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this text. GUIDELINE When writing the interpretation of a percentile in the context of the given data, make sure the sentence contains the following information: • Information about the context of the situation being considered • The data value (value of the variable) that represents the percentile • The percentage of... |
th percentile for points scored per player in a game is eight. Interpret the 40th percentile in the context of this situation. Example 2.23 A middle school is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the ... |
60 minutes a day. However, 15 students is a small sample, and the principal should survey more students to be sure of his survey results. 2.4 | Box Plots Box plots, also called box-and-whisker plots or box-whisker plots, give a good graphical image of the concentration of the data. They also show how far the extreme v... |
to start a box plot with a scaled number line. Otherwise, the box plot may not be useful. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 103 Example 2.24 The following data are the heights of 40 students in a statistics class: 59, 60, 61, 62, 62, 63, ... |
Q1: First quartile = 64.5. Q2: Second quartile or median = 66. Q3: Third quartile = 70. 104 Chapter 2 | Descriptive Statistics To construct the box plot: Press 4:Plotsoff. Press ENTER. Arrow down and then use the right arrow key to go to the fifth picture, which is the box plot. Press ENTER. Arrow down to Xlist: Press... |
class held during the evening are as follows: 98, 78, 68, 83, 81, 89, 88, 76, 65, 45, 98, 90, 80, 84.5, 85, 79, 78, 98, 90, 79, 81, 25.5. a. Find the smallest and largest values, the median, and the first and third quartile for Mr. Ramirez's class. b. Find the smallest and largest values, the median, and the first and... |
percent. d. Figure 2.16 e. The first data set has the wider spread for the middle 50 percent of the data. The IQR for the first data set is greater than the IQR for the second set. This means that there is more variability in the middle 50 percent of the first data set. 2.25 The following data set shows the heights in... |
The mean is the most common measure of the center. NOTE The words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is arithmetic mean and average is technically a center location. However, in practice among non statisticians, average is ... |
used to represent the median. The next example illustrates the location of the median and the value of the median. Example 2.27 Data indicating the number of months a patient with a specific disease lives after taking a new antibody drug are as follows (smallest to largest): 3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16,... |
M = 24 108 Chapter 2 | Descriptive Statistics 2.27 The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median. 3, 4, 5, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14,... |
, 4, 5, 5, 7, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12 Find the mode. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 109 Example 2.30 Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 ea... |
is a number calculated from a sample. Statistic examples include the mean, the median, and the mode 110 Chapter 2 | Descriptive Statistics ¯ as well as others. The sample mean x is an example of a statistic that estimates the population mean μ. Calculating the Mean of Grouped Frequency Tables When only grouped data is... |
2.28 53.25 59.5 65.5 71.5 77.5 83.5 89.5 95.5 • Calculate the sum of the product of each interval frequency and midpoint. ∑ f m 53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25 • μ = ∑ f m ∑ f = 1460.25 19 = 76.86 112 Chapter 2 | Descriptive Statistics 2.31 Maris conducted a stu... |
the left because it is pulled out to the left. A skewed left distribution has more high values. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 113 Figure 2.19 The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than... |
. b. Terry’s mean is 3.7, Davis’s mean is 2.7, and Maris’s mean is 4.6. c. Terry’s median is 3, Davis’s median is 3, and Maris’s median is four. It would be helpful to manually calculate these descriptive statistics, using the given data sets and then compare to the graphs. This OpenStax book is available for free at h... |
four minutes. Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average. The standard deviation can ... |
. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters. The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of fiv... |
