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Statistics Exam 2
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Terms in this set (64)
Experiment
An experiment is an act or process that leads to a single outcome that cannot be predicted with certainty.
Sample point
A sample point(or simple event) is the most basic outcome of an experiment.
Sample Space
The sample space of an experiment is the collection of all its sample points
Probability Rules For Sample Points
(i)Probability of each sample point lies between 0 and 1 (ii) Sum of the probabilities of all sample points of S is equal to 1
Assigning probabilities to sample points (Classic Probability Model) Finite Sample Space: n sample points Equally Likely Points:
each point in sample space S has equal probability
P(A)
sum of the probabilities of all outcomes or sample points that are in the event A
Steps for calculating event probabilities
1.Define the experiment
2. List sample points
3. Assign probabilities to sample points
4. Determine what is in the event
5. Sum the sample point probabilities to get the event probability
Combinations Rule
Choose a sample of size n without replacement from a set of N elements. Then, the number of different samples that can be selected is: (N/n)= N!/n!(N-n)!
Properties of probabilities(P)
Given that P(A) is the probability that event A has occurred,
1.0 < P(A) < 1
2.If P(A) = 1, this means A always happens
3.If P(A) = 0, this mean A never happens
Unions and Intersections
are compound events- they are formed by the composition of 2 or more other events.
Union of events A and B: (A∪B)
Event that occurs if either A or B or both occur on a single performance of an experiment
Intersection of events A and B: (A∩B)
Event that occurs if both A and B occur
Problem solving tips
●List all outcomes of the sample space
●Use tree diagram
●Use cross table
●Use Venn Diagram
●Use laws of probability
●Review all examples presented in class (Redo these examples at home)
Additive Rule:
P(A u B) = P(A) + P(B) - P(A u(upside down) B)
Mutually Exclusive Events
If A u(upside down) B = 0(slashed), then A and B are said to be mutually exclusive and P(A u(upside down) B) = 0.Note: If A and B are Mutually Exclusive, then P(A u B) = P(A) + P(B)
Conditional Probability Formula
To find the conditional probability that event a occurs given that event b occurs, divide the probability that both A and B occur by the probability that B occurs: that is, P(A/B)= P(A Upside down B)/ P(B)
We assume P(B) does not = 0)
Multiplicative Rule
P(B/A)= P(A Upside down B)/P(A) -> P(A Upside down)= P(A)P(B/A)
P(A/B)= P(A Upside down B)/ P(B) -> P(A Upside down B)= P(B)P(A/B)
Independent Events
Events A and B are independent events if the occurrence of B does not alter the probability that A has occurred P(A/B)= P(A)
Events that are not independent are said to be dependent
Random Variable
a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point
Discrete
Random variables that assume a countable number of values
Continuous
Random variables that can assume values corresponding to any of the points contained in an interval
probability distribution of the random variable
A formula (or a set of expressions), a table, or a graph that gives these information of a discrete random variable
Probability distribution of a discrete random variable
a graph, table or formula that specifies the probability associated with each possible value that the random variable can assume
Requirements for Probability distribution of a discrete random variable
p(x) is greater than or equal to 0 for all values of x
summation of p(x) = 1
summation of p(x) is over all possible values of x
mean or expected value
u=E(x)= summation xp(x)
Variance
o^2= E[(x-u)^2]= summation(x-u)^2p(x)=summationx^2p(x)-u^2
Standard Deviation
of a discrete random variable is equal to the square root of the variance of 0= sqrto^2
Bernoulli Experiments:
experiments that result in only two possible responses (success/failure)
Bernoulli distribution
Let X represents the number of success in a Bernoulli experiment. Assume that the probability of success is p. Then the probability distribution of X can be shown as
x 0 1p(x)q pq=1-p
Binomial Random Variable
Definition IF
1) The experiment consists of n identical trials
2)There are only 2 possible outcomes on each trial. We will denote one outcome by s(success) and f(failure)
3)The probability of s remains the same from trial to trial. This probability denoted by p, and the probability of f is denoted by q=1-p
4) The trials are independent
The number of S's in n's trial is a binomial random variable.
Binomial Probability Distribution
p(x)= (n/x)p^xq^n-x (x=0,1,2,...........,n)
Where
p=probability of a success on a single trial
q=1-p
n= number of trials
x= number of successes in n trials
n-x = number of failures in n trials
(n/x)= n!/x!(n-x)!
