samples 18. Find the margin of error for a survey of 800 U.S. women. 19. If you wanted your margin of error to be 0.5 %, how large a sample would you need?

11 20. A telephone survey contacts a random sample of 1000 Los Angeles telephone numbers, of which 58% are unlisted. a. What is the population under investigation? b. What is the parameter of interest? c. What is the sample? d. What is the sample statistic? Expected Value and Fair Games 21. Card Games: Each of the following games is played by shuffling the deck thoroughly, then selecting one card from the deck. Which game seems like the best one to play? The worst? Why? Game #1: You must pay \$1 to play. If you draw a face card you win \$5, otherwise you lose. Game #2: You must pay \$2 to play. If you draw a red card you win \$3, if you draw a black jack you win \$25, otherwise you lose. Game #3: You do not pay to play. If you draw a red number card you pay \$3. If you draw a black number card, you win \$2. If you draw a Jack or King you win \$5. If you draw a Queen you win \$15. If you draw an Ace you pay \$ Dice Games: Each of the following games is played by rolling two fair dice. Which game seems like the best one to play? The worst? Why? Game #1: You must pay \$0.50 to play. If you roll doubles you win \$2, otherwise you lose. Game #2: You must pay \$2 to play. If you roll a sum of 7 or more you win \$3. If you roll a sum of 5 you win \$5. Otherwise, you lose. Game #3: You do not pay to play. If you roll a sum greater than nine, you win \$10. If you roll snake eyes (double ones) you win \$25. Otherwise you pay \$4.

13 12. A call-in poll conducted by USA Today concluded that Americans love Donald Trump. USA Today later reported that 5640 of the 7800 calls for the poll came from the offices owned by one man, Cincinnati financier Carl Lindner, who is a friend of Donald Trump. The results of this poll are probably A. surprising, but reliable since it was conducted by a nationally recognized organization. B. biased, but only slightly since the sample size was quite large. C. unbiased because the calls were randomly made at varying times. D. biased, overstating the popularity of Donald Trump. E. biased, understating the popularity of Donald Trump. 13. Explain, in your own words, the difference between an experiment and an observational study. For parameters and statistics: 14. Listed below are the names of 20 students who are juniors. Use the random numbers listed below to select five of them to be in your sample. Clearly explain your method Adam 02 - Chris 03 - Dave 04 Deirdre 05 - Drew 06 - Ellen 07 - Eric 08 - Joan 09 - John 10 - Judi 11-Joy 12- Kenny 13 - Laura 14 - Mary 15 - Paul 16 - Peter 17 - Rachel 18- Rob 19 - Sara 20 - Stacey Listed at the right are the names of twenty full-time clerks on the retail staff. Use the random numbers listed below to select four of them to be in your sample. Clearly explain your method. 01- Larry 02- Marissa 03- Wayne 04- Diane 05- Carol 06- Joshua 07- Barbara 08- Allan 09- Rich 10- Sharyn 11 - Brian 12-Kelly 13 - Nicole 14 - John 15- June 16- Frank 17 - Steve 18 - Andrea 19- Matthew 20 - Erin a) Your population of interest consists of 8230 individuals. To select a Simple Random Sample using a random digit table, you should label the individuals with sequences of how many digits? b) You are selecting an individual from a group numbered 001 to 346. You enter the table at a random spot, at which point the listed digits begin as: Which individual is the first one selected for your sample? 17. Explain how to use a graphing calculator to simulate a softball player at bat, if the player has a batting average of.400 and she goes to bat 5 times in a game. For expected value and fair games: 18. An urn contains 10 balls, three white and seven red. You win \$5 if you draw a white ball and \$2 if you draw a red ball. If you pay nothing, what is the expected value of this game? Should you pay \$2 to play the game? Should you pay \$3 to play the game? 19. A mysterious card-playing squirrel offers you the opportunity to join in his game. The rules are: To play you must pay him \$2. If you pick a spade from a shuffled pack, you win \$9. Find the expected value you win (or lose) per game.

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21 "#\$%&'()*&+,-#*./0,-1,23& "#\$%%&'\$%%(#)**#++,- #./)01#2,301",2#4"#51""26*7)"4)#81/1#,12,10#4"#9),-#324".#,-1#51""26*7)"4)#:62,19#;<# m = 1330 # )"0# 2,)"0)/0#0174),4;"# s = 253A#### F-),#G#;<#++,- #./)01/2#2=;/10#H# )E;71#+I%%J##KKKKKKKKK# #####E1*;8#+%%%J##KKKKKKKKK# # E1,811"#+\$%%#)"0#+&%%J#KKKKKKKKK# # # # # # F-),#5::>#2=;/1#8;3*0#E1#),#,-1#I%,- #C1/=1",4*1J##KKKKKKKKKKKKKKKK# # L;1M2#5::>#2=;/1#8)2#),#,-1#(&,- #C1/=1",4*1A##F-),#8)2#-42#2=;/1J#KKKKKKKKKKKKKKKK# # # B-1#N#2=;/12#;"#,-1#:,)"0<;/0'O4"1,#4",1**4.1"=1#,12,#)/1#)CC/;D49),1*6#";/9)**6#042,/4E3,10#84,-#)#91)"#91)"# m = 100 # )"0#2,)"0)/0#0174),4;"# s = 15A#### F-),#G#;<#C1;C*1#-)71#)"#NH# #./1),1/#,-)"#(IJ##KKKKKKKKK# #####*122#,-)"#+\$IJ#KKKKKKKKK# # E1,811"#+\$%#)"0#+I%J#KKKKKKKKK# # # # # F-),#G#;<#C1;C*1#-)71#)"#N#./1),1/#,-)"#+P%#;/#*122#,-)"#Q%J##KKKKKKKKKKKKKKKKKKKK# # # # # # # # # F-),#N#2=;/1#8;3*0#E1#),#,-1#R%,- #C1/=1",4*1J##KKKKKKKKKKKKKKKK# # F-),#N#2=;/1#8;3*0#E1#),#,-1#\$I,- #C1/=1",4*1J#KKKKKKKKKKKKKKKK#