first. The variance is the average of the squares of the deviations (the x – x¯ values for a sample or the x – μ values for a population). The symbol σ2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s2 represents the sample variance; t... |
measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The dif... |
summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However, you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. The calculator instructions appear at the end of this ... |
950625 3 ×.950625 = 2.851875 The total is 9.7375. The last column simply multiplies each squared deviation by the frequency for the corresponding data value. The sample variance, s2, is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1): s2 = 9.7375 20 − 1 =.5125 ... |
6, 3) into list L2. Use the arrow keys to move around. ◦ Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER. ◦ x¯ = 10.525. ◦ Use Sx because this is sample data (not a population): Sx=.715891. b. c. d. ( x¯ + 1s) = 10.53 + (1)(.72) = 11.25 ( x¯ – 2s)... |
the average squared deviation. and deviations frequencies sum of the the is The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data. Notice that instead of dividing by n = 20, the... |
53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100 a. Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places. b. Calculate the following to one decimal place using a TI-83+ or TI-84 calc... |
the exam scores (IQR = 29) are Ds, Cs, and Bs. The box plot also shows us that the lower 25 percent of the exam scores are Ds and Fs. Data Frequency Relative Frequency Cumulative Relative Frequency 33 42 49 53 55 61 63 67 68 69 72 73 74 78 80 83 88 90 Table 2.33.032.032.065.032.065.032.032.032.065.065.032.032.032.032.... |
17 1 6 10 7 0 2 Table 2.34 1 4 7 10 13 16 1 16 49 100 169 256 ¯ 2 x 7.58 7.58 7.58 7.58 7.58 7.58 fm2 Standard Deviation 1 96 490 700 0 512 3.5 3.5 3.5 3.5 3.5 3.5 For this data set, we have the mean, x¯ = 7.58, and the standard deviation, sx = 3.5. This means that a randomly selected data value would be expected to be... |
content/col30309/1.8 Chapter 2 | Descriptive Statistics 127 z = x − x¯ s z = x − μ σ Sample Population Table 2.36 As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is u... |
At least 75 percent of the data is within two standard deviations of the mean. • At least 89 percent of the data is within three standard deviations of the mean. • At least 95 percent of the data is within 4.5 standard deviations of the mean. • This is known as Chebyshev's Rule. A bell-shaped distribution is one that ... |
know? 5. In complete sentences, describe the shape of the histogram. 6. Are there any potential outliers? List the value(s) that could be outliers. Use a formula to check the end values to determine if they are potential outliers. 130 Chapter 2 | Descriptive Statistics Analyze the Data 1. Determine the following value... |
the sample and for by μ) a, ¯ ) is is μ = Sum of all values in the population Number of values in the population median a number that separates ordered data into halves; half the values are the same number or smaller than the median, and half the values are the same number or larger than the median The median may or m... |
Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. ... |
R(1.5) • Q1 – IQR(1.5) 2.4 Box Plots Box plots are a type of graph that can help visually organize data. Before a box plot can be graphed, the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can dis... |
deviation of a population, we would use the population mean, μ, and the formula σ = ∑ (x − μ)2 N or σ = ∑ f (x − μ)2 N. FORMULA REVIEW 2.3 Measures of the Location of the Data 2.5 Measures of the Center of the Data i = ⎛ ⎝ k 100 ⎞ ⎠(n + 1) where i = the ranking or position of a data value, k = the kth percentile, n = ... |
260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610 4. The following data are daily high temperatures in a town for one month: 61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95. For the ne... |
graph that shows the population percentage of competitors from each school. High School Science Competition Population Overall Student Population Alabaster 28.9% Concordia 7.6% Genoa 12.1% Mocksville 18.5% Tynneson 24.2% West End 8.7% Table 2.44 8.6% 23.2% 15.0% 14.3% 10.1% 28.8% 11. Use the data from the David County... |
