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Chapter 6 Notes AP STATS
Name_________________________________
Introduction –
Station Number -
Bottled Water cup? --
Looking at the class data:
1)
2)
3)
4)
How many students in class guessed correctly?
What percent of the class is this?
What’s the probability that an individual student would guess correctly?
How many correct identifications would you need to see to be convinced that the students in your class aren’t
just guessing?
5) Design and carry out a simulation to answer this question. What do you conclude about your classes ability to
distinguish tap water from bottled water?
6.1 Discrete and Continuous Random Variables
Definition:
A random variable takes numerical values that describe the outcomes of some chance process. The probability
distribution of a random variable gives its possible values and their probabilities.
Suppose we toss a fair coin three times. The
Sample space is
HHH HHT HTH THH HTT THT TTH TTT
Let X = The number of heads obtained
X=0 X=1 X=2 X=3
Probability Distribution of X:
What is the probability that we get at least one head in three tosses of a coin?
P(X > 1) =
Interpret P(x > 1) in context. This probability means that if we were to repeatedly flip 3 coins and record the number of
heads, then in the long-run about _____% of the time we will get 1 or more heads.
There are two main types of random variables, discrete and continuous!
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Discrete Random Variables and Their Probability Distributions
Example: APGAR SCORES: BABIES HEALTH AT BIRTH Discrete random variables
In 1952, Dr. Virginia Apgar suggested five criteria for measuring a baby’s health at birth: skin color, heart rate, muscle
tone, breathing, and response when stimulated. She developed a 0-1-2 scale to rate a newborn on each of the five
criteria. A baby’s Apgar score is the sum of the rating on each of the five scales, which gives a whole-number vale from 0
to 10. Apgar scores are still used today to evaluate the health of newborns.
What Apgar scores are typical? To find out, researchers recorded the Apgar scores of over 2 million newborn babies in a
single year. Imagine selecting one of these newborns at random. Define the random variable X = Apgar score of a
randomly selected baby one minute after birth. The table below gives the probability distribution for X.
(a) Show that the probability distribution for X is legitimate.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)?
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1. TOSS 4 TIMES. Suppose you toss a fair coin 4 times. Let X = the number of heads you get.
a) Find the probability distribution of X.
b) Make a histogram of the probability distribution. Describe what you see.
c) Find P(X < 3) and interpret the result.
5. BENFORD’s LAW. Faked numbers in tax returns, invoices, or expense account claims often display patterns that
aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a
clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often
follow a model known as Benford’s law. Call the first digit of a randomly chosen record X for short. Benford’s law gives
this probability model for X. (note that the first digit cannot be 0):
First Digit X: 1
2
3
4
5
6
7
8
9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
a) Show that this is a legitimate probability distribution
b) Make a histogram of the probability distribution. Describe what you see.
c) Describe the event X > 6 in words. What is P(X> 6)
d) Express the event “the first digit is at most 5” in terms of X. What is the probability of this event?
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7. Refer to problem above. The first digit of a randomly chosen expense account claim follows Benford’s law. Consider
the events A = First digit is 7 or greater and B = first digit is odd.
a) What outcomes make up the event A? What is P(A)?
b) What outcomes make up the event B? What is P(B)?
c) What outcomes make up the event “A or B”? What is P(A or B)? Why is this probability not equal to P(A)+P(B)?
Mean (Expected Value) of a Discrete Random Variable
Consider the random variable X = Apgar Score
Compute the mean of the random variable X and interpret it in context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
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9. Keno. Keno is a favorite game in casinos, and similar games are popular with the states that operate lotteries. Balls
numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls are chosen at random. Players
select numbers by marking a card. The simplest of the many wagers available is "mark 1 Number." your payoff is $3 on a
$1 bet if the number you select is one of those chosen. Because 20 of 80 numbers are chosen, your probability of
winning is 20/80 or 0.25. Let X = the amount you gain on a simple play of the game.
a) Make a table that shows the probability distribution of X.
b) Computer the expected value of X. Explain what this result means for the player.
13. Benford's law and fraud. A not so clever employee decided to fake his monthly expense report. He believed that
the first digits of his expense accounts should be equally likely to be any of the number from 1 to 9. In that case, the
first digits, Y, of a randomly selected expense amount would have the probability distribution shown in the histogram.
a) Explain why the mean of the random variable Y is located at the solid line in the diagram.
b) The first digits of randomly selected expense amounts actually follow Benford's law. What's the expected value of the
first digit? Explain how this information could be sued to detect a fake expense account.
First Digit X: 1
2
3
4
5
6
7
8
9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
c) What's P(Y >6)? According to Benford's law, what proportion of first digits in the employee's expense amounts should
be greater than 6? how could this information be sued to detect a fake expense report?