22 Normal Probability Distribution A normal distribution is a continuous, symmetric, bell-shaped distribution of a variable. The properties of a normal distribution, are: 1. A normal distribution curve is bell-shaped 2. The mean, median, and mode are equal and are located at the center of the distribution 3. A normal distribution curve is unimodal (i.e., it has only one mode) 4. The curve is symmetric about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center. 5. The curve is continuous, that is, there are no gaps or holes. For each value of X, there is a corresponding value of Y. 6. The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis but it gets increasingly closer 7. The total area under a normal distribution curve is equal to 1.00, or 100% The Empirical Rule states that: The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%.

23 Using the Empirical Rule For each problem set, label the normal curve with the appropriate values, and use the curve to answer the questions. 1. The mean score on the midterm was an 82 with a standard deviation of 5. Find the probability that a randomly selected person: a. scored between 77 and 87 b. scored between 82 and 87 c. scored between 72 and 87 d. scored higher than 92 e. scored less than The mean SAT score is 490 with a standard deviation of 100. Find the probability that a randomly selected student: a. scored between 390 and 590 b. scored above 790 c. scored less than 490 d. scored between 290 and The mean weight of college football players is 200 pounds with a standard deviation of 30. Find the probability that a randomly selected player: a. weighs between 170 and 260 b. weighs less than 170 c. weighs over 290 d. weighs less than 140 e. weighs between 140 and The average life of a car tire is 28,000 miles with a standard deviation of Find the probability that a randomly selected tire will have a life of: a. between 19,000 and 37,000 miles b. less than 25,000 miles c. between 31,000 and 37,000 miles d. over 22,000 miles e. below 31,000 miles

24 Working with Z Scores 1. The mean score on the SAT (math & verbal) is 1500, with a standard deviation of 240. The ACT, a different college entrance examination, has a mean score of 21 with a standard deviation of 6. (a) If Bobby scored 1740 on the SAT, how many points above the SAT mean did he score? (b) If Kathy scored 30 on the ACT, how many points above the ACT mean did she score? (c) Is it sensible to conclude that because Bobby's difference is bigger that he outperformed Kathy on the admissions test? Explain. (d) Determine how many standard deviations above the mean Bobby scored by dividing your answer to part (a) by the standard deviation of the SAT scores. (e) Determine how many standard deviations above the mean Kathy scored by dividing your answer to part (b) by the standard deviation of the ACT scores. The values you just calculated are called z-scores. A z-score is an indication of how many standard deviations above or below the mean a given data point is located. Z-scores are used to convert data from different scales to a common scale so a more accurate comparison can be made between them. Z-scores are locations on the Standard Normal Curve, which has a mean of 0 and a standard deviation of 1. To find a z-score, use the formula: where x is the given value, m is the mean, and s is the standard deviation. Now mark the values that you found in parts (d) and (e) on the standard normal curve above. (f) Who had the higher z-score on their admissions test? (g) Who performed better on his or her admissions test compared to his or her peers? (h) Calculate the z-score for Peter who scored 1380 on the SAT. Calculate the z-score for Kelly who scored 15 on the ACT. (i) Does Peter or Kelly have the higher z-score? (j) What does it mean to have a negative z-score? Can you think of situations in which it would be beneficial to have a negative z-score?

25 2. The weight of an average 3 month child is 12.5 pounds with a standard deviation of 1.5 pounds. Benjamin is a healthy 3 month old child who weighs 13.9 pounds. (a) Determine the z-score for Benjamin's weight at 3 months. (b) Interpret what the z-score means in context" in a sentence. (c) The weight of an average 6 month old is pounds with a standard deviation of 2.0 pounds. If Benjamin had the same z-score at 6 months as he did at 3 months, determine how much a 6 month old Benjamin would weigh. 3. The height of women aged 20 to 29 are approximately normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches. John and his sister June both play basketball for N. C. State University. John is 81 inches tall; June is 74 inches tall. (a) Compared to their respective peers, who is the tallest? (b) How tall would a woman be who has a z-score of 1.5? (c) If a man has a z-score of -0.5 and a woman has a z-score of 1.2, which is tallest? 4. The following table shows the scores of Toni on six different scales of an aptitude test. Also shown are the means and standard deviations of these scales. Test Mean Standard Deviation Score Z-Score Clerical Ability Logical Reasoning Mechanical Ability Numerical Reasoning Spatial Relations Verbal Fluency (a) Calculate the z-scores for each. (b) On which test did Toni score the highest? On which did Toni score the lowest?

26 Using Z-Scores to Pick a Winner The decathlon is an event in track and field that consists of 10 events. Suppose two competitors tie in each of the first eight events. In the ninth event, the high jump, one competitor clears the bar 1 in. higher. Then in the final event, the 1500-meter run, the other competitor runs 5 seconds faster. Who wins? To determine who wins, we have to know whether it is harder to jump an inch higher or run 5 seconds faster. We have to be able to compare two fundamentally different activities involving different units. Standard deviations to the rescue If we knew the mean performance (by world-class athletes) in each event, and the standard deviation, we could compute how far each performance was from the mean in standard deviation units (that is, the z-scores). So consider the three athletes performances shown below in a three event competition. Note that each placed first, second, and third in an event. Competitor Event 100 m dash Shot put Long Jump A 10.1 sec B 9.9 sec C 10.3 sec m dash Shot Put Long Jump mean 10 sec standard deviation 0.2 sec 3 6 Who gets the gold medal? Who turned in the most remarkable performance of the competition? Explain your reasoning using mathematics.

