Honors Statistics Aug 23-8:26 PM 3. Review OTL C6#2 Aug 23-8:31 PM 1
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Ask about 1 and 5 Write a program to generate random numbers. I've decided to give them free will. A Skip 4, 12, 16 Apr 25-10:55 AM Toss 4 times Suppose you toss a fair coin 4 times. Let X = the number of heads you get. First List the Sample Space... HHHH THHH HHHT THHT HHTH THTH HHTT THTT HTHH TTHH HTHT TTHT HTTH TTTH HTTT TTTT (a) Find the probability distribution of X. (b) Make a histogram of the probability distribution. Describe what you see. 0.5 frequency 0.4 0.3 0.2 0.1 (c) Find P(X 3) and interpret the result. 0 1 2 3 4 Number of heads 15 P( X 3) = + + + = = 0.9375 16 The probability that 4 tosses of a coin results in 3 or fewer heads is 0.9375 Nov 28-12:03 AM 4
Kids and toys In an experiment on the behavior of young children, each subject is placed in an area with five toys. Past experiments have shown that the probability distribution of the number X of toys played with by a randomly selected subject is as follows: > (a) Write the event plays with at most two toys in terms of X. > What is the probability of this event? P(x 2) = 0.03 + 0.16 + 0.30 = 0.49 > (b) Describe the event X > 3 in words. > The probability that the young child plays with more than 3 toys. > What is its probability? P(X > 3) = 0.17 + 0.11 = 0.28 > What is the probability that X 3? P(X 3) = 0.28 + 0.23 = 0.51 Nov 28-12:08 AM Kids and toys Refer to Exercise 4. Calculate the mean of the random variable X and interpret this result in context. µ x = 0(0.03) + 1(0.16) + 2(0.30) + 3(0.23) + 4(0.17) + 5(0.11) = 2.68 If many, many children participated in this experiment, the mean number of toys that randomly selected children would play with will average 2.68 toys. (The expected number of toys a randomly selected young child will play with is 2.68.) This statement is optional. Nov 28-12:16 AM 5
Kids and toys Refer to Exercise 4. Calculate and interpret the standard deviation of the random variable X. Show your work. σ 2 x = (0-2.68) 2 (0.03) + (1-2.68) 2 (0.16) + (2-2.68) 2 (0.30) + (3-2.68) 2 (0.23) + (4-2.68) 2 (0.17) + (5-2.68) 2 (0.11) = 1.7176 σ x = 1.7176 = 1.31057 The standard deviation of X is σ x = 1.31057 The number of toys a randomly selected young child will play with will typically differ from the mean (2.68) by about 1.31 toys. Nov 28-12:22 AM Benford s law Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford s law. 7 Call the first digit of a randomly chosen record X for short. Benford s law gives this probability model for X (note that a first digit can t be 0): (a) Show that this is a legitimate probability distribution. all individual probabilities are between 0 and 1 0.301 + 0.176 + 0.125 + 0.097 + 0.079 + 0.067 + 0.058 + 0.051 + 0.046 = 1 (b) Make a histogram of the probability distribution. Describe what you see. See histogram above. The distribution is NOT symmetric. It is skewed to the right. The data should be analyzed using the 5 number summary. (c) Describe the event X 6 in words. What is P(X 6)? What is the probability that the first digit in a legitimate legal document is 6 or greater? P(X 6) = 0.067 + 0.058 + 0.051 + 0.046 = 0.222 (d) Express the event first digit is at most 5 in terms of X. What is the probability of this event? P(X < 6) = 1 - P(X 6) = 1-0.222 = 0.778 Nov 14-5:53 PM 6
Benford s law Refer to Exercise 5. The first digit of a randomly chosen expense account claim follows Benford s law. Consider the events A = first digit is 7 or greater and B = first digit is odd. (a) What outcomes make up the event A? What is P(A)? P(X 7) = 0.058 + 0.051 + 0.046 = 0.155 (b) What outcomes make up the event B? What is P(B)? P(X is odd) = 0.301 + 0.125 + 0.079 + 0.058 + 0.046 = 0.609 (c) What outcomes make up the event A or B? What is P(A or B)? Why is this probability not equal to P(A) + P(B)? P(X 7 or X is odd) = 0.609 + 0.155 - (0.058 + 0.046) = 0.66 Both 7 and 9 are included in each event and must their sum must be subtracted because they were counted twice ( the general probability addition rule) Nov 28-12:14 AM Benford s law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from 1 to 9. In that case, the first digit Y of a randomly selected expense amount would have the probability distribution shown in the histogram. > (a) Explain why the mean of the random variable