Mean, Variance, and Standard Deviation for Binomial Random Variable
Mean: u=np
Variance: o^2=npq
Standard Deviation: o= sqrt(npq)
u=np o^2=npq
The Poisson Random Variable
The Poisson distribution is useful in describing the number of events that will occur during a specific period of time or in a specific area or volume.
Characteristics of The Poisson Random Variable
1) The experiment consists of counting the number of times a certain event occurs during a given unit of time or in a given area or volume ( or weight, distance, and any other unit of measurement)
2) The probability that an event occurs in a given unit of time, area, or volume is the same for all units
3) The number of events that occur in one unit of time, area, or volume is independent of the number that occur in other units.
4) The mean or expected number of events in each unit is denoted by greek letter lamba
Probability Distribution
p(x)= lamba^x e^-lamba/x!
Continuous Random Variable
can assume any numerical value within some interval or intervals
5.1 continuous probability distributions
A continuous random variable can assume any numerical value within some interval or intervals.
Assume X is a continuous random variable.
The probability distribution of x is characterized by its probability density function f(x), a function of x.
Probability density function for continuous random variables possess different shapes, depending on the relative frequency distribution of real data that it models.
Discrete vs. Continuous
With discrete distributions, our probabilities are massed at the possible values of x.
With continuous distributions, our probabilities are found by looking for the area under the probability density function between 2 points.
Defining p(a < x < b)
The areas under f(x) above x-axis correspond to probabilities for x. For example, the area A beneath the curve between the two points a and b is the probability that x assumes a value between a and b, P(a < x < b).
Defining p(a < x < b)
Remark: For any value c, the area under the curve on single point c is 0. Therefore, the probability associated with c is equal to 0, i.e. P(x = c) = 0.
Hence,
P(a < x < b) = P(a ≤ x ≤ b).
uniform probability distribution
Continuous random variables that have equally likely outcomes over their range of possible values possess a
Assume X is a uniform random variable on [c, d].
Then, the probability density function
𝑓(𝑥) = 1/(d-c) , (𝑐 ≤ 𝑥 ≤ 𝑑)
, 𝑐 ≤ 𝑎 < 𝑏 ≤ 𝑑
Mean: 𝜇 = (𝑐+𝑑)/2
Standard deviation: 𝜎 = (𝑑−𝑐)/√12
One of the most commonly observed continuous random variables has a bell-shaped (or mound-shaped) probability distribution.
It is known as a normal random variable.
Its probability distribution is called a normal distribution.
The normal distribution is symmetric about its mean µ.
Its spread is determined by the value of its standard deviation 𝜎.
Normal Probability - Summary
To find the probability for a normal random variable
Convert x values to z values
Draw the standard normal curve
Mark z scores on the horizontal axis
Shade the desired area and use normal table to find the area
To find the Z-score for a given area/probability
Draw standard normal curve
Identify Z on the horizontal axis and shade the given area
Identify the area between 0 and desired z score
Use normal table to find z score for the given area
To find the X-score for a given area/probability
First find the z score (described above) for the given area, then find x using X = μ + σ z
1. It is estimated that 40% of clowns like scaring children. A sample of 15 clowns are indicted. Assuming the binomial requirements are met, what is the probability that exactly 7 of these clowns like scaring children?
A.0.610
B.0.213
C.0.177
D.0.787
E.0.390
C
2. The time it takes a bachelor/bachelorette to break off their engagement after the finale episode is normally distributed with an average of 268 days and a standard deviation of 15 days. What proportion of engagements last less than 250 days?
A.0.8849
B.0.3856
C.0.1151
D.0.6211
C
3. The contingency table shows the results of a survey that asked 200 rappers about their microphone preferences.
A. B. Total
East Coast 51. 34. 85
West Coast. 47. 68. 115
Total. 98. 102. 200
Suppose a rapper is selected at random. What is the probability they are East Coast and prefer A?
A.132/200
B.183/200
C.51/200
D.187/200
E.98/200
C
4. The time it takes for you to read this question is a __________ random variable; the number of marshmallows you can fit into your mouth is a __________ random variable.
A.discrete; continuous
B.continuous; discrete
C.continuous; continuous
D.discrete; discrete
B
5. Suppose event A = get an A on Exam 2 and event B = eat Bojangles the morning of Exam 2. If event A and event B are independent, which of the following must be true?