27

28 The Normal Probability Distribution menu for the TI-83+/84+ is found under DISTR (2nd VARS). NOTE: A mean of zero and a standard deviation of one are considered to be the default values for a normal distribution on the calculator, if you choose not to set these values. The Normal Distribution functions: 1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve. Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation) 2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) 3: invnorm inv = Inverse Normal Probability Distribution Function This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invnorm (probability, mean, standard deviation)

29 Example 1: (answers will be rounded to the nearest thousandth) Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90 th percentile for this distribution. 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax: normalcdf(lower bound, upper bound, mean, standard deviation) Answer: The probability is %. 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x Type 1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is %. 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is -1 x Type -1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is 3.772%. 1d: Find the 90 th percentile for this distribution. (Given a probability region to the left of a value (as a decimal), determine the value using invnorm.) Syntax: invnorm (probability, mean, standard deviation) Answer: The x-value is

30 Example 2: Graph and investigate the normal distribution curve where the mean is 0 and the standard deviation is 1. Go to the Y = menu. normalpdf (x, 0, 1) Adjust the WINDOW. Guideline is: Xmin = mean - 3 SD Xmax = mean + 3 SD Xscl = SD Ymin = 0 Ymax = 1/(2 SD) Yscl = 0 GRAPH. Using TRACE, simply type the desired x value and the point will be plotted. Investigate: What happens to the curve as the standard deviation increases? Double the standard deviation and see what happens to the graph. When graphing 2 normal curves, the window will need to be adjusted. Xmin = mean - 3 (largest SD) Xmax = mean + 3 (largest SD) Xscl = largest SD Ymin = 0 Ymax = 1/(2 smallest SD) Yscl = 0 Observe that as the standard deviation increases, the more spread out the graph becomes. Now, the area under the curve between particular values represents the probabilities of events occurring within that specific range. This area can be seen using the command ShadeNorm(. To find ShadeNorm(, go to DISTR and right arrow to DRAW. Choose #1:ShadeNorm(. ShadeNorm (lower bound, upperbound, mean, standard deviation) Enter parameters -1,1 to see the area, approximately 68%. Since the calculator defaults to a mean of 0 and standard deviation of 1, it was not necessary to enter these values in this example, but is is a good idea to get in the habit of entering all 4 parameters.

31 Example 3: Graph and examine a situation where the mean score is 46 and the standard deviation is 8.5 for a normally distributed set of data. Go to Y= Adjust the window. GRAPH. Examine: What is the probability of a value falling between the mean and the first standard deviation to the right? Answer: approximately 34% Example 4: The lifetime of a battery is normally distributed with a mean life of 40 hours and a standard deviation of 1.2 hours. Find the probability that a randomly selected battery lasts longer than 42 hours. Let's get a visual look at the situation by examining the graph. The location of 42 hours indicates that our answer is going to be quite small. (Be sure to set an appropriate window) Go to Y=. GRAPH. Now: What is the probability of a value falling to the right of 42 hours (between 42 hours and infinity)? Answer: approximately 4.8% ShadeNorm( go to DISTR and right arrow to DRAW. Choose #1:ShadeNorm(. ShadeNorm (lower bound, upperbound, mean, standard deviation) ENTER. The percentage is read from the Area =.

32 Normal Distribution Practice 1 For all questions, assume that the distribution is normal. Shade the appropriate area of the normal curve, then solve using the calculator. Show the calculator steps 1. A survey found that mean length of time that Americans keep their cars is 5.3 years with a standard deviation of 1.2 years. If a person decides to purchase a new car, find the probability that he or she has owned the old car for a) less than 2.5 years b) between 3 and 6 years c) more than 7 years d) The length of time John keeps his car is in the 90 th percentile. Determine how long John keeps his car. a) b) c) d) 2. The average waiting time at Walgreen s drive-through window is 7.6 minutes, with a standard deviation of 2.6 minutes. When a customer arrives at Walgreen s, find the probability that he will have to wait a) between 4 and 6 minutes b) less than 3 minutes c) more than 8 minutes d) Only 8% of customers have to wait longer than Mrs. Sickalot. Determine how long Mrs. Sickalot has to wait. a) b) c) d)

33 3. The scores on an Algebra II test have a mean of 76.4 and a standard deviation of Find the probability that a student will score a) above 78 b) below 60 c) between 80 and 85 d) Mr. Reeves scales his tests so that only 5% of students can receive an A. What is the minimum score Andrea can make on this test and still get an A?. a) b) c) d) 4. The average life of automobile tires is 30,000 miles with a standard deviation of 2000 miles. If a tire is selected and tested, find the probability that it will have a lifetime a) between 25,000 and 28,000 miles b) between 27,000 and 32,000 miles c) over 35,000 miles d) The tire company will replace tires whose tread life falls in the lowest 150% of all tires of this model. What is the lifetime of a tire that qualifies for replacement? a) b) c) d) 5. The mean height of an American man is 69 with a standard deviation of 2.4. If a man is selected at random, find the probability that he will be a) between 68 and 71 tall b) shorter than 67 c) taller than 72 d) If Jose is in the 75 th percentile, how tall is he? a) b) c) d)

34 Normal Distribution Practice 2 For all questions, assume that the distribution is normal. Shade the appropriate area of the normal curve, then solve using the calculator. Show the calculator steps 1. The percentage impurity of a chemical can be modelled by a normal distribution with a mean of 5.8 and a standard deviation of 0.5. Obtain the probability that a sample of the chemical has percentage impurity between 5 and Melons sold on a market stall have weights that are normally distributed with a mean of 2.18 kg and a standard deviation of 0.25 kg. For a melon chosen at random, find the probability that its weight lies between 2 kg and 2.5 kg. 3. A teacher travels from home to work by car each weekday by one of two routes, X or Y. For route X, her journey times are normally distributed with a mean of 30.4 minutes and a standard deviation of 3.6 minutes. Calculate the probability that her journey time on a particular day takes between 25 minutes and 35 minutes. 4. Soup is sold in tins which are filled by a machine. The actual weight of soup delivered to a tin by the filling machine is always normally distributed about the mean weight with a standard deviation of 8g. The machine is set originally to deliver a mean weight of 810g. (a) Determine the probability that the weight of soup in a tin, selected at random, is less than 800g. (b) Determine the probability that the weight of soup in a tin, selected at random, is between 795 g and 820 g.