138 Chapter 2 | Descriptive Statistics 19. Construct a frequency polygon from the frequency distribution for the 50 highest-ranked countries for depth of hunger. Depth of Hunger Frequency 230–259 260–289 290–319 320–349 350–379 380–409 410–439 Table 2.49 21 13 5 7 1 1 1 20. Use the two frequency tables to compare the ... |
293 60,029 61,293 61,467 63,602 63,432 116,128 115,390 118,765 116,128 118,765 119,700 123,711 123,578 Male Total Table 2.54 22. The following data sets list full-time police per 100,000 citizens along with incidents of a certain crime per 100,000 citizens for the city of Detroit, Michigan, during the period from 1961 ... |
have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race? b. The 20th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20th percentile in the context of the situation. c. A bicyclist in the 90th percent... |
not in the top 12 percent of all students in the state. What percentage of students from each high school are eligible in the local context? 33. Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34th percentile. The 34th percentile o... |
data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40 43. Calculate the mean. 142 Chapter 2 | Descriptive Statistics 44. Identify the median. 45. Identify the mode. Use... |
the mean and the median the exact same in this distribution? Why or why not? Figure 2.39 61. Describe the shape of this distribution. Figure 2.40 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 145 62. Describe the relationship between the mode and the... |
Harold rolls the ball and Jaiqua works the stopwatch. The differences in outcomes may be attributed to which type of variability? 72. Twenty people begin the same workout program on the same day and continue for three months. During that time, all participants worked out for the same amount of time and did the same nu... |
.5 Table 2.63 14 32 15 23 2 HOMEWORK 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs 78. Student grades on a chemistry exam were 77, 78, 76, 81, 86, 51, 79, 82, 84, and 99. a. Construct a stem-and-leaf plot of the data. b. Are there any potential outliers? If so, which scores are they? Why do you cons... |
. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows: Number of Books Frequency Relative Frequency 0 1 2 3 4 5 6 8 10 12 16 12 8 6 2 2 Table 2.65 Publisher A Number of Books Frequency Relative Fr... |
ogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis. c. Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis. d. Compare the two graphs: i. List two similarities between the graphs. ii. List two differences betwee... |
.2 24.5 24.3 30.1 24.0 21.0 22.5 28.0 Washington, DC 22.2 Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Table 2.71 26.6 29.6 22.7 26.5 28.2 29.6 28.4 29.4 Kentucky Louisiana Maine Maryland 31.3 31.0 26.8 27.1 North Dakota 27.2 Ohio Oklahoma Oregon 29.2 30.4 26.8 Massachusetts 23.0 Pennsylvania 28.6 Michigan... |
percentage think that middle class is from $25,000 to $50,000? c. Construct a histogram of the data. i. Should all bars have the same width, based on the data? Why or why not? ii. How should the < 20,000 and the 100,000+ intervals be handled? Why? d. Find the 40th and 80th percentiles. e. Construct a bar graph of the ... |
down the example. b. What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart? 156 Chapter 2 | Descriptive Statistics 92. Given the following box plots, answer the questions. Figure 2.47 a. In complete sentences, explain why each statement is false. ... |
158 Chapter 2 | Descriptive Statistics 2.5 Measures of the Center of the Data 95. Scientists are studying a particular disease. They found that countries that have the highest rates of people who have ever been diagnosed with this disease range from 11.4 percent to 74.6 percent. Percentage of Population Diagnosed Numb... |
quartile = 1,447.5 FTES • n = 29 years 98. A sample of 11 years is taken. About how many are expected to have an FTES of 1,014 or above? Explain how you determined your answer. 99. Seventy-five percent of all years have an FTES a. at or below ______. b. at or above ______. 100. The population standard deviation = ____... |
to other instruments of the same type? Which cost is the highest when compared to other instruments of the same type? Justify your answer. 109. An elementary school class ran one mile with a mean of 11 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in eight minutes. A j... |
free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 161 112. Santa Clara County, California, has approximately 27,873 Japanese Americans. Table 2.80 shows their ages by group and each age-group's percentage of the Japanese American community. Age-Group Percentage of Community 0–17 18–24 25–3... |