Y is located at the solid red line in the figure. The mean is the balance point of the distribution. So it is located in the center of a uniform or symmetric distribution histogram in this case at 5. > (b) The first digits of randomly selected expense amounts actually follow Benford s law (Exercise 5). According to Benford s law, what s the expected value of the first digit? Explain how this information could be used to detect a fake expense report. µx = 1(0.301) + 2(0.176) + 3(0.125) + 4(0.097) + 5(0.079) + 6(0.067) + 7(0.058) + 8(0.051) + 9(0.046) = 3.441 To detect a fake expense report, compute the sample mean of the first digits of the numbers on the report. A value closer to 3.441 suggests a truthful report but a value closer to 5 (the more uniform distribution) suggest a false report. > (c) What s P(Y > 6) in the above distribution? According to Benford s law, what proportion of first digits in the employee s expense amounts should be greater than 6? How could this information be used to detect a fake expense report? P(Y > 6) = 0.058 + 0.051 + 0.046 = 0.155 For a uniform distribution the P(Y > 6) = 0.3 To detect a fake expense report, compute the percent of the first digits that are greater than 6 on the report. A value closer to 0.155 suggests a truthful report but a value closer to 0.3 (the more uniform distribution) suggest a false report. Nov 28-12:18 AM 7
Benford s law and fraud Refer to Exercise 13. It might also be possible to detect an employee s fake expense records by looking at the variability in the first digits of those expense amounts. > (a) Calculate the standard deviation σy. This gives us an idea of how much variation we d expect in the employee s expense records if he assumed that first digits from 1 to 9 were equally likely. σ 2 x = (1-5) 2 (0.10) + (2-5) 2 (0.10) + (3-5) 2 (0.10) + (4-5) 2 (0.10) + (5-5) 2 (0.10) + (6-5) 2 (0.10) + (7-5) 2 (0.10) + (8-5) 2 (0.10) + (9-5) 2 (0.10) = 6.66 σ x = 6.66 = 2.58 > (b) Now calculate the standard deviation of first digits that follow Benford s law (Exercise 5). Would using standard deviations be a good way to detect fraud? Explain. σ 2 x = (1-3.44) 2 (0.301) + (2-3.44) 2 (0.176) + (3-3.44) 2 (0.125) + (4-3.44) 2 (0.097) + (5-3.44) 2 (0.079) + (6-3.44) 2 (0.067) + (7-3.44) 2 (0.058) + (8-3.44) 2 (0.051) + (9-3.44) 2 (0.046) = 6.06052 σ x = 6.06052 = 2.42 Because the standard deviations are so close 2.58 and 2.42 it would be difficult to determine fake reports from legitimate reports using the standard deviation. Nov 28-12:22 AM Pair-a-dice Suppose you roll a pair of fair, six-sided dice. Let T= the sum of the spots showing on the up-faces. (a) Find the probability distribution of T. Probability model for the sum of two fair dice (b) Make a histogram of the probability distribution. Describe what you see. 0.16 0.12 0.08 0.04 2 3 4 5 6 7 8 9 10 11 12 (c) Find P(T 5) and interpret the result. (d) Find the expected value of the distribution. Interpret this value in context. Nov 27-10:56 PM 8
Pair-a-dice Suppose you roll a pair of fair, six-sided dice. Let T= the sum of the spots showing on the up-faces. (a) Find the probability distribution of T. Probability model for the sum of two fair dice (b) Make a histogram of the probability distribution. Describe what you see. frequency of sum 0.16 0.12 0.08 0.04 2 3 4 5 6 7 8 9 10 11 12 sum of dice (c) Find P(T 5) and interpret the result. 1 2 3 6 30 P(T 5) = 1 - P(T < 5) = 1 - ( + + ) = 1 - = = 0.8333 36 36 36 36 36 The probability that a random roll of dice will add to more than 5 is 0.8333 or 83.3% (d) Find the expected value of the distribution. Interpret this value in context. 1 2 3 4 5 6 µt = 2( ) + 3( ) + 4( ) + 5( ) + 6( ) + 7( ) + 36 36 36 36 36 36 5 4 3 2 1 252 8( ) + 9( ) + 10( ) + 11( ) + 12( ) = = 7 36 36 36 36 36 36 If a pair of fair dice are rolled many, many times and the sum of the dots is calculated the sum will average a total of 7. The expected sum of a pair of randomly rolled dice is 7. Nov 27-10:56 PM You choose a 3 digit number. The lottery commission announces the The "box" pays $83.33 if the number you choose has the same digits as on the box. (Assume a number with three different digits is chosen) let X = Dec 5-1:10 PM 9