35 5. The weight, X grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 percent should be graded as small, the largest 20 percent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as: (i) small (ii) medium. 6. Jars of bolognese sauce, sold by a supermarket, are stated to have contents of weight 500 g. The weights, in grams, of the actual contents of jars in a large batch are normally distributed with mean 506 and standard deviation 5. Find the weight which is exceeded by the contents of 99.9% of the jars in this batch. 7. The distance, in kilometers, travelled to work by the employees of a city council may be modelled by a normal distribution with mean 7.5 and standard deviation 2.5. Find d such that 10% of the council s employees travel less than d kilometers to work. 8. An airline operates a service between Manchester and Paris. The flight time may be modelled by a normal distribution with mean 85 minutes and standard deviation 8 minutes. In order to gain publicity for the service, the airline decides to refund fares when a flight time exceeds q minutes. Find the value of q such that the probability of fares being refunded on a particular flight is

37 Percentiles & Z-scores Review of normal curve properties A normal curve is a density curve that is symmetric, single-peaked and bell-shaped. When data fits a normal pattern, one can standardize values and compare distributions. The standardized value of x is, where m is the mean and s is the standard deviation. This standardized value is called the z-score and it corresponds to the number of standard deviations a data point is above or below the mean. The p-th percentile of a distribution is the data value such that p percent of the observations fall at or below it. Part 1 Given x-values, finding percentages Problem: Too Good candy bars The average (mean) number of calories in a bar is 210 and has a standard deviation of 10. The number of calories per bar is approximately normally distributed. What percent of candy bars contain between 200 and 220 calories? Solution: Select normalcdf(200, 220, 210, 10) What is the answer to the problem? Try It The length of useful life of a fluorescent tube used for indoor gardening is normally distributed. The useful life has a mean of 600 hours and a standard deviation of 40 hours. Determine the probability that a. a tube chosen at random will last between 620 and 680 hours. b. such a tube will last more than 740 hours Texas Instruments Incorporated Percentiles & Z-scores

38 Percentiles & Z-scores Part 2 Given percentiles, finding x-values Problem: Mike is in the 99 th percentile for his height. U.S. men have an average height of 69.3 inches with a standard deviation of 2.8 inches. How tall is he? Solution: Select the invnorm (.99, 69.3, 2.8) What is the answer to the problem? Try It 1. The lifetimes of zip drives marketed by Zippers, Inc. are normally distributed, with a mean lifetime of 11 months and a standard deviation of 3 months. Zippers plans to offer a new warranty guaranteeing the replacement of failed zip drives during the warranty period. It can afford to replace up to 4 percent of its drives. How many months of warranty should the company offer with these drives? Round your answer to the nearest month. 2. Final grade averages are typically approximately normally distributed with a mean of 72 and a standard deviation of Your professor says that the top 8% of the class will receive A; the next 20%, B; the next 42%, C; the next 18%, D; and the bottom 12%, F. a. What average must you exceed to obtain an A? b. What average must you exceed to receive a grade better than a C? c. What average must you obtain to pass the course? 2008 Texas Instruments Incorporated Percentiles & Z-scores

39 Percentiles & Z-scores Part 3 Given z-scores, finding percentiles and x-values Problem: Find the corresponding percentile and x-value that has a z-score = 2.3 with mean = 100 and standard deviation = 10. Solution: Finding percentile using a standardized normal curve. Calculate normcdf( 1E99, 2.3, 0, 1). Or Calculate normcdf( 1E99, 77, 100, 10). Solution: Finding x-value. Calculate invnorm(percentile, 100, 10), where percentile is the value you found in the first part of the solution. Or 100 ((2.3)(10) = 77 What are the answers to the problem? Try It 1. In a field, the heights of sunflowers are normally distributed with a mean of 72 inches and standard deviation of 4 inches. Find the corresponding percentile and x-value for a sunflower that has a z-score of The shoe sizes of a men s basketball team are normally distributed with a mean of 11.5 and a standard deviation of Find the corresponding percentile and x-value for a player that has a z-score of A machine is programmed to fill 10-oz containers with a cleanser. However, the variability inherent in any machine causes the actual amounts of fill to vary. The distribution is normal with a standard deviation of 0.02 oz. What must the mean amount be in order for only 5% of the containers receive less than 10-oz? 4. The weights of ripe watermelons grown at Mr. Smith s farm are normally distributed with a standard deviation of 2.8 lb. Find the mean weight of Mr. Smith s ripe watermelons if only 3% weigh less than 15 lb Texas Instruments Incorporated Percentiles & Z-scores

