STAT 2023 - Okstate - Week 2
If E is an experimental outcome, the P(E) denotes the probability that E will occur. P(E) must satisfy the following conditions
- 0≤P(E) ≤ 1. If E can never occur, then P(E) = 0. If E is certain to occur, the P(E) = 1 - the probabilities of all outcomes must sum to 1
If the population is described by a normal distribution (symmetric "bell-shaped" curve), with mean μ and standard deviation σ, then...
- 68.26% of the population lie within one standard deviation of the mean, that is [μ−σ,μ+σ]. - 95.44% of the population lie within two standard deviations of the mean, that is [μ−2σ,μ+2σ]. - 99.73% of the population lie within three standard deviations of the mean, that is [μ−3σ,μ+3σ].
Define: box and wicker plot
- Box: Q1, Md, and Q3 - Fences: 1.5 x IQR away from the quartiles - Whiskers: dashed lines that below Q1 and above Q3, to the smallest/largest measurement within fences - Stars (*): outliers beyond fences
Define: probability distributions
A probability model describing random variable
Q: If events A and B are mutually exclusive, calculate P(A|B). a.) Cannot be determined b.) 0 c.) 1 d.) 0.50
A: 0 Mutually exclusive events have intersection = 0. Therefore, the conditional probability is also 0.
Q: Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. If one item is drawn from each container, what is the probability that both items are not defective? a.) 0.3750 b.) 0.3846 c.) 0.1500 d.) 0.6154 e.) 0.2000
A: 0.3750 (5/8)(3/5) = .375
Q: Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. If one item is drawn from each container, what is the probability that only one of the items is defective? a.) 0.2250 b.)0.6250 c.)0.2500 d.)0.4750 e.)0.1500
A: 0.4750 Container 1 P(Defective) = 3/8 = .375; Container 2 P(Defective) = 2/5 = .400 P(Both defective) = .375 × .400 = .150; P(Neither defective) = (1 - .375) × (1 - .40) = .375 P(Only one defective) = 1 - (.150 + .375) = .475
Q: If A and B are independent events, P(A) = .2, and P(B) = .7, determine P(A∪B). a.) 0.90 b.) 0.14 c.) 0.76 d.) 0.50 e.) 0.24
A: 0.76 P(A∪B)=P(A) + P(B)−P(A∩B) P(A∪B)=(.7)+(.2)−(.7)(.2)=.76
Q: The average lateness for one of the top airline companies is 10 minutes. The variance of the lateness measure is calculated as 9. An airplane arrived 13 minutes after the stated arrival time. Calculate the z-score for the lateness of this particular airplane. a.) .33 b.) .58 c.) 1.33 d.) .44 e.) 1.00
A: 1.00 Z=(13−109)/√9 = 1
Q: In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70): 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68.What is the variance? a.) 18 b.) 4.73 c.) 22.40 d.) 324 e.) 6.76
A: 22.40 62, 67, 68, 68, 68, 68, 70, 71, 73, 75, 80 (62 − 70)= −8 squared = 64 (67 − 70)= −3 squared = 9 (68 − 70)= −2 squared = 4 (68 − 70)= −2 squared = 4 (68 − 70)= −2 squared = 4 (68 − 70)= −2 squared = 4 (70 − 70)= 0 squared = 0 (71 − 70)= 1 squared = 1 (73 − 70)= 2 squared = 4 (75 − 70)= 5 squared = 25 (80 − 70)= 10 squared = 100 (64 + 9 + 4 + 4 + 4 + 4 + 0 + 1 + 4 + 25 + 100) = 224 / 10 = 22.40
Q: In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70): 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68.What is the standard deviation? a.) 18 b.) 4.73 c.)22.40 d.) 324 e.) 6.76
A: 4.73 You take the square root of 22.40 = 4.73
Q: What is the probability of rolling a seven with a pair of fair dice? a.) 6/36 b.) 3/36 c.) 1/36 d.) 8/36 e.) 7/36
A: 6/36 Set up sample spaces: 36 total; 6 have combination adding to 7.
Q; In a statistics class, 10 scores were randomly selected, with the following results:74, 73, 77, 77, 71, 68, 65, 77, 67, 66.What is the 10th percentile? a.) 65.5 b.) 67.3 c.) 66.75 d.) 73.85 e.) 77.0
A: 65.5 65, 66, 67, 68, 71, 73, 74, 77, 77, 77 (10/100)n = (10/100)(10) = 1th position = 65 + 66 / 2 = 65.5
Q: In a hearing test, subjects estimate the loudness (in decibels) of a sound, and the results are: 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68.What is the mode? a.) 70 b.)75 c.) 68 d.) 71 e.) 80
A: 68 68 appears most frequently in this data set.
Q: In a statistics class, the following 10 scores were randomly selected: 74, 73, 77, 77, 71, 68, 65, 77, 67, 66.What is the mean? a.) 71.5 b.)72.0 c.)77.0 d.)71.0 e.) 73.0
A: 71.5 74 + 73 + 77 + 77 + 71 + 68 + 65 + 77 + 67 + 66 ÷ 10 = 71.5
Q: In a statistics class, the following 10 scores were randomly selected: 74, 73, 77, 77, 71, 68, 65, 77, 67, 66.What is the median? a.) 71.5 b.) 72.0 c.) 77.0 d.) 71.0 e.) 73.0
A: 72.0 65 + 66 + 67 + 68 + 71 + 73 + 74 + 77 + 77 + 77 (71 + 73) ÷ 2 = 72
Q: The simultaneous occurrence of events A and B is represented by the notation ________. a.) AUB b.) A|B c.) AnB d.) B|A
A: AnB This is when events A and B occur at the same time.
