Download Exercise 1 - OCVTS MATES-STAT

NAME: __________________________________ BLOCK:___________DATE:________
5.1 Random Variables and Probability Distributions
TERMINOLOGY
Statistical experiment
Random variable, x
Discrete random variable
Continuous random variable
Discrete probability
distribution
DEFINITION
Any process by which an observation or measurement is
made.
A quantity being measured in a statistical experiment.
A random variable that can take on only countable
values (e.g. 0,1,2,3 . . . )
A random variable that can take on any value in an
interval.
Lists each possible value the random variable can
assume, together with its probability.
Exercise 1:
An industrial psychologist administered a personality inventory test for passive-aggressive
traits to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely
passive and 5 extremely aggressive. A score of 3 indicated neither trait. Construct a
probability distribution for the random variable x. Then graph the distribution using a
histogram.
Score, x
1
2
3
4
5
TOTAL
f
P(x)
24
33
42
30
21
150
0.16
0.22
0.28
0.20
0.14
1.00
Title:
1
NAME: __________________________________ BLOCK:___________DATE:________
Formula
Comments
Mean of discrete probability distribution.
𝜇 = ∑ 𝑥 ∙ 𝑃(𝑥)
𝜎 2 = ∑(𝑥 − 𝜇)2 ∙ 𝑃(𝑥)
𝜎 2 = ∑[ 𝑥 2 ∙ 𝑃(𝑥)] − 𝜇 2
𝜇 = 𝐸(𝑥) = ∑ 𝑥 ∙ 𝑃(𝑥)
Variance of discrete probability distribution. Defining
formula.
Variance of discrete probability distribution.
Computational formula.
Expected value of x. Mean of discrete probability
distribution.
Exercise 2: Find the mean of the personality inventory test. Write an interpretive
statement.
Score, x
f
P(x)
1
2
3
4
5
24
33
42
30
21
TOTAL
150
x * P(x)
0.16
0.22
0.28
0.20
0.14
1.00
0.16
0.44
0.84
0.80
0.70
2.94
Mean Score: _________________
Interpretive Statement:________________________________________________________
Exercise 3: Find the standard deviation of the distribution. Use the defining formula.
L1
Score, x
L2
f
P(x)
1
24
2
33
3
42
4
30
5
21
TOTAL
150
(x -µ)2
(x -µ)2 * P(x)
0.16
0.22
0.28
0.20
0.14
1.00
2
NAME: __________________________________ BLOCK:___________DATE:________
Exercise 4: Find the standard deviation of the distribution. Use the computational formula.
Score, x
f
P(x)
x * P(x)
1
24
0.16
2
33
0.22
3
42
0.28
4
30
0.20
5
21
0.14
TOTAL
150
1.00
x2 * P(x)
Exercise 5: Find the standard deviation of the distribution. Use the graphing calculator
1-Var Stats L1, L2
𝑥̅ = 2.94
𝜎 = 1.271377206
Exercise 6: If you bet $1 in Kentucky’s Pick 4 Lottery, you either lose $1, or gain $4999.
The winning prize is $5000, but your $1 bet is not returned, so the net gain is $4999. The
game is played by selecting a four-digit number between 0000 and 9999. If you bet $1 on
1234, what is your expected gain (or loss)?
To find the gain for each prize, subtract the price of the ticket from the value
of the prize.
Description
Winner
Losing Ticket
Payout
Cost
$5,000
$1
$0
$1
Gain (x)
Fraction
Decimal
P(x)
P(x)
Expected
Value
X*P(X)
TOTAL
3
NAME: __________________________________ BLOCK:___________DATE:________
5.2 Binomial Distributions
Variable
n
p
q
r
Probability Description
Exactly r successes in n
trials
Binomial Distribution
Meaning
number of trials
probability of success
probability of failure
number of successes in n trials
Probability Expression
𝑃(𝑟) =
𝐶𝑝
𝑛 𝑟
𝑟 𝑛−𝑟
𝑞
𝑛!
𝑝𝑟
(𝑛 − 𝑟)! 𝑟!