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46 Observational Study or Experiment? For each situation, determine whether the research conducted is an observational study or an experiment. Explain your reasoning. 1. In an attempt to study the health effects of air pollution, a group of researchers selected 6 cities in very different environments; some from an urban setting (e.g. greater Boston), some from a heavy industrial setting (e.g. eastern Ohio), some from a rural setting (e.g. Wisconsin). Altogether they selected 8000 subjects from the 6 cities, and followed their health for the next 20 years. At this time their health prognoses were compared with measurements of air pollution in the 6 cities. 2. Among a group of women aged 65 and older who were tracked for several years, those who had a vitamin B 12 deficiency were twice as likely to suffer severe depression as those who did not. 3. Forty volunteers suffering from insomnia were divided into two groups. The first group was assigned to a special no-desserts diet while the other continued desserts as usual. Half of the people in these groups were randomly assigned to an exercise program, while the others did not exercise. Those who ate no desserts and engaged in exercise showed the most improvement. 4. A study in California showed that students who study a musical instrument have higher GPAs than students who do not, 3.59 to Of the music students, 16% had all A s, compared with only 5% among the students who did not study a musical instrument. 5. Scientists at a major pharmaceutical firm investigated the effectiveness of an herbal compound to treat the common cold. They exposed each subject to a cold virus, and then gave him or her either the herbal compound or a sugar solution known to have no effect. Several days later, they assessed the patient s condition, using a cold severity scale of 0 to In 2001, a report in the Journal of the American Cancer Institute indicated that women who work nights have a 60% greater risk of developing breast cancer. Researchers based these findings on the work histories of 763 women with breast cancer and 741 women without the disease. 7. To research the effects of dietary patterns on blood pressure in 459 subjects, subjects were randomly assigned to three groups and had their meals prepared by dietitians. Those who were fed a diet low in fat and cholesterol lowered their systolic blood pressure by an average of 6.7 points when compared with subjects fed a control diet. 8. Some people who race greyhounds give the dogs large doses of vitamin C in the belief that the dogs will run faster. Investigators at the University of Florida tried three different diets in random order on each of five racing greyhounds. They were surprised to find that when the dogs ate high amounts of vitamin C, they ran more slowly.

47 Sampling Methods Identify the sampling method: simple random, cluster, stratified, convenience, voluntary response, or systematic. 1. Every fifth person boarding a plane is searched thoroughly. 2. At a local community College, five math classes are randomly selected out of 20 and all of the students from each class are interviewed. 3. A researcher randomly selects and interviews fifty male and fifty female teachers. 4, A researcher for an airline interviews all of the passengers on five randomly selected flights. 5. Based on 12,500 responses from 42,000 surveys sent to its alumni, a major university estimated that the annual salary of its alumni was 92, A community college student interviews the first 100 students to enter the building to determine the percentage of students that own a car. 7. A market researcher randomly selects 200 drivers under 35 years of age and 100 drivers over 35 years of age. 8. All of the teachers from 85 randomly selected nation s middle schools were interviewed. 9. To avoid working late, the quality control manager inspects the last 10 items produced that day. 10. The names of 70 contestants are written on 70 cards, The cards are placed in a bag, and three names are picked from the bag sophomores, 35 juniors and 49 seniors are randomly selected from 230 sophomores, 280 juniors, 577 seniors at a certain high school. 12. To ensure customer satisfaction, every 35th phone call received by customer service will be monitored. 13. Calling randomly generated telephone numbers, a study asked 855 U.S. adults which medical conditions could be prevented by their diet. 14. A pregnancy study in Chicago, randomly selected 25 communities from the metropolitan area, then interviewed all pregnant women in these communities. Are these samples representative? Explain your thinking. 15. To determine the percentage of teenage girls with long hair, Teen magazine published a mailin questionnaire. Of the 500 respondents, 85% had hair shoulder length or longer (USA Today, July 1, 1985). 16. A college psychology professor needs subjects for a research project to determine which colors average American adults find restful. From the list of all 743 students taking introductory psychology at her school, she selects 25 students using a random number table. 17. To evaluate the reliability of cars owned by its subscribers, Consumer Reports magazine publishes a yearly list of automobiles and their frequency-of-repair records. The magazine collects the information by mailing a questionnaire to subscribers and tabulating the results from those who return it. 18. Oranges from an orchard need to be samples to see if they are sweet enough for juice. The orchard has 25,000 orange trees. Each tree has at least 500 oranges. Claire decides to randomly choose 800 trees and test one orange from each tree.

48 Sampling Bias Activity Read through each scenario and determine whether or not bias exists in the sampling method. If so, identify the source of the bias. Then make suggestions on how this sampling method could be improved in order to select an unbiased sample. Scenario #1: A company is interested in opening a gym on its premises for all employees. The company operates 3 shifts: morning, evening, and late night. They ask the first 25 people reporting to work during the morning and evening shifts if they would use the gym, and what hours they would like the gym to be open. Scenario #2: A newspaper is interested in determining whether working women support federal aid for child care. A reporter attends a conference designed for women in professional careers and randomly selects 40 women attending the conference to complete the survey. Scenario #3: A company makes products typically used by the elderly. The company manager provides samples of various products to residents of the retirement home where his mother lives. The residents use the products for several weeks and then complete a survey, giving their opinions of the company s products. Scenario #4: The school board is interested in taxpayers opinions on cutting funding for fine arts. They decide to ask parents since they would have an interest in school funding. A school board member attends a local high school chorus concert and interviews several parents as they are leaving the concert. Scenario #5: The Republican Party sends out a survey to 500 registered Republicans in 3 states in the Northeast to determine the issues that the Republican Party should focus on for the next election. Scenario #6: A television rating service is interested in finding out what shows are being watched most often during prime time viewing hours. To obtain a sample of households, the service dials numbers taken at random from telephone directories. Scenario #7: WRAL announces during the morning show that they are interested in listeners opinions on the new toll road between Holly Springs and I-40. Listeners are asked to call in and answer a few questions regarding the amount of tolls being charged on the road. Scenario #8: A clothing company wants to know what color leggings teenagers will buy. The company decides to spend one day in the junior departments of five randomly selected stores in randomly selected cities and ask every teenager who enters what color leggings they buy.