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HOMEWORK PROBLEM #14
14. Life Insurance. A life insurance company sells a term insurance policy to a 21 year old male that pays $100,000 if
the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in
mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount Y that
the company earns on this policy is $250.00 per year, less the $100,000 that is must pay if the insured dies. Here is a
partially competed table that shows information about risk of mortality and the values of Y = profit earned by the
company:
A) Fill in the missing values of Y.
b) Find the missing probability.
c) calculate mean µx. Interpret this value in context.
Day 2 Standard Deviation and Variance. Continuous Random Variables
Recall: Variance -- an average of the standard deviation of the vales of the variable X from its mean µx
Recall: standard Deviation is the Square Root of the variance. -- the measure of how much the values of the variable X
tend to vary, on average, from the mean.
Example: Consider the random variable X = Apgar Score
Compute the standard deviation of the random variable X and interpret it in context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
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18. It would be quite risky for you to insure the life of a 21 year old friend under the terms of Exercise 14. There is a
high probability that your friend would live and you would gain $1250 in premiums. But if he were to die, you would
lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many
thousands of 21 year old men.
b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more
variable the return is, the riskier the investment. We can measure the great risk of insuring a single person's life in
Exercise 14 by computing the standard deviation of the income Y that the insurer will receive. Find σY using the
distribution and mean found in Exercise 14.
Common AP ERROR:
Students often lose credit for not showing adequate work -or for not showing any work at all. To get full credit, students
need to go beyond just reporting commands used on a
graphing calculator. In this case, showing the first couple of
terms with an ellipsis (...) for the calculation of the mean or
standard deviation is sufficient.
Calculator steps to find Expected Mean and
Standard Deviation
19. Housing in San Jose. How do rented housing units differ from units occupied by their owners? Here are the
distributions of the number of rooms for owner occupied units and renter-occupied units in San Jose, California?
Number of Rooms
1
2
3
4
5
6
7
8
9
10
Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035
Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003
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Let X = the number of rooms in a randomly selected owner-occupied unit and Y - the number of rooms in a randomly
chosen center-occupied unit.
a) Make histograms suitable for comparing the probability distribution of X and Y. Describe any differences you that you
observe.
b) Find the mean number of rooms for both types of housing unit. Explain why this difference makes sense.
c) Find the standard deviation of both X and Y. Explain why this difference makes sense.
Example: The heights of young women closely follow the Normal distribution with mean µ = 64 inches and standard
deviation σ = 2.7 inches. This is a distribution for a large set of data. Now choose one young woman at random. call
her height Y. What is the probability that a randomly chosen young woman has height between 68 and 70 inches?
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Exam Tip: When you solve problems involving random variables, start by defining the random variable of interest.
For example, let X = the Apgar score of a randomly selected baby or let Y = the height of a randomly selected
young women. Then state the probability you are trying to find in terms of the random variable:P(68<Y<70) or
P(X>7)
Another Example: Weights of Three-Year-Old Females
The weights of three-year-old females closely follow a Normal distribution with a mean of  = 30.7 pounds and a
standard deviation of 3.6 pounds. Randomly choose one three-year-old female and call her weight X. Find the
probability that the randomly selected three-year-old female weighs at least 30 pounds.
Using your calculator:
23. ITBS Scores. The Normal distribution with mean µ = 6.8 and standard deviation σ = 1.6 is a good description of the
Iowa Test of Basic Skills (ITBS) vocabulary scores of seventh grade students in Gary, Indiana. call the score of a randomly
chosen student X for short. Find P(X>9) and interpret the result.
25. Did you vote? A sample survey contacted an SRS of 663 registered voters in Oregon shortly after an election and
asked respondents whether they had voted. Voter records shows that 56% of registered voters had actually voted. We
will see later than in repeated samples of size 663, the proportion in the sample who voted (V) will vary according to the
Normal distribution with mean μ = 0.56 and standard deviation σ = 0.019
a) If the respondents answer truthfully, what is P(0.52 < V< 0.60)? This is the probability that the sample proportion V
estimates the population proportion 0.56 within ± 0.04.
b) In fact, 72% of the respondents said they had voted (V= 0.72). if respondents answered truthfully, what is P(V>0.72)?
This probability is so small that it is good evidence that some people who did not vote claimed that they did vote.
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Section 6.2 Linear Transformations
Review Questions 27 - 30.
Choose an American household at random and let the random variable X be the number of cars (including SUVs and
light trucks) they own. there is the probability model if we ignore the few households that own more than 5 cars.
Number of Cars X 0
1
2
3
4
5
Probability
0.09 0.36 0.35 0.13 0.05 0.02
27. A housing company builds houses with two car garages. What percent of the households have more cars than the
garage can hold?
A. 13%
B. 20%
C. 45%
D. 55%
E. 80%
28. What's the expected number of cars in a randomly selected American household?
A. Between 0 and 5
B. 1.00
C. 1.75
D. 1.84
E. 2.00
29. A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wagers: Draw one card at
random from the deck. You win $10 if the card drawn is an ace. Otherwise you lose $1. If you make this wager very
many times, what will be the mean amount you win?