15 d. 35 115. What is the mode? a. 19 b. 19.5 c. 14 and 20 d. 22.65 116. Is this a sample or the entire population? sample a. b. entire population c. neither 117. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows: Number of Movies Frequenc... |
included every team member who ever played for a California-based football team, would the above data be a sample of weights or the population of weights? Why? h. i. Assume the population was a California-based football team. Find i. ii. iii. iv. the population mean, μ, the population standard deviation, σ, and the we... |
Let X = the length (in days) of an engineering conference. a. Organize the data in a chart. b. Find the median, the first quartile, and the third quartile. c. Find the 65th percentile. d. Find the 10th percentile. e. Construct a box plot of the data. f. The middle 50 percent of the conferences last from ________ days ... |
5 b. 80 c. 3 d. 4 125. The number that is 1.5 standard deviations below the mean is approximately ________. a. 0.7 b. 4.8 c. –2.8 d. cannot be determined This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 167 126. Suppose that a publisher conducted a surv... |
gons, and Time Series Graphs Bureau of Labor Statistics, U.S. Department of Labor. (n.d.). Consumer price index. Retrieved from https://www.bls.gov/ cpi/ CIA World Factbook. http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (n.d.). Demographics: Children under the age of 5 years underweight. Available at Centers fo... |
.census.gov/main/www/cen1990.html U.S. Census Bureau. (n.d.). Data. Retrieved from http://www.census.gov/ 2.4 Box Plots West Magazine. (n.d.). Retrieved from https://westmagazine.net/ 2.5 Measures of the Center of the Data CIA World Factbook. r.aspx?t=50&v=2228&l=en (n.d.). Obesity – adult prevalence rate. Available at... |
in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster. b. 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster). 29 When waiting in line at the DMV... |
right. The median is 87.5, and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right. 53 When the data are symmetrical, the mean and median are close or the same. 55 The distribution... |
number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}. 174 Chapter 2 | Descriptive Statistics Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}. Figure 2.60 b. Figure 2.61 This OpenStax b... |
30309/1.8 Chapter 2 | Descriptive Statistics 177 the overall patterns for the graphs are the same. e. Check student's solution. f. Compare the graph for the singles with the new graph for the couples: i. ▪ Both graphs have a single peak ▪ Both graphs display six class intervals ▪ Both graphs show the same general patte... |
50 percent of buyers are more variable than the ages of the lower 50 percent. b. The black sports car is most likely to have an outlier. It has the longest whisker. c. Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a... |
obtain a standard deviation of: s x = 12.95. • The obesity rate of the United States is 10.58 percent higher than the average obesity rate. • Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the disease percentage that is one standard deviation from the mean. The U.S. disease rate is slightl... |
content/col30309/1.8 Chapter 3 | Probability Topics 181 3 | PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction Chapter Objectives By the end of this chapter, the student should be able to do the following: • Understand a... |
hands. Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person in your class rode a bus within the last month and so on. Discuss your ans... |
An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like A and B are commonly used to represent events. For example, if the experiment is to flip three fair coins, event A might be getting at most one head. The probability of an event A is written P(A), and 0 ≤ P(A) ≤ 1.P(... |
runs the red light. Experimental Probability of Event A P(A) = Number of times event A occurs. Total number of trials While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two po... |
the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling c... |
reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) is P(A|B) = P(A AND B) P(B) where P(B) is greater than zero. For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = {2, 3} and B = {2, 4, 6}. P(A|... |