Q: If a population distribution is skewed to the right, then, given a random sample from that population, one would expect that the ________. a.) median would be greater than the mean b.) mode would be equal to the mean c.) median would be less than the mean d.) median would be equal to the mean
A: Median would be less than the mean The median in this case would be a better representation of the population - showing where most of the numbers congregate.
Q: Consider a standard deck of 52 playing cards, a randomly selected card from the deck, and the following events: R = red, B = black, A = ace, N = nine, D = diamond, and C = club. Are R and C mutually exclusive? a.) Yes, mutually exclusive b.) No, not mutually exclusive
A: Yes, mutually exclusive Clubs are only black card so you cannot have a red club.
Q: If events A and B are independent, then the probability of simultaneous occurrence of event A and event B can be found with ________. a.) P(A)•P(B) b.) P(A)•P(B|A) c.) P(B)•P(A|B) d.) All of these choices are correct
A: all of these choices are correct All of these notations depict independent events.
Q: Which of the following is influenced the least by the occurrence of extreme values in a sample? a.) mean b.)median c.) geometric mena d.) weighted mean
A: median The median looks at the middle of a sample or population and does not take into effect low or high values.
Q: Determine whether these two events are mutually exclusive: consumer with an unlisted phone number and a consumer who does not drive. a.) mutually exclusive b.) not mutually exclusive
A: not mutually exclusive In this case, one does not determine the other.
Q: The set of all possible outcomes for an experiment is called a(n) ________. a.) sample space b.) event c.) experiment d.) probability
A: sample space We must define the sample space outcomes so that on any single repetition of the experiment, only one sample space outcome will occur.
Define: the multiplicative rule
By definition: P(A|B)= P(A∩B) / P(B) then, P(A ∩ B) = P(A | B)P(B)
Define: complement
Given an event A, the complement of A is the event consisting of all sample space outcomes that are not in set/event A. The complement of A is denoted by A ̄ interpreted as "A does not occur". Probability rule: P(A ̄) = 1 - P(A)
Define: intersection
Given two events A and B, the intersection of A and B is the event that occur if both A and B simultaneously occur. The intersection is denoted by A ∩ B.
Define: Union
Given two events A and B, the union of A and B is the event that occur if A or B or both occur. The intersection is denoted by A ∪ B.
Define: addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Define: conditional probability
The probability of the event A, given the condition that the event B has occurred, is written as P(A | B), pronounced "the probability of A given B". This is called the conditional probability of A given B.
Define: independent events
Two events A and B are independent if and only if P(A|B)=P(A) or P(B |A)=P(B). That is, the probability of A is not affected by the event B.
Define: mutually exclusive
Two events A and B are mutually exclusive if they have no outcomes in common, that is, the two events cannot occur simultaneously, and P(A ∩ B) = 0
Define: probability model
a mathematical representation of random phenomenon
Define: probability
a measure of the chance that an experimental outcome will occur when an experiment is carried out.
Define: central tendency
a measure that represents the center or middle of the data. ex: mean
Define: population parameter
a number calculated from all the population measurements that describes some aspects of the population
Define: sample statistic
a number calculated using the sample measurements that describes some aspects of the sample
Define: event
a set of one or more sample space outcomes
Define: median
a value such that 50% of all measurements, after having been arranged in numerical order, lie above (or below) it, denoted by Md.
Define: pth percentile
a value such that p percent of the measurements fall below the value and (100-p) percent of the measurements fall above the value. step 1: arrange the measurements in increasing order step 2: calculate the index i = (p/100) x n step 3: (a). If i is not an integer, round up to the next integer greater than i, which denotes the position of the pth percentile. (b). If i is an integer, the pth percentile is the average of the measurement at position i and the measurement at position i + 1. first quartile Q1 = 25 percentile second quartile q2 is the 50th percentile or the median third quartile is the 75th percentile
Define: random variable
a variable whose value is numeric and is determined by the outcome of an experiment
Define: experiment
any process of observation with an uncertain outcome
Define: mean
average, denoted by μ for the population and x̄ for the sample. If the population measurements are x1, x2 ..., xn, then the mean is: x̄/μ= x1+ x2+...+xn ÷ N
Define: z score
indicates the relative location of a value in the population or sample. Z = (x - mean)÷ standard deviation
Define: standard deviation
is the square root of the variance σ= √σ^2 s= √s^2
Define Interquartile Range (IQR)
q3 - q1
Define: variance
the average of the squared deviations of the individual measurements from the mean. σ^2 = {(x1 −μ)^2 +(x2 −μ)^2 +···+(xN −μ)^2 } ÷ N σ^2 = {(x1 −μ)^2 +(x2 −μ)^2 +···+(xN −μ)^2 } ÷ (n - 1)
Define: sample space outcomes
the experimental outcomes in the sample space
Define: range
the largest measurement minus the smallest measurement. measures the interval spanned by all the data
Define: mode
the measurement that occurs most frequently. If there are two modes, the data is called bimodal. If there are more than two modes, that data is called multimodal.
Define: experiment outcomes
the possible outcomes for an experiment
Define: sample space
the set of all possible outcomes
Define: probability of an event
the sum of the probabilities of the sample space outcomes