𝑃(𝑟 ≤ 𝑟𝑚𝑎𝑥 ) = 𝑃(0) + 𝑃(1) +
𝑃(2) + ⋯ +
𝑃(𝑟𝑚𝑎𝑥 ) 𝑏𝑖𝑛𝑜𝑚𝑐𝑑𝑓(𝑛, 𝑝, 𝑟𝑚𝑎𝑥 )
TI-84 Command
𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓(𝑛, 𝑝, 𝑟)
=
At most 𝑟𝑚𝑎𝑥 successes
in n trials
𝑏𝑖𝑛𝑜𝑚𝑐𝑑𝑓(𝑛, 𝑝, 𝑟𝑚𝑎𝑥 )
Exercise 1: Microfracture knee surgery has a 75% chance of success on patients with
degenerative knees. The surgery is performed on three patients. Find the probability of the
surgery being successful on exactly two patients, Use the binomial formula.
3
𝑛 = 3, 𝑝 = , 𝑟 = 2
4
=2
Exercise 2: Repeat Exercise 1 using the graphing calculator binompdf(n,p,r) function.
Exercise 3: Create a binomial probability distribution for microfracture knee surgery
performed on three patients. Use n=3, p = 0.75. Use the binompdf(n,p,r) function. Place r
values in L1, and L2 = binompdf(3,0.75,L1).
r
P(r)
0
1
2
3
4
NAME: __________________________________ BLOCK:___________DATE:________
Exercise 4: A survey indicates that 41% of women in the U.S. consider reading their
favorite leisure-time activity. You randomly select four U.S. women and ask them if reading
is their favorite leisure-time activity. Find the probability that at least two of them respond
yes.
Solution: n=4, p=0.41
5.3 Additional Properties of Binomial Distributions
Exercise 1: About thirty percent of working adults spend less than 15 minutes each way
commuting to their jobs. You randomly select six working adults. What is the probability that
exactly three of them spend less than 15 minutes each way commuting to work? Use a table
to find the probability. Use a table to find the probability.
Solution: n=6, p=0.30, r=3
The probability that exactly three of the six workers spend less than 15 minutes each way
commuting to work is 0.185.
Exercise 2: Fifty-nine percent of households in the U.S. subscribe to cable TV. You
randomly select six households and ask each if they subscribe to cable TV. Construct a
probability distribution for the random variable x. Then graph the distribution.
Solution: Solution:
n = 6, p = 0.59, q = 0.41
Find the probability for each value of r
r
P(r)
0
1
2
3
4
5
6
Graph: Binomial Distribution
5
NAME: __________________________________ BLOCK:___________DATE:________
Binomial Distribution Properties
Comments
Mean of binomial probability distribution. Expected
value.
Variance of binomial probability distribution. Expected
value of squares of deviations.
Standard deviation of binomial probability distribution.
Formula
𝜇 = 𝑛𝑝
𝜎 2 = 𝑛𝑝𝑞
𝜎 = √𝑛𝑝𝑞
Exercise 3: In Pittsburgh, about 56% of the days in a year are cloudy. Find the mean,
variance, and standard deviation for the number of cloudy days during the month of June.
Interpret the results and determine any unusual values. Values that are more than 2.5 standard
deviations from the mean are considered unusual.
Solution:
Mean:
Variance:
Standard Deviation:
Unusual Values:
Exercise 4: If 22% of U.S. households have a Nintendo game, compute the probability that
a) exactly 5 of 12 randomly chosen families will have Nintendo games.
b) at most, 5 randomly chosen families will have Nintendo games
n=
p=
12
0.22
L1
r
L2 =binompdf(12,0.22,L1)
P(r)
0
1
2
3
4
5
TOTAL
a)
b)
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NAME: __________________________________ BLOCK:___________DATE:________
Exercise 5: If 22% of U.S. households have a Nintendo game, compute the probability that
at most, 5 of 12 randomly chosen families will have Nintendo games. Use the binomial
cumulative density function.
Exercise 6: Suppose a satellite requires 3 solar cells for its power, the probability that any
one of these cells will fail is 0.15, and the cells operate and fail independently. We want to
find the smallest number of cells the satellite should have to be 99% sure that there will be
adequate power (i.e. at least three cells work). How many cells do we need to meet the
specification that 𝑟 ≥3?
n=
p=
?