, 13, 14, 15, 16, 17, 18, 19} b. A = {2, 4, 6, 8, 10, 12, 14, 16, 18}, B = {14, 15, 16, 17, 18, 19} c. P(A) = number of outcomes in A number of outcomes in S = 9 19, P(B) = P(A) = number of outcomes in B number of outcomes in S = 6 19 d. The set A AND B contains all outcomes that lie in both sets A and B, so A AND B = ... |
(B|A) = ________; are the probabilities equal? Example 3.2 A fair, six-sided die is rolled. The sample space, S, is {1, 2, 3, 4, 5, 6}. Describe each event and calculate its probability. a. Event T = the outcome is two. b. Event A = the outcome is an even number. c. Event B = the outcome is less than four. d. The compl... |
4 Let’s denote the events M = the subject is male, F = the subject is female, R = the subject is right-handed, L = the subject is left-handed. Compute the following probabilities: a. P(M) b. P(F) c. P(R) d. P(L) e. P(M AND R) f. P(F AND L) g. P(M OR F) h. P(M OR R) 188 Chapter 3 | Probability Topics i. P(F OR L) j. P(... |
52 =.8269 (rounded to four decimal places) P(FandL) P(L) = 0.04 0.13 P(LandF) P(F) = 0.04 0.48 =.3077 (rounded to four decimal places) =.0833 (rounded to four decimal places) 3.2 | Independent and Mutually Exclusive Events Independent and mutually exclusive do not mean the same thing. Independent Events Two events are ... |
drawing blue on the first draw is 4 7. Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now three blue and three white marbles. So, the probability of drawing blue is now 3 6. Removing the first marble without replacing it influences the probabilities =... |
shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit. Three cards are picked at random. a. Suppose you know that the picked cards are Q of spades, K ... |
sample four cards without replacement. Which of the following outcomes are possible? Answer the same question for sampling with replacement. a. QS, 1D, 1C, QD b. KH, 7D, 6D, KH c. QS, 7D, 6D, KS 192 Chapter 3 | Probability Topics Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at t... |
}, P(B AND C) = 0. B and C are mutually exclusive. (B and C have no members in common because you cannot have all tails and all heads at the same time.) • Let D = event of getting more than one tail. D = {TT}. P(D) = 1 4. • Let E = event of getting a head on the first roll. This implies you can get either a head or tai... |
box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events: a. Let F = the event of getting the white ball twice. b. Let G = the event of getting two balls of different colors. c. Let H = the e... |
8 Let event A = learning Spanish. Let event B = learning German. Then A AND B = learning Spanish and German. Suppose P(A) = 0.4 and P(B) =.2. P(A AND B) =.08. Are events A and B independent? Hint—You must show one of the following: • P(A|B) = P(A) • P(B|A) • P(A AND B) = P(A)P(B) Example 3.9 Let event G = taking a math... |
Stax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 | Probability Topics 195 • R = a red marble • G = a green marble • O = an odd-numbered marble • The sample space is S = {R1, R2, R3, R4, R5, R6, G1, G2, G3, G4}. S has 10 outcomes. What is P(G AND O)? Example 3.10 Let event C = taking an E... |
5 8. P(R AND B) = 0. You cannot draw one card that is both red and blue. • P(E) = 3 8. There are three even-numbered cards, R2, B2, and B4. 196 Chapter 3 | Probability Topics • P(E|B = 2 5. There are five blue cards: B1, B2, B3, B4, and B5. Out of the blue cards, there are two even cards; B2 and B4. • P(B|E) = 2 3 are... |
L) = 0.50 • P(F AND L) = 0.45 • P(L|F) = 0.75 NOTE The choice you make depends on the information you have. You could use the first or last condition on the list for this example. You do not know P(F|L) yet, so you cannot use the second condition. Solution 1 Check whether P(F AND L) = P(F)P(L). We are given that P(F AN... |
T6. e. Event A = heads (H) on the coin followed by an even number (2, 4, 6) on the die. A = {________}. Find P(A). f. Event B = heads on the coin followed by a three on the die. B = {________}. Find P(B). g. Are A and B mutually exclusive? Hint—What is P(A AND B)? If P(A AND B) = 0, then A and B are mutually exclusive... |
d. Are F and S mutually exclusive? e. Are F and S independent? 3.3 | Two Basic Rules of Probability In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not. The Multiplication Rule If A and B are two event... |
A OR B) = P(A) + P(B) − P(A AND B) becomes P(A OR B) = P(A) + P(B). Draw one card from a standard deck of playing cards. Let H = the card is a heart and S = the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or... |