A.P(A) = P(B)
B.P(A|B) = P(A)
C.P(A|B) = P(B)
D.P(AꓵB) = P(AꓴB)
E.P(B|A) = P(A)
B
6. Which of the following characteristics is false regarding the binomial distribution?
A.There are only two possible outcomes on each trial
B.The trials are dependent
C.The random variable x identifies the number of successes in n trials
D.The probability of success remains constant from trial to trial
B
7. NFL quarterback Tom Brady likes to have his game balls inflated so that the mean is 12.51 psi with a standard deviation of 0.46 psi. If this measurement (psi is a pressure unit) follows a normal distribution, determine what proportion of his game balls have psi more than 11.93.
A.0.1038
B.0.3962
C.0.8962
D.0.6038
C
8. Find a value of the standard normal random variable z, called z_0, such that P(z≥ z_0 )=0.70.
A.-0.98
B.-0.47
C.-0.81
D.-0.53
D
9. Consider this probability distribution. Find the probability that x is no greater than 3.
x. 0. 1 2. 3 4 5.
p(x) 0.26 0.49 ?? 0.06 0.04 0.02
A.0.94
B.0.13
C.0.06
D.0.88
E.0.12
A
10. The contingency table shows the results of a survey that asked 200 rappers about their microphone preferences.
A. B. Total
East Coast 51. 34. 85
West Coast. 47. 68. 115
Total. 98. 102. 200
Given that a rapper is East Coast, what is the probability they prefer microphone A?
A.85/98
B.98/200
C.85/200
D.51/85
D
11. It is estimated that 80% of UF graduates are still in love with Tim Tebow. A sample of 4 UF graduates are taken into custody. Assuming the binomial requirements are met, what is the probability that exactly 2 of them are still in love with Tim Tebow?
A.0.154
B.0.500
C.0.640
D.0.320
E.0.460
A
12. The time it takes for an STA2023 student to answer a uniform distribution question is uniformly distributed over the interval 30 seconds to 90 seconds. What is the probability it takes a student less than 40 seconds to answer a uniform distribution question?
A.0.6677
B.0.8571
C.0.4212
D.0.3146
E.0.1667
E
13. The salaries of dogcatchers are normally distributed with a mean of $28,000 and a standard deviation of $3,000. What salary cuts off the poorest 10% of dogcatchers?
A.$23,065
B.$31,840
C.$24,160
D.$32,935
C
14. The contingency table shows the results of a survey that asked 200 rappers about their microphone preferences.
A. B. Total
East Coast 51. 34. 85
West Coast. 47. 68. 115
Total. 98. 102. 200
Suppose a rapper is selected at random. What is the probability they are East Coast or prefer A?
A.132/200
B.183/200
C.51/200
D.187/200
E.98/200
A
15. The office Keurig is broken and is uniformly dispensing between 1 fl. oz. and 3 fl. oz. of fluid. What is the mean amount of fluid dispensed?
A.1.15
B.2
C.2.5
D.0.5
B
16. A section of an exam contains two multiple-choice questions, each with three answer choices (listed "A", "B", and "C"). List all the outcomes of the sample space.
A.{AB, AC, BA, BC, CA, CB}
B.{AA, AB, AC, BA, BB, BC, CA, CB, CC}
C.{AA, AB, AC, BB, BC, CC}
D.{A, B, C}
C
17. True or false: The expected value of a discrete random variable must be one of the values in which the random variable can result.
A.True
B.False
A
18. The probability a UCF student has received a parking ticket is 0.17. In a sample of 40 students, assuming the binomial distribution requirements are met, what is the mean and standard deviation of the number of students who have received a parking ticket?
A.mean: 6.8; standard deviation: 2.38
B.mean: 6.8; standard deviation: 2.61
C.mean: 40; standard deviation: 2.61
D.mean: 40; standard deviation: 2.38
A
19. According to a television doctor 80% of college students procrastinate. Assuming the binomial modeling requirements are met, for a random sample of 8 college students, what is the probability less than 5 procrastinate?
A. 0.056
B. 0.203
C. 0.944
D. 0.640
E. 0.500
A
20. The weight of a sperm whale's brain is normally distributed with a mean of 15.5 pounds and a standard deviation of 3.6 pounds. What is the probability that a randomly selected sperm whale has a brain that weighs between 10 and 20 pounds?
A.0.9973
B.0.1056
C.0.4040
D.0.3944
E.0.8314
E
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