50 Margin of Error A survey gathers information from a sample of the population and then the results are used to reflect the opinions of the larger population. The reason that researchers and pollsters use a sample of the population is that it is cheaper and easier to poll a few people rather than everybody. One key to successful surveys is finding the appropriate size for the sample that will give accurate results without spending too much time or money. When you do a poll or survey, you're making a very educated guess about what the larger population thinks. When pollsters report the margin of error for their surveys, they are stating their confidence mathematically in the data they have collected. If a poll has a margin of error of 2.5 percent, that means that if you ran that poll 100 times asking a different sample of people each time the overall percentage of people who responded the same way would remain within 2.5 percent of your original result in at least 95 of those 100 polls. The margin of error is what statisticians call a confidence interval. The margin of error for a 95% confidence interval can be calculated by using the formula, where n is the sample size. This graph shows the relationship Between Sample Size (n) and Margin of Error (M.E.). As the sample size increases, the margin of error decreases. Notice that the amount by which the margin of error decreases is substantial between samples sizes of 200 and This implies that the accuracy of the estimate is strongly affected by the size of the sample. In contrast, the margin of error does not substantially decrease at sample sizes above Therefore, pollsters have concluded that it is not worth it to spend additional time and money for samples that contain more than 1500 people. Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter Olympics in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed. To determine how accurately the results of surveying 900 American teens truly reflect the results of surveying all 31 million American teens, a margin of error should be given. The margin of error would be,. Since the actual statistic could be larger or smaller than the true amount, the margin of error can be expressed as ± 3%. Pollsters would report that the favorite ski category for 20% of American teens is ski jumping with a margin of error of ± 3% at a 95% level of confidence.

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52 Margin of Error Practice 1 Assume a 95% confidence interval for the margin of error. 1. Find the margin of error for a survey of 100 American teens. 2. Find the margin of error for a survey of 900 American teens. 3. Find the margin of error for a survey of 9,000 American teens. 4. Find the margin of error for a survey of 90,000 American teens. 5. Draw a conclusion about the margin of error based on the size of the sample. Why do you think this is so? 6. If you want to cut your margin of error in half, what would you have to do to the sample size? Why? 7. If you wanted your margin of error to be 5%, how large a sample would you need? 8. If you wanted your margin of error to be 2%, how large a sample would you need? 9. If you wanted your margin of error to be 0.35%, how large a sample would you need? 10. In a survey of 750 people, 14% said they watch TV more than 12 hours per week. a) What is the margin of error for this survey? b) Give an interval that has a 95% confidence of containing the true proportion of all people who watch TV more than 12 hours per week.

53 Margin of Error Practice 2 Assume a 95% confidence interval for the margin of error. #1-4: a) Find the margin of error for each sample. Round each margin of error to the nearest %. b) Find an interval that has a 95% confidence of containing the true population proportion % of 350 students 2. 41% of 2,000 viewers 3. 6% of 525 artists 4. 97% of 4,000 readers 5. In a pre-election poll in a close race, how many people would you need to poll to get: a) a margin of error of 1.5%. b) a margin of error of 4% 6. In a pre-election poll the interval 45.7% to 50.7% has a 95% confidence of containing the proportion of people who prefer candidate A. a) What is the mean % and the margin of error? b) What was the sample size of the poll?

54 What is the Difference Between a Statistic and a Parameter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

55 Population Parameters and Sample Statistics Practice 1 1. The National Endowment of the Arts conducted a survey titled "Reading at Risk" on the reading habits of approximately 17,000 adults. Of those surveyed, only 57% read a book in a. What is the population under investigation? b. What is the parameter of interest? c. What is the sample? d. What is the sample statistic? 2. A recent survey by the alumni of a major university indicated that the average salary of 10,000 of its 300,000 graduates was \$125,000. a. What is the population under investigation? b. What is the parameter of interest? c. What is the sample? d. What is the sample statistic? 3. A major metropolitan newspaper selected a simple random sample of 1,600 readers from their list of 100,000 subscribers. They asked whether the paper should increase its coverage of local news. Forty percent of the sample wanted more local news. a. What is the population under investigation? b. What is the parameter of interest? c. What is the sample? d. What is the sample statistic? Determine whether the numerical value is a parameter or a statistic (and explain): 4. The average salary of all assembly-line employees at a certain car manufacturer is \$33, The average late fee for credit card holders was found to be \$ % of physics majors at NC State University were double majoring in math. 7. The average IQ score of college students is 115 with a standard deviation of Acme Corporation manufactures light bulbs. The CEO claims that an Acme light bulb lasts 300 days. 9. The owner of a fish market determined that the average weight for a catfish is 3.2 pounds. 10. The Pew Research Center reported that 24 percent of adults did not read a single book last year. 11. For incoming freshmen in 2013 at NC State University, the average SAT Math score is 625.

56 Population Parameters and Sample Statistics Practice 2 1. Why is a parameter is fixed, while a statistic varies? 2. A sample of 1200 randomly selected adults in the U.S. resulted in 33.2% reporting an allergy. e. What is the population under investigation? f. What is the parameter of interest? g. What is the sample? h. What is the sample statistic? 3. The admissions office at USC randomly identifies 100 students and obtains their total textbook costs. The average cost for the 100 students is calculated. a. What is the population under investigation? b. What is the parameter of interest? c. What is the sample? d. What is the sample statistic? Determine whether the numerical value is a parameter or a statistic (and explain): 4. 28% of all elementary school children can be classified as obese % of the kindergarten teachers at Apex Elementary say that knowing the alphabet is an essential skill for kindergarteners % of senators in the 109th Congress of the United States are Republicans % of American adults who do not use the Internet at home, work, school, or on a mobile device. For each scenario, describe the population and the sample. 8. The leaders of a large company want to know whether on-site day care would be considered a valuable employee benefit. They randomly selected 200 employees and asked their opinion about on-site day care. 9. A farmer saw some bollworms in his cotton field. Before deciding whether or not to reduce the number of bollworms by spraying his field, he selected 50 plants and checked each carefully for bollworms. 10. A manufacturer received a large shipment of bolts. The bolts must meet certain specifications to be useful. Before accepting shipment, 100 bolts were selected, and it was determined whether or not each met specifications.