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The probability distribution for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes. In fact, all continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability. To see that this is true, consider a specific outcome from the random number generator of the previous example, such as P(Y = 0.7). The probability of this event is the area under the density curve that s above the point 0.70000 on the horizontal axis.but this vertical line segment has no width, so the area is 0. For that reason, P(0.3 Y 0.7) = P(0.3 Y < 0.7) = P(0.3 < Y < 0.7) = 0.4 CAUT ION ALL continuous probability models assign probability 0 to every individual outcome. P(x=3) = 0 In many cases, discrete random variables arise from counting something for instance, the number of siblings that a randomly selected student has. Continuous random variables often arise from measuring something for instance, the height or time to run a mile for a randomly selected student. Nov 28-9:33 PM Apr 27-11:47 AM 13
b) P(X HOMEWORK PROBLEM d) What important fact about continuous random variables does comparing Nov 21-11:51 AM Dec 1-2:08 PM 14
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CLASS EXAMPLE Weights of 3-year-old females The weights of 3-year-old females closely follow a Normal Distribution with a mean of µ = 30.7 pounds and a standard deviation of σ = 3.6 pounds. Randomly choose one 3-year-old female and call her weight X. What is the probability that a randomly selected 3-year-old female weights at least 30 pounds? What is the probability that a randomly selected 3-year-old female weights between 28 and 33 pounds? Dec 2-3:52 PM Nov 28-9:30 PM 18
Discuss with students to guide them when doing #20 in the homework assignment Apr 27-9:24 AM Owned Units: The histogram of owned housing units is symmetric. It is Nov 21-12:26 PM 19
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Working out Choose a person aged 19 to 25 years at random and ask, In the past seven days, how many times did you go to an exercise or fitness center or work out? Call the responsey for short. Based on a large sample survey, here is a probability model for the answer you will get: 8 (a) Show that this is a legitimate probability distribution. 0.68 + 0.05 + 0.07 + 0.08 + 0.05 + 0.04 + 0.01 + 0.02 = 1 (b) Make a histogram of the probability distribution. Describe what you see. frequency 0.5 0.4 0.3 0.2 0.1 0 1 2 3 (c) Describe the event Y < 7 in words. What is P(Y < 7)? 5 6 7 Number of workout days What is the probability that a randomly selected person aged 19 to 25 went to the gym less than seven days this week? P(Y < 7) = 1 - P(Y = 7) = 1-0.02 = 0.98 (d) Express the event worked out at least once in terms of Y. What is the probability of this event? P(Y 1) = 1 - P(Y = 0) = 1-0.68 = 0.32 Nov 29-10:57 AM 27, 28, 29, 30 Nov 21-8:16 PM 22
Working out Refer to Exercise 6. Consider the events A = works out at least once and B = works out less than 5 times per week. (a) What outcomes make up the event A? What is P(A)? outcomes = 1,2,3,4,5,6,7 P(Y 1) = 1 - P(Y = 0) = 1-0.68 = 0.32 (b) What outcomes make up the event B? What is P(B)? outcomes = 0,1,2,3,4 P(Y < 5) = 0.68 + 0.05 + 0.07 + 0.08 + 0.05 = 0.93 (c) What outcomes make up the event A and B? What isp(a and B)? Why is this probability not equal to P(A) P(B)? P(A and B) = 0.05 + 0.07 + 0.08 + 0.05 = 0.25 The events working out at least once and working out less than 5 times per week are not INDEPENDENT events. So Multiplication cannot be used to determine the probability of P(A and B) Nov 29-11:00 AM Keno Keno is a favorite game in casinos, and similar games are popular with the states that operate lotteries. Balls numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls are chosen at random. Players select numbers by marking a card. The simplest of the many wagers available is Mark 1 Number. Your payoff is $3 on a $1 bet if the number you select is one of those chosen. Because 20 of 80 numbers are chosen, your probability of winning is 20/80, or 0.25. Let X= the net amount you gain on a single play of the game. OR Based on what you "get" back OR µ x = -1(0.75) + 2(0.25) = -0.25 Nov 29-11:02 AM 23