0.85
n
x
P(r >=3 ) = 1 - P( r <=2)
Y1 = 1 - binomcdf(x,0.85,2)
Expected Successes
µ = np
3
4
5
6
7
Strategy:
Define function. Create table. Guess + check.
Show at least 3 guesses and three checks.
Conclusion:
We will need at least _____ cells/satellite to be ___%
sure that all satellites will be operational
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NAME: __________________________________ BLOCK:___________DATE:________
5.4 The Geometric and Poisson Probability Distributions
Exercise 1
Susan is taking Western Civilization on a pass/fail basis. Historically, the passing rate for
this course has been 77% each term. Let n = 1, 2, 3 … represent the number of times a
student takes this course until a passing grade is received. Assume the attempts are
independent.
a) Construct a tree diagram for n = 1, 2, and 3 attempts. List the outcomes and their
associated probabilities.
b) What is the probability that Susan passes on the first try?
c) What is the probability that Susan passes on the second try?
d) What is the probability that Susan passes on the third try?
e) What is the probability that Susan passes on the nth try?
n
Factors
P(n)
1
2
3
Tree Diagram:
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NAME: __________________________________ BLOCK:___________DATE:________
Exercise 2: Susan is taking Western Civilization on a pass/fail basis. Historically, the
passing rate for this course has been 77% each term. Let n = 1, 2, 3 … represent the number
of times a student takes this course until a passing grade is received. Assume the attempts
are independent. Use a graphing calculator to compute the following probabilities:
Solution: 𝑝 = 0.77, 𝑞 = 0.23
Probability Description
Probability
Expression
TI-84 Command
success on exactly the 3rd try
success on or before the 3rd try
success after the 3rd try
Exercise 3: From experience, you know that the probability that you will make a sale on
any given telephone call is 0.23. Find the probability that your first sale on any given day
will occur on your fourth or fifth sales call.
Solution: 𝑝 = 0.23, 𝑞 = 0.77
Probability Description
success on exactly the 4th try
success on exactly the 5th try
success on the 4th or 5th try
Probability Expression
TI-84 Command
Poisson Distribution
Probability Description Probability Expression
TI-84 Command
−𝜆
𝑟
The probability of exactly r
𝑝𝑜𝑖𝑠𝑠𝑜𝑛𝑝𝑑𝑓(𝜆, 𝑟)
𝑒 ∙𝜆
𝑃(𝑟)
=
successes (r = 0, 1, 2, 3 …) over
𝑟!
the same interval of time
(volume, area, etc.)
The probability of at most
𝑝𝑜𝑖𝑠𝑠𝑜𝑛𝑐𝑑𝑓(𝜆, 𝑟)
𝑃(𝑟 ≤ 𝑟𝑚𝑎𝑥 )
𝑟𝑚𝑎𝑥 successes
Parameter
Formula
Mean
𝜇= 𝜆
Variance
𝜎2 = 𝜆
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NAME: __________________________________ BLOCK:___________DATE:________
Exercise 4 The mean number of accidents per month at a certain intersection is 3. What is
the probability that in any given month four accidents will occur at this intersection?
Solution:
Exercise 5 On average boat fishermen on Pyramid Lake catch 0.667 fish per hour.
Suppose you decide to fish the lake on a boat for 7 hours.
a) What is S?
S=
b) What is the expected number of fish caught over the 7-hour period?
c) Construct a table for r = 0, 1, 2, . . ., 8 fish
λ
L1
L2
4.7
r
P(r)
0
1
2
3
4
5
6
7
8
d) Use your table to compute the following
Probability Description
At least 4 fish?
Probability Expression
TI-84 Command
More than 7 fish?
Between 2 and 5 fish
inclusive
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NAME: __________________________________ BLOCK:___________DATE:________
Exercise 6 In Kentucky’s Pick 4 game, you pay $1 to select a sequence of four digits, such
as 2283. If you play this game once every day, find the probability of winning exactly once
in 365 days.
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