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Oct. 2). Teen drivers: Get the facts. Retrieved from http://www.cdc.gov/ Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html Daily Mail. (2011, June 9). One born every minute: the maternity unit where mothers are THREE to a bed. Retrieved http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-pl... |
.com/index.php?option=ethq_prediction earthquakes: Live (2012). World earthquake news and highlights. Retrieved from SOLUTIONS 1 x P(x) 0 1 2 3 4 5 6.12.18.30.15.10.10.05 Table 4.38 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 | Discrete Random Variables 317 3.10 +.05 =.15 5... |
the law that was passed. 47 1,2,β¦ 49 1.4 51 X = the number of business majors in the sample. 53 2, 3, 4, 5, 6, 7, 8, 9 55 6.26 57 0, 1, 2, 3, 4, β¦ 59.0485 61.0214 63 X = the number of United States teens who die from motor vehicle injuries per day. 65 0, 1, 2, 3, 4,... 67 no 71 The variable of interest is X, or the ga... |
Chapter 4 | Discrete Random Variables 319 75 a. Software Company x 5,000,000 1,000,000 β1,000,000 Table 4.42 P(x).10.30.60 Hardware Company x 3,000,000 1,000,000 β1,000,00 Table 4.43 P(x).20.40.40 Biotech Firm x 6,000,000 0 P(x).10.70 β1,000,000.20 Table 4.44 b. $200,000; $600,000; $400,000 c. d. e. third investment b... |
a. X = the number of college and universities that offer online offerings. b. 0, 1, 2, β¦, 13 c. X ~ B(13, 0.96) d. 12.48 e..0135 f. P(x = 12) =.3186 P(x = 13) = 0.5882 More likely to get 13. 97 a. X = the number of fencers who do not use the foil as their main weapon b. 0, 1, 2, 3,... 25 c. X ~ B(25,.40) d. 10 This Op... |
. X = the number of adults in America who are surveyed until one says he or she will watch the Super Bowl. b. X ~ G(.40) c. 2.5 d. e..0187.2304 107 a. X = the number of pages that advertise footwear b. X takes on the values 0, 1, 2,..., 20 c. X ~ B(20, 29 192 ) d. 3.02 e. no f..9997 g. X = the number of pages we must s... |
.8944 =.1056 119 Let X = the number of defective bulbs in a string. Using the Poisson distribution: β’ ΞΌ = np = 100(.03) = 3 β’ X ~ P(3) β’ P(x β€ 4) = poissoncdf(3, 4) β.8153 Using the binomial distribution β’ X ~ B(100,.03) β’ P(x β€ 4) = binomcdf(100,.03, 4) β.8179 The Poisson approximation is very goodβthe difference betw... |
variables. (credit: Rev Stan) Introduction Chapter Objectives By the end of this chapter, the student should be able to do the following: β’ Recognize and understand continuous probability density functions in general β’ Recognize the uniform probability distribution and apply it appropriately β’ Recognize the exponentia... |
individual value is zero. The area below the curve, above the x-axis, and between x = c and x = c has no width, and therefore no area (area = 0). Since the probability is equal to the area, the probability is also zero. β’ P(c < x < d) is the same as P(c β€ x β€ d) because probability is equal to area. We will find the a... |
Consider the function f(x) = 1 20 for 0 β€ x β€ 20. x = a real number. The graph of f(x) = 1 20 is a horizontal line. However, since 0 β€ x β€ 20, f(x) is restricted to the portion between x = 0 and x = 20, inclusive. 328 Chapter 5 | Continuous Random Variables Figure 5.5 f(x) = 1 20 for 0 β€ x β€ 20. The graph of f(x) = 1 ... |
as P(X < x) for continuous distributions, is called the cumulative distribution function or CDF. Notice the less than or equal to symbol. We can also use the CDF to calculate P(X > x). The CDF gives area to the left and P(X > x) gives area to the right. We calculate P(X > x) for continuous distributions as follows: P(... |
8 3.4 10.0 3.3 6.7 3.7 2.1 7.8 17.9 19.2 9.8 4.5 6.3 10.7 11.6 13.8 18.6 Table 5.1 The sample mean = 11.49 and the sample standard deviation = 6.23. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. This means that any smiling time from zero to and ... |
οΏ½οΏ½ β 1 23 = 16 23 Figure 5.11 b. Find the 90th percentile for an eight-week-old baby's smiling time. Solution 5.3 b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. β β β βbaseβ P(x < k) = 0.90 βheightβ β = 0.90 β β β = 0.90 (k β 0) β 1 23 This OpenStax book is available for f... |
(0, 20). What is P(2 < x < 18)? Find the 90th percentile. Example 5.4 The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. a. What is the probability that a person waits fewer than 12.5 minutes? Solution 5.4 a. Let X = the number of minutes a... |