Size of American households In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States: Legitimate Distributions... Let X = the number of people in a randomly selected U.S. household and Y = the number of people in a randomly chosen U.S. family. (a) Make histograms suitable for comparing the probability distributions of X and Y. Describe any differences that you observe. Dec 5-5:48 PM ` > (b) Find the mean for each random variable. Explain why this difference makes sense. µ x = 1(0.25) + 2(0.32) + 3(0.17) + 4(0.15) + 5(0.07) + 6(0.03) + 7(0.01) = 2.6 If many, many random households were selected, they would on average have 2.6 people dwelling in them. µ y = 1(0.0) + 2(0.42) + 3(0.23) + 4(0.21) + 5(0.09) + 6(0.03) + 7(0.02) = 3.14 If many, many random families were selected, they would on average have 3.14 people in the family. > (c) Find and interpret the standard deviations of both Xand Y. Nov 21-10:40 PM 24
> (c) Find and interpret the standard deviations of both X and Y. σ 2 x = (1-2.6) 2 (0.25) + (2-2.6) 2 (0.32) + (3-2.6) 2 (0.17) + (4-2.6) 2 (0.15) + (5-2.6) 2 (0.07) + (6-2.6) 2 (0.03) (7-2.6) 2 (0.01) = 2.02 σ x = 2.02 = 1.421 The standard deviation of X is σ x = 1.421 The number of people in a randomly selected household will typically differ from the mean (2.6) by about 1.421 people. σ 2 y = (1-3.14) 2 (0.0) + (2-3.14) 2 (0.42) + (3-3.14) 2 (0.23) + (4-3.14) 2 (0.21) + (5-3.14) 2 (0.09) + (6-3.14) 2 (0.03) (7-3.14) 2 (0.02) = 1.5604 σ x = 1.5604 = 1.249 The standard deviation of X is σ y = 1.249 A the number of people in a randomly selected family will typically differ from the mean (3.14) by about 1.249 people. Dec 4-7:00 AM THIS IS ON THE CLASS WORKSHEET! Random numbers Let Y be a number between 0 and 1 produced by a random number generator. Assuming that the random variable Y has a uniform distribution, find the following probabilities: HOMEWORK PROBLEM #22 Random numbers a) P(Y 0.4) = (0.4)(1) = 0.40 b) P(Y < 0.4) = (0.4)(1) = 0.40 c) P(0.1 < Y 0.15 or 0.77 Y < 0.88) = (0.05)(1) + (0.11)(1) = 0.16 d) What important fact about continuous random variables does comparing your answer to a and b illustrate? The area of a line segment (Y = 0.4) is equal to zero Nov 29-11:16 AM 25
Running a mile A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute. 10 Choose a student at random from this group and call his time for the mile Y. Find P(Y < 6) and interpret the result. N(7.11, 0.74) This interprets (in the context of this problem)... The probability of randomly choosing a random student who can run the mile in less than 6 minutes is approximately 6.68% or 6.7 out of 100. Nov 29-11:19 AM Ace! Professional tennis player Rafael Nadal hits the ball extremely hard. His first-serve speeds follow a Normal distribution with mean 115 miles per hour (mph) and standard deviation 6 mph. Choose one of Nadal s first serves at random. Let Y = its speed, measured in miles per hour. N(115,6) (a) Find P(Y > 120) and interpret the result. 0.2033 This interprets (in the context of this problem)... The probability of randomly choosing one of Nadal's first serves that is faster than 120 mph is approximately 20.33% or 20 out of 100. (b) What is P(Y 120)? Explain. The answer is equal to P(Y > 120) 0.2033 Because the P(Y = 120) is zero. Nov 29-11:21 AM 26
(c) Find the value of c such that P(Y c) = 0.15. Show your work. 0.15 c =? z = -1. 04 c = -1.04(6) + 115 c = 108.76 This interprets (in the context of this problem)... The probability of randomly choosing one of Nadal's first serves that is slower than 108.76 mph is approximately 15% or 15 out of 100. Dec 1-9:51 PM Pregnancy length The length of human pregnancies from conception to birth follows a Normal distribution with mean 266 days and standard deviation 16 days. Choose a pregnant woman at random. Let X = the length of her pregnancy. N(266, 16) x = 240 (a) Find P(X 240) and interpret the result. (b) What is P(X > 240)? Explain. 1-0.0516 0.9484 This interprets (in the context of this problem)... The probability of randomly choosing one pregnant lady that carries her baby at least 240 days is approximately 94.84% or 95 out of 100 pregnant women. The answer is equal to P(X > 240) 0.9484 Because the P(X = 240) is zero. Dec 4-7:12 AM 27
(c) Find the value of c such that P(X c) = 0.20. Show your work. N(266, 16) Look up 0.80 so this side is 0.80 0.20 c =? z = 0.?? c = 0.84(16) + 266 c = 279.44 This interprets (in the context of this problem)... The probability of randomly choosing one pregnant lady that carries her baby at least 279.44 days is approximately 20% or 20 out of 100 pregnant women. Dec 4-7:12 AM Multiple choice: Select the best answer for Exercises 27 to 30. Exercises 27 to 29 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: What s the expected number of cars in a randomly selected American household? (a) 1.00 (b) 1.75 (c) 1.84 (d) 2.00 (e) 2.50 µ x = 0(0.09) + 1(0.36) + 2(0.35) + 3(0.13) + 4(0.05) + 5(0.02) = 1.75 Dec 4-7:12 AM 28