57 Population Parameters with M&M S Claim: Mars, Incorporated claims that when producing milk chocolate M&M S, 16% of the total number produced is green in color. Question of Interest: Is the claim that 16% of milk chocolate M&M S produced by Mars, Inc. are green a valid claim? Each group of students will perform the following investigation using one sample of M&M S. The total number of M&M S in one sample is. 1. Calculate the percent of green M&M S in your sample. % of green M&M S 2. Record the % you found in #1 above on the dot plot on the board. 3. Did every sample have the same proportion of green M&M S? 4. What value is at the center of the dot plot constructed by using one sample per group? Is this value close to/far from the 16% claimed by Mars, Incorporated? 5. Does the information on the dot plot seem to support Mars claim? Why or why not? 6. Define each of the following in the context of the investigation you are performing with one sample of M&M S. Population Population parameter Sample Sample Statistic Each group of students will perform the following investigation using two samples of M&M S. The total number of M&M S in two samples is. 7. Calculate the percent of green M&M S two samples. % of green M&M S 8. Record the % you found in #1 above on the dot plot on the board. 9. What value is at the center of the dot plot constructed by using two samples per group? 10. What type of changes occurred in the dot plot when you used two samples of M&M S instead of one? 11. Mars, Inc. claimed that 16% of milk chocolate M&M S produced are green. Does the dot plot for two samples seem to support this claim more or less than the dot plot for one sample?

58 Are You Smarter than a Fifth Grader? 1. Which state lies to the south of Georgia? 2. If it's noon in Boston, what time is it in New York? 3. What is the postal abbreviation for New Hampshire? 4. What is the capital of the state of Mississippi? 5. Which ocean is found west of the California coast? 6. San Diego is in which state? 7. What was the last name of flight pioneers Orville and Wilbur? 8. In 1993 Michael Jordan gave up basketball to try which sport? 9. What color are the stars on the United States of America Flag? 10. What is the postal abbreviation for Vermont? 11. Which city is the home of Jazz? 12. What state is called the Sioux State? 13. In basketball where do the Suns come from? 14. Which baseball team are Giants? 15. Which two mid-atlantic states have New in their names? 16. What is the capital of Alabama? 17. Which Gulf lies to the south of Florida? 18. Which Island is the smallest state of the Union? 19. Which city has an area called Haight-Ashbury? 20. Who was the youngest US President to die in office? 21. The Sony company originated in which country? 22. Which Bill formed Microsoft? 23. Foods will not brown in what type of oven? 24. Which fruit gave its name to a desk top computer in 1984? 25. What was the favorite food of the Teenage Mutant Ninja Turtles? 26. Who was the first ex-movie actor to become President of the United States? 27. In football, where do the Vikings come from? 28. In sports, what is an MVP? 29. What is the postal abbreviation for Alaska? 30. What was the name of Michael Jackson's famous chimpanzee companion?

59

60 Designing Simulations Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. By observing simulated outcomes, researchers gain insight on the real world. When designing a simulation, you need to make sure you understand and answer the following questions: What is the problem being simulated? What are the possible outcomes? What is the probability of each outcome? Are there any assumptions? What question is the simulation trying to answer? What random device will be used to do the simulation, and how will it be used? Will it be a coin, spinner, playing cards, number cube, random digit table, or the random number generator? o Coin: What does each side of the coin represent? o Spinner: What does each section represent? o Playing Cards: What does each color, number, or suit represent? o Dice: What does each face represent? o Random Digit Table: What does each number represent? o Random Number Generator: What does each number represent? What is one trial in this simulation? Coin: How many times do you need to flip? Spinner: How many times do you need to spin? Playing Cards: How many cards do you need to choose? Dice: How many times do you need to throw? Random Digit Table: How many digits do you need to look at? Random Number Generator: How many digits do you need to look at? How many trials will be conducted? Will the process be repeated 10, 20, 30 times? What are the results of the simulated trials? State the specific results of YOUR trial you should get a fraction. My results show that out of were. What predictions can be made based on these results? This should include your conclusion based on the simulation you conducted. Based on my results the chances of are.

61 Design a Simulation Read the following situations and determine if/how each of random devices could be used to conduct a simulation. 1. A high school basketball player makes 50% of his shots from the three point line. If he takes 13 shots during a game predict the number of baskets he will make. Random Device Can it be used? How? Coin Spinner Playing Cards Dice Random Number Table Random Number Generator 2. In a family of 5, what is the probability that three of the members are male? Random Device Can it be used? How? Coin Spinner Playing Cards Dice Random Number Table Random Number Generator

62 3. In a random survey, 68% of high school students report that they have less than an hour of homework on any given night. What is the probability that a high school student selected at random will have less than an hour of homework on any given night. Random Device Can it be used? How? Coin Spinner Playing Cards Dice Random Number Table Random Number Generator 4. The probability that a student gets an A or B on a Chapter Test is 1/3. What is the probability that a student chosen at random does not get an A or a B on the test? Random Device Can it be used? How? Coin Spinner Playing Cards Dice Random Number Table Random Number Generator