hours and 521 hours inclusive. a. Find a and b and describe what they represent. b. Write the distribution. c. Find the mean and the standard deviation. d. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? e. What is the 65th percentile for the duration of ... |
(0.5, 4). Use the conditional formula P(x > 2|x > 1.5) = P(x > 2 AND x > 1.5) P(x > 1.5) = P(x > 2) P(x > 1.5) = 2 3.5 2.5 3.5 = 0.8 = 4 5 5.5 Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let X = the time, in minutes, it takes a student to finish a... |
than at x = 0. Because X ~ U(1.5, 4), x cannot be less than 1.5. Figure 5.19 Uniform distribution between 1.5 and four with shaded area between 1.5 and three representing the probability that the repair time x is less than three Solution 5.6 c. This OpenStax book is available for free at http://cnx.org/content/col3030... |
6 The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Let X = the time needed to change the oil on a car. a. Write the random variable X in words. X = __________________. b. Write the distribution. c. Graph the distribution. d. Find P (x > 19). e.... |
(β0.25)(5) = 0.072. The probability that the postal clerk spends five minutes with the This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 5 | Continuous Random Variables 341 customers is 0.072. The graph is as follows: Figure 5.22 Notice the graph is a declining curve. When x = 0, f... |
follows: This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 5 | Continuous Random Variables 343 P(x < k) = 0.50 and P(x < k) = 1 β e β 0.25k Therefore, 0.50 = 1 β eβ0.25k and eβ0.25k = 1 β 0.50 = 0.5. Take natural logs: ln(eβ0.25k) = ln(0.50). So, β0.25k = ln(0.50). Solve for k: k ... |
0.40)). 344 Chapter 5 | Continuous Random Variables Example 5.9 On the average, a certain computer part lasts 10 years. The length of time the computer part lasts is exponentially distributed. a. What is the probability that a computer part lasts more than seven years? Solution 5.9 a. Let x = the amount of time (in ye... |
0.5934 = 0.0737. The probability that a computer part lasts between nine and 11 years is 0.0737. 346 Chapter 5 | Continuous Random Variables On the home screen, enter e^(β0.1*9) β e^(β0.1*11). 5.9 On average, a pair of running shoes can last 18 months if used every day. The length of time running shoes last is exponen... |
the previous customer? f. Is an exponential distribution reasonable for this situation? Solution 5.11 a. Since we expect 30 customers to arrive per hour (60 minutes), we expect on average one customer to arrive every two minutes on average. b. Since one customer arrives every two minutes on average, it will take six m... |
ier than others. 5.11 Suppose that on a certain stretch of highway, cars pass at an average rate of five cars per minute. Assume that the duration of time between successive cars follows the exponential distribution. a. On average, how many seconds elapse between two successive cars? b. After a car passes by, how long ... |
more likely to break down at any particular time than a brand new part. In other words, the part stays as good as new until it suddenly breaks. For example, if the part has already lasted ten years, then the probability that it lasts another seven years is P(X > 17|X > 10) = P(X > 7) = 0.4966. Example 5.12 Refer to Ex... |
k! time between events follows the exponential distribution. (k! = k*(kβ1*)(kβ2)*(kβ3)β¦3*2*1). Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of Suppose X has the Poisson distribution with mean Ξ». Compute P(X = k) by entering 2nd, VARS(DISTR), C: poissonpdf(Ξ», k). To ... |
Continuous Random Variables 351 Figure 5.32 c. Let X = the number of calls per minute. As previously stated, the number of calls per minute has a Poisson distribution, with a mean of four calls per minute. Therefore, X ~ Poisson(4), and so P(X = 5) = 45 eβ4 5! poissonpdf(4, 5) = 0.1563 β 0.1563. (5! = (5)(4)(3)(2)(1))... |
__________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ _... |
so, which values are they? Either way, justify your answer numerically. (Recall that any data that are less than Q1 β 1.5(IQR) or more than Q3 + 1.5(IQR) are potential outliers. IQR means interquartile range.) Compare the Data 1. For each of the following parts, use a complete sentence to comment on how the value obta... |