The standard deviation of X is σ X = 1.08. If many households were selected at random, which of the following would be the best interpretation of the value 1.08? (a) The mean number of cars would be about 1.08. (b) The number of cars would typically be about 1.08 from the mean. (c) The number of cars would be at most 1.08 from the mean. (d) The number of cars would be within 1.08 from the mean about 68% of the time. (e) The mean number of cars would be about 1.08 from the expected value. Dec 4-7:12 AM About what percentage of households have a number of cars within 2 standard deviations of the mean? (a) 68% (b) 71% (c) 93% (d) 95% (e) 98% C we know the following... µ x = 1.75 σ x = 1.08 but it is not stated that the distribution is approximately NORMAL so use the table above... mean + 2 St. Dev. = 1.75 + 2(1.08) = 3.91 mean - 2 St. Dev. = 1.75-2(1.08) = -0.41 between 0 and 3.91 cars (must use 3) 0.09 + 0.36 + 0.35 + 0.13 = 0.93 What is the probability that they have more than 5 cars????? Dec 4-11:45 AM 29
ITBS scores The Normal distribution with mean µ = 6.8 and standard deviation σ = 1.6 is a good description of the Iowa Test of Basic Skills (ITBS) vocabulary scores of seventh-grade students in Gary, Indiana. Call the score of a randomly chosen student X for short. Find P(X 9) and interpret the result. N(6.8, 1.6) Nov 29-11:20 AM 0.0838 This interprets (in the context of this problem)... The probability of randomly choosing a random seventh grade student who scores a 9 or higher on the IOWA test is approximately 8.38% or 8.4 out of 100. May 5-8:15 PM 30
Dec 4-11:22 AM Honors Statistics POP Quiz Chapter 6 Section 1 Nov 1-1:41 PM 31
Spell-checking Spell-checking software catches nonword errors, which result in a string of letters that is not a word, as when the is typed as teh. When undergraduates are asked to write a 250-word essay (without spell-checking), the numberx of nonword errors has the following distribution: (a) Write the event at least one nonword error in terms of X. What is the probability of this event? 1) =.2 +.3 +.3 +.1 =.9 (b) Describe the event X 2 in words. What is its probability? What is the probability that X < 2? what is the probability that two or less nonword errors are caught? P(x 2) =.1 +.2 +.3 =.6 P(x < 2 ) =.1 +.2 =.3 c. Find the expected value of the distribution. Interpret this value in context. µx = 0(0.1) + 1(0.2) + 2(0.3) + 3(0.3) + 4(0.1) = 2.1 selected, the number of non word error will average 2.1 per The expected number of nonword errors in a randomly selected undergraduate 250 word essay is 2.1. d. Find the standard deviation of the distribution. Interpret this value in context. σ 2 x = (0-2.1) 2 (0.1) + (1-2.1) 2 (0.2) + (2-2.1) 2 (0.3) + (3-2.1) 2 (0.3) + (4-2.1) 2 (0.1) = 1.29 σ x = 1.29 = 1.136 The standard deviation of X is σ x = 1.136 A randomly selected undergrad essay will typically differ from the mean (2.1) by about 1.136 nonword errors. Nov 21-12:23 PM N(42,2) Sep 20-8:20 PM 32
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Sep 20-8:20 PM ITBS scores The Normal distribution with mean µ = 6.8 and standard deviation σ = 1.6 is a good description of the Iowa Test of Basic Skills (ITBS) vocabulary scores of seventh-grade students in Gary, Indiana. Call the score of a randomly chosen student X for short. Find P(X 9) and interpret the result. N(6.8, 1.6) This interprets (in the context of this problem)... The probability of randomly choosing a random seventh grade student... Nov 29-11:20 AM 36
A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: Draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise, you lose $1. If you make this wager very many times, what will be the mean amount you win? 48 µ x = -1( ) + 10( ) = -0.1538 52 4 52 (a) About $1, because you will lose most of the time. (b) About $9, because you win $10 but lose only $1. (c) About $0.15; that is, on average you lose about 15 cents. (d) About $0.77; that is, on average you win about 77 cents. (e) About $0, because the random draw gives you a fair bet. Dec 4-11:45 AM Apr 29-10:28 AM 37