Topics P(A OR B) = P(A) + P(B) − P(A AND B) =.65 +.65 −.585 =.715 Carlos makes either the first goal or the second goal with probability.715. c. Are A and B independent? Solution 3.15 c. No, they are not, because P(B AND A) =.585. P(B)P(A) = (.65)(.65) =.423.423 ≠.585 = P(B AND A) So, P(B AND A) is not equal to P(B)P(... |
.8 Chapter 3 | Probability Topics 201 Solution 3.16 c. There are 40 advanced swimmers who practice four times per week, so the probability is 40 150. d. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and being an intermediate swimmer mutually excl... |
S) =.25(.65) =.1625 b. P(M OR S) = P(M) + P(S) − P(M AND S) =.2 +.65 −.1625 =.6875 c. No, P(M|S) =.25 and P(M) =.2. d. No, P(M AND S) =.1625. 202 Chapter 3 | Probability Topics 3.17 A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) =.40, ... |
) =.85; P(N|B) =.02. So, P(N|B) does not equal P(N). f. Are having the disease and testing negative mutually exclusive? Solution 3.18 f. No. P(B AND N) =.0029. For B and N to be mutually exclusive, P(B AND N) must be zero. 3.18 A school has 200 seniors of whom 140 will be going to college next year. Forty will be going... |
.5. a. Find P(B′). b. Find P(D AND B). c. Find P(B|D). d. Find P(D AND B′). e. Find P(D|B′). 3.4 | Contingency Tables A two-way table provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a contingency table. The table help... |
� ⎠ = number who had no violation number in study = 685 755 ≈.9073. c. Find the number of participants who satisfy both conditions. P(Person had no violation in the last year AND uses a cell phone while driving) = number who had no violation AND uses cell phone while driving number in study = 280 755 ≈.3709 d. To find ... |
coastline and 16 prefer hiking near lakes and streams. So, we know there are 45 − 18 − 16 = 11 female students who prefer hiking on mountain peaks. Continue reasoning in this way to complete the table. 206 Chapter 3 | Probability Topics Sex The Coastline Near Lakes and Streams On Mountain Peaks Total Female 18 Male To... |
for free at http://cnx.org/content/col30309/1.8 Chapter 3 | Probability Topics 207 Solution 3.21 d. 1. P(F) = 45 100 2. P(P) = 25 100 3. P(F AND P) = number of hikers that are both female AND prefers mountain peaks number of hikers in study = 11 100 4. P(F OR P) = P(F) + P(P) − P(F AND P) = 45 100 + 25 100 - 11 100 = ... |
Choice 1 12 3 12 4 12 1 6 1 6 2 6 19 60 41 60 1 b. What is the probability that Alissa does not catch Muddy? Solution 3.22 b. 41 60 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 | Probability Topics 209 c. What is the probability that Muddy chooses Door One OR Door Two given... |
(2010 AND Crime + 2,852.9 4,520.7 − 701 4,520.7 =.7165, d. 113.7 511.8 =.2222, e. 314.7 1,222.2 =.2575 3.23 Table 3.11 relates the weights and heights of a group of individuals participating in an observational study. 210 Chapter 3 | Probability Topics Ages Tall Medium Short Totals 28 51 25 14 28 9 Under 18 18 20 12 18... |
second set of branches represents the second draw. Regardless of the choice on the first draw, there are again eight ways to draw a blue marble and 3 ways to draw a red one. Read down each branch to see the total number of possible outcomes. For example, there are 8 ways to get a blue marble on the first draw, and eig... |
(RR) = P(R)P(R) = ⎛ ⎝ 3 11 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 3 11 = 9 121. 212 Chapter 3 | Probability Topics c. Calculate P(RB OR BR). Solution 3.24 c. The tree diagram shows that there are 24 ways to draw RB and 24 ways to draw BR. There are 121 possible outcomes, so P(RB or BR) = 24 + 24. = 48 121 121 The events RB and BR are mutually ex... |
(BB). Solution 3.24 f. P(BB) = 64 121 g. Using the tree diagram, calculate P(B on the 2nd draw GIVEN R on the first draw). Solution 3.24 g. P(B on 2nd draw|R on 1st draw) = 8 11 There are 9 + 24 outcomes that have R on the first draw (9 RR and 24 RB). The sample space is then 9 + 24 = 33. Twenty-four of the 33 outcomes... |
�� ⎝ 8 10 ⎞ ⎠ + (________)(________) = 48 110 Solution 3.25 b. P(RB OR BR) = P(RB) + P(BR) = P(R on 1st) P(B on 2nd) + P(B on 1st) P(R on 2nd) = ⎛ ⎝ 3 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 8 10 + ⎛ ⎝ 8 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 3 10 = 48 110 c. Because this is a conditional probability, we restrict the sample space to consider only those outcomes that ... |
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