+ k|X > x) = P(X > k). Poisson distribution a distribution function that gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event; if there is a known average of Ξ» events occurring per ... |
include a and b). All values x are equally likely. We write X βΌ U(a, b). The mean of X is ΞΌ = a + b. The standard deviation of X is 2 Ο = (b β a)2 12 of X is P(X β€ x is continuous.. The probability density function of X is f (x) = 1 for a β€ x β€ b. The cumulative distribution function b β a Figure 5.36 The probability ... |
β’ standard deviation Ο = (b β a)2 12 β’ P(c < X < d) = (d β c real number between a and b (in some instances, X can take on the values a and b). a = smallest X, b = largest X 5.3 The Exponential Distribution (Optional) Exponential: X ~ Exp(m) where m = the decay parameter X ~ U(a, b) The mean is ΞΌ = a + b 2. The standa... |
Continuous Probability Functions 1. Which type of distribution does the graph illustrate? Figure 5.37 2. Which type of distribution does the graph illustrate? Figure 5.38 3. Which type of distribution does the graph illustrate? Figure 5.39 360 Chapter 5 | Continuous Random Variables 4. What does the shaded area repres... |
.5 2.8 1.8 4.5 1.9 1.9 3.1 1.6 Table 5.4 The sample mean = 2.50 and the sample standard deviation = 0.8302. The distribution can be written as X ~ U(1.5, 4.5). 16. What type of distribution is this? 17. In this distribution, outcomes are equally likely. What does this mean? 18. What is the height of f(x) for the contin... |
_______ iv. Label for x-axis (words): _______ v. Label for y-axis (words): _______ 41. Find the average age of the cars in the lot. 42. Find the probability that a randomly chosen car in the lot was less than four years old. a. Sketch the graph, and shade the area of interest. Figure 5.46 b. Find the probability. P(x <... |
distribution function? 59. Draw the distribution. 60. Find P(x < 4). 61. Find the 30th percentile. 62. Find the median. 63. Which is larger, the mean or the median? Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is s... |
probability and percentile problem, draw the picture. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 5 | Continuous Random Variables 367 74. Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from o... |
__________. State this result in a probability question, similarly to Parts g and h, draw the picture, and find the probability. 77. A subway train arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution. ... |
goes from 25 to 45 minutes. a. Define the random variable. X = ________ b. X ~ ________ c. Graph the probability distribution. d. The distribution is ______________ (name of distribution). It is _____________ (discrete or continuous). e. ΞΌ = ________ f. Ο = ________ g. Find the probability that the time is at most 30 ... |
At least how many miles does the truck driver travel on the 10 percent of days with the highest mileage? 5.3 The Exponential Distribution (Optional) 86. Suppose that the length of long-distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to eig... |
. The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about five years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement. Is X continuous or discrete? a. Define the random varia... |
g. Sketch a new graph, shade the area corresponding to the 40th percentile and find the value. h. Find the average value of x. 95. Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. a. Find the probability that a light bulb lasts less than one year. b. Find the probability t... |
is exponential. Assume that the current year is 2013 a. What is the probability that the next earthquake occurs within the next three months? b. Given that six months has passed without an earthquake in Papua New Guinea, what is the probability that the next three months will be free of earthquakes? c. What is the pro... |
Retrieved from http://www.baseball-reference.com/bullpen/No-hitter U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/ 372 Chapter 5 | Continuous Random Variables World Earthquakes. (2013). Earthquake data for Papua New Guinea. Retrieved from http://www.world-earthquakes.com/ Zhou, Rick. (2013). Exponen... |