63 Random digit table Line

64 Examples: Simulation Using a Random Digit Table 1. A teacher must choose two students from a group of ten to participate in a certain activity. To avoid favoritism, she assigns numeric labels to each of her students as follows: Number Student Number Student 0 Amanda 5 Lynn 1 Bill 6 Malcombe 2 Daniel 7 Neal 3 Emilio 8 Samantha 4 Jacob 9 Tracy Beginning on line 102 of the Table of Random Digits, the first two digits are 73. The two students chosen for the activity would be Neal (#7) and Emilio (#3). Suppose the teacher wanted to select four students from the group of students in the table. The first 4 numbers are 7, 3, 6, 7, but 7 has already been chosen so we need to move to the next number, which is 6. That has also been chosen, so we pick yet another one: 4. The students are Neal (#7), Emilio (#3), Malcombe (#6), and Jacob (#4). 2. Now consider a group of 100 students. The best way to use 100 numbers is to assign a 2 digit numbers to each student. (If you use three digit numbers you have to skip a lot of numbers in the random digit table to pick 5.) Using the table of random digits starting at line 106, the five labels would you select are 68, 41, 73, 50, To gather data on a 1200-acre pine forest in Louisiana, the U.S. Forest Service laid a grid of 1410 equally spaced circular plots over a map of the forest. A ground survey visited a sample of 10% of these plots. Starting on line 104 of random digit table, choose the first 5 plots. You ll need 4 digit numbers (since you have extra numbers, there s no need to use 0000, but you could use if desired). The first five numbers chosen are bolded. Note that numbers that are not between 0000 and 1410 are skipped

65 Now you try: 4. A medical study of heart surgery investigates the effect of a drug called a beta-blocker on the pulse rate of the patient during surgery. The pulse rate will be measured at a specific point during the operation. The investigators will use 20 patients facing heart surgery as subjects. You have a list of these patients, numbered 1 to 20, in alphabetical order. Half of these subjects will receive the beta-blocker during surgery. Use line 107 from the random digits table to determine which patients will receive the treatment. "There are 120 students in the BETA club. The national convention is to be held in Las Vegas this year, but only 20 students are interested in attending. Because of expenses, Mr. Ed Visor can only take 3 students. She has asked you to help him randomly pick the 3 students who will get to go to Las Vegas. The twenty students who wish to go are: Moore Phillips Barnes Shaw Cook Garris Jones Allen Scott Flynn Norris Jacobs Long Edwards Bennett Dixon Alex Thomas Young Glenn Determine a method to select the 3 students who will get to go to Las Vegas, then use line 101 to pick the students.

66 Practice - Simulations with a Random Digit Table 1. A club contains 33 students and 10 faculty members. The students are: Aisen DuFour Kittaka May Rokop Thyen Albrecht Dwivedi Kuhn MacDonald Sommer Wang Bhagava Gartner Lee Neukam Stuy Yerant Bonini Hollensburg Lipp Patel Terrell Chakra Huang Lundberg Pham Thomas Ding Joseph Marshall Ranaweera Thompson The faculty members are: Beck Burse Laumeyer Nesbitt Twining Brown Diamente Mitchener Sohalski Wernke The club can send 4 students and 2 faculty members to a convention and decides to choose those who will go by random selection. a. Use the random digit table to choose a sample of 4 students. Start on line 109. List the numbers and the names below. b. Use the random digit table to choose a sample of 2 faculty members. Start on line 112. List the numbers and the names below. 2. You are a marketing executive for a clothing company. Choose a SRS of 10 of the 440 retail outlets in New York that sell your company's products. Describe how you would label the retail outlets and select your sample using the random digit table starting on line Your school will send a delegation of 35 seniors to a student life convention. 200 girls and 150 boys are eligible to be chosen. If a sample of 20 girls and a separate sample of 15 boys are each selected randomly, it gives each senior the same chance to be chosen to attend the convention. Beginning at line 105 in the random digits table below, select the first four senior girls to be in the sample. Explain your procedures clearly.

67 4. Five boxes, each containing 24 cartons of strawberries, are delivered in a shipment to a grocery store. The produce manager always selects a few cartons randomly to inspect. He knows better than to just look at some of the cartons on the top or only in one box, because sometimes the rotten ones are on the bottom. Today he wishes to select a total of 6 cartons to inspect. He has the boxes arranged in order and has a set way to count the cartons inside each box. Explain the process used to make the random selection using a random digit table starting at line Five of the employees at the Stellar Boutique are going to be selected to go to training in Las Vegas for four days. Everyone wants to go of course, so the owner has decided to make the selection randomly. She has decided to send two managers and three sales representatives. The employees' names are listed in the table below. Managers Angela Barbara Elise Gigi Malena Rosie Tammie Veronica Sales Reps Sales Reps Sales Reps Alfie Irma Ray Betty Joe Sandy Carrie Katarina Shirley Cathy Lynn Suzi Darcy Marcie Tawny Fred Nancy Wendy Heidu Orville Zoe Explain the process she can follow to use a random digit table to select the employees who will get to go to the training. Select the managers first, then select the sales representatives. Use the random digit table starting on line The manager at Big-N-Yummy-Burger wishes to know his employees' opinions regarding the work environment. He has 56 employees and plans to select 12 employees at random to complete a survey. Explain the process he can follow to use the random digit table, starting at line 110, to select an SRS of size 12. Which are the first five employees numbers selected?

68 7. Use the random digit table, starting on line 112, to select an SRS of five of the fifty U.S. States. Explain your process thoroughly and report the five states that you chose. Repeat this a second time, but begin on a different line on the random digit table. Compare your lists to another classmate's lists. Did you end up with any of the same states in your samples? Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming 8. Washington High School has had some recent problems with students using steroids. The district decides that it will randomly test student athletes for steroids and other drugs. The boy's hockey team is to be tested. There are 13 players on the varsity team and 21 players on the junior varsity team. Use a table of random digits starting at line 122, to choose a stratified random sample of 3 varsity players and 5 junior varsity players to be tested. Remember to clearly describe your process.

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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