Check student's solution b. P(x < 5,730) = 0.5001 71 a. Check student's solution b. k = 2947.73 73 Age is a measurement, regardless of the accuracy used. 75 a. X ~ U(1, 9) 374 Chapter 5 | Continuous Random Variables b. Check studentβs solution c. f (x) = 1 8 where 1 β€ x β€ 9 d. five e. 2.3 f. g. h. 15 32 333 800 2 3 i.... |
125. b. P(400 < X < 650) = 650 β 400 700 β 300 = 250 400 = 0.625 c. 0.10 = width 700 β 300 farthest 10 percent of days., so width = 400(0.10) = 40. Since 700 β 40 = 660, the drivers travel at least 660 miles on the 87 a. X = the useful life of a particular car battery, measured in months. b. X is continuous. c. X ~ Exp... |
377 Figure 5.57 d. We want to find 0.02 = P(T < t) = 1 β e β t 8. Solving for t, e β t 8 = 0.98, so β t 8 = ln(0.98), and t = β8ln(0.98) β 0.1616 years, or roughly two months. The warranty should cover light bulbs that last less than 2 months. Or use ln(area_to_the_right) = 0.1616. ( β m) = ln(1 β 0.2) β 1 8 e. We mus... |
isson with mean Ξ» = 3. Then P(X > 3) = 1 β P(X β€ 3) = 0.3528. 99 a. 100 9 = 11.11 b. P(X > 10) = 1 β P(X β€ 10) = 1 β Poissoncdf(11.11, 10) β 0.5532. 378 Chapter 5 | Continuous Random Variables c. The number of people with Type B blood encountered roughly follows the Poisson distribution, so the number of people X who a... |
β 0.0311. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 6 | The Normal Distribution 379 6 | THE NORMAL DISTRIBUTION Figure 6.1 If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described as normally dist... |
distribution. This means there are an infinite number of normal probability distributions. One distribution of special interest is called the standard normal distribution. Your instructor will record the heights of both men and women in your class, separately. Draw histograms of your data. Then draw a smooth curve thr... |
we have a data set with a mean of 5 and standard deviation of 2. We want to determine the number of standard deviations the score of 11 falls above the mean. We can find this answer (or z-score) by writing or we can solve for z. z = 11 β 5 2 = 3 5 + (z)(2) = 11, 2z = 6 z = 3 We have determined that the score of 11 fal... |
the right of ΞΌ, and when z is negative, x is to the left of or below ΞΌ. Or, when z is positive, x is greater than ΞΌ, and when z is negative, x is less than ΞΌ. The absolute value of z indicates how far the score is from the mean, in either direction. 6.1 What is the z-score of x, when x = 1 and X ~ N(12, 3)? 382 Chapte... |
(5, 6) represents weight gains for one group of people who are trying to gain weight in a six-week period and Y ~ N(2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y = 4 are each two standard deviations to the right of their means, they r... |
multiplying by 2. The empirical rule is also known as the 68β95β99.7 rule. Figure 6.3 Example 6.3 The mean height of 15-to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let X = the height of a 15-to 18-year-old ma... |
6.34 cm. Let Y = the height of 15-to 18-year-old males from 1984β1985, and y = the height of one male from this group. Then Y ~ N(172.36, 6.34). The mean height of 15-to 18-year-old males from Chile in 2009β2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Le... |
x values lie within two standard deviations of the mean. Therefore, about 95 percent of the x values lie between β2Ο = (β2)(6) = β12 and 2Ο = (2)(6) = 12. The values 50 β 12 = 38 and 50 + 12 = 62 are within two standard deviations from the mean 50. The z-scores are β2 and +2 for 38 and 62, respectively. This OpenStax ... |
z-scores are β3 and 3. 6.6 The scores on a college entrance exam have an approximate normal distribution with mean, Β΅ = 52 points and a standard deviation, Ο = 11 points. a. About 68 percent of the y values lie between what two values? These values are ________________. The z-scores are ________________, respectively.... |
Others show the mean to z area. The method used will be indicated on the table. We will discuss the z-table that represents the area under the normal curve to the left of z. Once you have located the z-score, locate the corresponding area. This will be the area under the normal curve, to the left of the z-score. This ... |
tail of the normal curve. We are calculating the area between 65 and 1099. In some instances, the lower number of the area might be β1E99 (= β1099). The number β1099 is way out in the left tail of the normal curve. We chose the exponent of 99 because this produces such a large number that we can reasonably expect all ... |
into those that are the same or lower than k and those that are the same or higher. Ninety percent of the test scores are the same or lower than k, and 10 percent are the same or higher. The variable k is often called a critical value. We know the mean, standard deviation, and area under the normal curve. We need to f... |
things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment 390 Chapter 6 | The Normal Distribution are normally distributed and the standard deviation for the times is half an hour. a. Find the probability that a hou... |
for entertainment is 1.66 hours. 6.9 The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three. Find the probability that a golfer scored between 66 and 70. Example 6.10 In the United States smartphone users between the ages of 13 and 55+ between the ages of 13 and... |
mean and standard deviation of 36.9 years and 13.9 years, respectively. Using this information, answer the following questions. βRound answers to one decimal place. a. Calculate the interquartile range (IQR). Solution 6.11 a. IQR = Q3 β Q1 Calculate Q3 = 75th percentile and Q1 = 25th percentile. Recall that we can use... |
of mandarin oranges from this farm have diameters between ______ and ______. Solution 6.12 b. 1 β 0.20 = 0.80. Outside of the middle 20 percent will be 80 percent of the values. The tails of the graph of the normal distribution each have an area of 0.40. Find k1, the 40th percentile, and k2, the 60th percentile (0.40 ... |
_______ Table 6.1 Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil. Scale the axes. Figure 6.10 a. Calculate the following: xΒ― = _______ s = _______ b. Draw a smooth curve through the tops of the bars of the histogram. Write one to two complete sentences to describe the gene... |
to determine if data from the experiment follow a continuous distribution. Collect the Data Measure the length of your pinkie finger, in centimeters. 1. Randomly survey 30 adults for their pinkie finger lengths. Round the lengths to the nearest 0.5 cm. _______ _______ _______ _______ _______ _______ _______ _______ __... |
ribe the Data and Theoretical Distribution, explain why or why not. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 6 | The Normal Distribution 399 KEY TERMS normal distribution a continuous random variable (RV) where ΞΌ is the mean of the distribution and Ο is the standard deviat... |
οΏ½), where Β΅ is the mean and Ο is the standard deviation 6.1 The Standard Normal Distribution Standard Normal Distribution: Z ~ N(0, 1). Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation) Calculator function for the kth percentile: k = invNorm ... |
(8, 1). What value of x has a z-score of β2.25? 17. Suppose X ~ N(9, 5). What value of x has a z-score of β0.5? 18. Suppose X ~ N(2, 3). What value of x has a z-score of β0.67? 19. Suppose X ~ N(4, 2). What value of x is 1.5 standard deviations to the left of the mean? 20. Suppose X ~ N(4, 2). What value of x is two st... |
lie within two standard deviations, left and right, of the mean of that distribution? 33. About what percent of x values lie between the second and third standard deviations, both sides? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 6 | The Normal Distribution 401 34. Suppose ... |
.543, what is the area to the left of x? Use the following information to answer the next four exercises: X ~ N(54, 8) 50. Find the probability that x > 56. 51. Find the probability that x < 30. 52. Find the 80th percentile. 53. Find the 60th percentile. 54. X ~ N(6, 2) Find the probability that x is between three and ... |
-score for a patient who takes 10 days to recover? a. 1.5 b. 0.2 c. 2.2 d. 7.3 62. The length of time it takes to find a parking space at 9 a.m. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. If the mean is significantly greater than the standard deviation, which of t... |
βs systolic blood pressure is 1.75 above the average systolic blood pressure of men his age. iv. Kylesβs systolic blood pressure is 1.75 standard deviations above the average systolic blood pressure for men. b. Calculate Kyleβs blood pressure. This OpenStax book is available for free at http://cnx.org/content/col30309/... |
7.99 d. 4.32 Use the following information to answer the next three exercises: The length of time it takes to find a parking space at 9 a.m. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. 70. Based on the given information and numerically justified, would you be surp... |
. Let X = percentage of fat calories. a. X ~ _____ (_____, _____) b. Find the probability that the percentage of fat calories a person consumes is more than 40. Graph the situation. Shade in the area to be determined. c. Find the maximum number for the lower quarter of percent of fat calories. Sketch the graph and writ... |
, and write the probability statement. d. Find the probability that a randomly selected district had between 1,800 and 2,000 votes for the candidate. e. Find the third quartile for votes for the candidate. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 6 | The Normal Distributio... |
Table 6.3 a. Calculate the sample mean and the sample standard deviation. b. Construct a histogram. c. Draw a smooth curve through the midpoints of the tops of the bars. d. In words, describe the shape of your histogram and smooth curve. e. Let the sample mean approximate ΞΌ and the sample standard deviation approximat... |
585 71,594 72,000 72,922 73,379 74,500 75,025 76,212 78,000 80,000 80,000 82,300 Table 6.4 a. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums. b. Construct a histogram. c. Draw a smooth curve through the midpoints of the tops of the bars of the histogram. d. e. Le... |
for the number of times the coin lands on heads is Β΅ = 20 and Ο = 4βverify the mean and standard deviation. Solve the following: a. There is about a 68 percent chance that the number of heads will be somewhere between ___ and ___. b. There is about a ____chance that the number of heads will be somewhere between 12 and... |
dt09_147.asp NBA.com. (2013). NBA Media Ventures. Retrieved from http://www.nba.com StatCrunch. viewreport.php?reportid=11960 (2010). Blood pressure of males and females. Retrieved from http://www.statcrunch.com/5.0/ The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/ Wikipedia. (2013). List of List_of... |
. 3, 0.1979 59 a. Check studentβs solution b. 0.70, 4.78 years 61 c 63 a. Use the z-score formula. z = β0.5141. The height of 77 inches is 0.5141 standard deviations below the mean. An NBA player whose height is 77 inches is shorter than average. b. Use the z-score formula. z = 1.5424. The height 85 inches is 1.5424 st... |
-old in a rural area is unsupervised during the day. b. X ~ N(3, 1.5) c. The probability that the child spends less than one hour a day unsupervised is 0.0918. d. The probability that a child spends over 10 hours a day unsupervised is less than 0.0001. e. 2.21 hours 79 a. X = the distribution of the number of days a pa... |
and x2 = Β΅ β zΟ = 10 β 2(3) = 4. 95 percent of the defective cars will fall between four and 16 z = Β±3: x1 = Β΅ + zΟ = 10 + 3(3) = 19 and x2 = Β΅ β zΟ = 10 β 3(3) = 1. 99.7 percent of the defective cars will fall between one and 19 87 n = 190; p = 1 5 = 0.2; q = 0.8 ΞΌ = np = (190)(0.2) = 38 Ο = npq = (190)(0.2)(0.8) = 5... |
οΏ½. The first alternative says that if we collect samples of size n with a large enough n, calculate each sample's mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. The second alternative says that if we again collect samples of size n that are ... |
distribution do these means appear to be representing? Draw the graph for the means using two dice. Do the sample means show any kind of pattern? Draw the graph for the means using five dice. Do you see any pattern emerging? Finally, draw the graph for the means using ten dice. Do you see any pattern to the graph? Wha... |
associated with it from that of the random variable X. The mean xΒ― is the Β― value of X in one sample. z = xΒ― β ΞΌ x Ο x n β β β β, Β―. ΞΌX is the average of both X and X Ο xΒ― = Οx n = standard deviation of X Β― and is called the standard error of the mean. To find probabilities for means on the calculator, follow these st... |
οΏ½ β value = 90 + 2 β β 15 25 β β = 96. The value that is two standard deviations above the expected value is 96. The standard error of the mean is Οx n = 15 25 = 3. Recall that the standard error of the mean is a description of how far (on average) that the sample mean will be from the population mean in repeated simpl... |
normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the population. Find the probability that the sample mean is between two hours and three hours. To find percentiles for means on the calculator, follow these steps. 2nd DIStR 3:invNorm k... |
a sample of 100 randomly selected gamers. If your target market is 29- to 35-year-olds, should you continue with your development strategy? Example 7.4 The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of 60. a. What are the mean ... |
of X If you draw random samples of size n, then as n increases, the random variable Ξ£X consisting of sums tends to be normally distributed and ΣΧ ~ N[(n)(ΞΌΞ§), ( n )(ΟΞ§)]. The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal dis... |
Find P(Ξ£x > 7500) P(Ξ£x > 7500) = 0.0127 Figure 7.3 normalcdf(lower value, upper value, mean of sums, stdev of sums) The parameter list is abbreviated(lower, upper, (n)(ΞΌX, ( n) (ΟX)) normalcdf (7500,1E99,(80)(90), β β 80β β (15)) = 0.0127 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 ... |
1500 < Ξ£x < 1800) = normalcdf (1500, 1800, (50)(34), ( 50 ) (15)) = 0.7974 c. Let k = the 80th percentile. k = invNorm(0.80,(50)(34), ( 50 ) (15)) = 1789.3 422 Chapter 7 | The Central Limit Theorem 7.6 In a recent study reported Oct.29, 2012, the mean age of tablet users is 35 years. Suppose the standard deviation is 1... |
Central Limit Theorem It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the means. If you are being asked to find the probability of a sum or total, use the clt for sums. This also applies to percentiles for means an... |
b 2 (b β a)2 12 Ο X = = 1 + 5 2 (5 β 1)2 12 = = 3 = 1.15 In the formula above, the denominator is understood to be 12, regardless of the endpoints of the uniform distribution. For problems (a) and (b), let X Β― = the mean stress score for the 75 students. Then, Β― X ~ N β β3, 1.15 75 β β where n = 75. a. Find P( xΒ― < 2)... |
(75,200,(75)(3), ( 75) (1.15)). REMINDER The smallest total of 75 stress scores is 75, because the smallest single score is one. d. Find the 90th percentile for the total of 75 stress scores. Draw a graph. Solution 7.8 d. Let k = the 90th percentile. Find k where P(Ξ£x < k) = 0.90. k = 237.8 Figure 7.7 The 90th percent... |
is randomly selected. Find the probability that this individual customer's excess time is longer than 20 minutes. This is asking us to find P(x > 20). c. Explain why the probabilities in parts (a) and (b) are different. Solution 7.9 a. Find: P( xΒ― > 20) P( xΒ― > 20) = 0.79199 using normalcdf β β20,1E99,22, 22 80 β β Th... |
9 The 95th percentile for the sample mean excess time used is about 26.0 minutes for a random sample of 80 customers who exceed their contractual allowed time. 95 percent of such samples would have means under 26 minutes; only five percent of such samples would have means above 26 minutes. 7.9 Use the information in Ex... |
. 75th percentile = invNorm(0.75,200,5) = 203.37. c. P(1.75 < xΒ― < 1.85) = normalcdf(1.75,1.85,2,0.05) = 0.0013 d. Using the z-score equation, z = xΒ― β ΞΌ xΒ― Ο xΒ―, and solving for x, we get x = 2(0.05) + 2 = 2.1. e. The IQR is 75th percentile β 25th percentile = 203.37 β 196.63 = 6.74. 7.10 Based on data from the Nation... |
P( xΒ― > 50) = normalcdf(50, E99,30.9,1.8) β 0. For this sample group, it is almost impossible for the groupβs average age to be more than 50. However, it is still possible for an individual in this group to have an age greater than 50. c. P(Ξ£x β₯ 1,600) = normalcdf(1600,E99,1514.10,63) = 0.0864 d. P(Ξ£x β€ 1,595) = norma... |
complicated. Using the normal approximation to the binomial distribution simplified the process. To compute the normal approximation to the binomial distribution, take a simple random sample from a population. You must meet the following conditions for a binomial distribution: β’ There are a certain number, n, of indep... |
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