A. About -$1, because you will lose most of the time.
B. About $9, because you win $10 but lose only $1.
C. About -$0.15; that is, on average you lose about 15 cents
D. About $0.77; that is on average you win about 77₵
E. About $0, because the random draw gives you a fair bet.
30. The deck of 52 cards contains 13 hearts. Here is another wager: draw one card at random from the deck. If the card
drawn is a heart, you win $2. Otherwise you lose $1. Compare this wager (Call it Wager 2) with that of the pervious
exercise (call it Wager 2). Which one should you prefer?
A. Wager 1, because it has a higher expected value.
B. Wager 2, because it has a higher expected value.
C. Wager 1, because it has a higher probability of winning
D. Wager 2, because it has a higher probability of winning
E. Both wagers are equally favorable.
6.2 Linear Transformations
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Note: Multiplying a random variable by a constant b multiplies the variance by b2.
Example:
El Dorado Community College considers a student to be full-time if he or she is taking between 12 and 18 units. The
number of units X that a randomly selected El Dorado Community College full-time student is taking in the fall semester
has the following distribution.
Number of Units (X) 12
13
14
15
16
17
18
Probability
0.25 0.10 0.05 0.30 0.10 0.05 0.15
Find µx = ______________ σx2 = _________ σx = _______
At El Dorado Community College, the tuition for full-time students is $50 per unit. That is, if T = tuition charge for a
randomly selected full-time student, T = 50X. Here is the probability distribution for T and a histogram of the probability
distribution:
Tuition Charge (T) 600 650 700 750 800 850 900
Probability
0.25 0.10 0.05 0.30 0.10 0.05 0.15
Find µx = ______________ σx2 = _________ σx = _______
Notice that



the shapes of both distributions are the same
the mean of the distribution of T is 50 times bigger than the mean of X: 732.5 = 50(14.65)
the standard deviation of T is 50 times bigger than the standard deviation of X: 103 = 50(2.06)
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El Dorado Community College
In addition to tuition charges, each full-time student at El Dorado Community College is assessed student fees of $100
per semester. If C = overall cost for a randomly selected full-time student, C = 100 + T.
Here is the probability distribution for C and a histogram of the probability distribution:
Overall Cost (C)
Probability
700 750 800 850 900 950 1000
0.25 0.10 0.05 0.30 0.10 0.05 0.15
Find µx = ______________ σx2 = _________ σx = _______
Notice that



the shapes of both distributions are the same
the mean of the distribution of C is $100 larger than the mean of T: 832.5 = 100 + 732.5
the standard deviation of C is the same as the standard deviation of T
AP TEST COMMON ERROR:
When students are asked about the effect of a linear transformation on summary statistics, some students
forget that adding (subtracting) a constant to (from) every value in the distribution has no effect on measures of
spread, including the standard deviation, variance, range and interquartile range.
Scaling a Test
Problem: In a large introductory statistics class, the distribution of X = raw scores on a test was approximately normally
distributed with a mean of 17.2 and a standard deviation of 3.8. The professor decides to scale the scores by multiplying
the raw scores by 4 and adding 10.
(a) Define the variable Y to be the scaled score of a randomly selected student from this class. Find the mean and
standard deviation of Y.
(b) What is the probability that a randomly selected student has a scaled test score of at least 90?
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37. Get on the boat! A small ferry runs every half hour from one side of a large river to the other. The number of cars
X on a randomly chosen ferry trip has the probability distribution shown below. You can check that μx = 3.87 and σx =
1.29
Cars
0
1
2
3
4
5
Probability 0.02 0.05 0.06 0.16 0.27 0.42
a) the cost for the ferry trip is $5. Make a graph of the probability distribution for the random variable M = money
collected on a randomly selected ferry trip. Describe its shape.
b) Find and interpret μx
c) Compute and interpret σx
39-40 Ms. Hall gave her class a 10 question multiple choice quiz. Let X = the number of questions that a randomly
selected student in the class answered correctly. The computer output below gives information about the probability
distribution of X. To determine each student's grade on the quiz (out of 100). Ms. Hall will multiply his or her number of
correct answers by 10. let G = the grade of a randomly chosen student in the class.
N
30
Mean
7.6
Median
8.5
St. Dev
1.32
Min
4
Max
10
Q1
8
Q3
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39. a) Find the mean of G. Show your method
b) Find the standard deviation of G. Show your method.
c) How do the variance of X and the variance of G compare? Justify your answer.
40. a) Find the median of G. Show your method.
b) find the IQR of G. Show your method
c) What shape would the probability distribution of G have? Justify your answer.
41. Refer to exercise 37. The ferry company's expenses are $20 per trip. Define the random variable Y to be the
amount of profit (money collected minus expenses) made by the ferry company on a randomly selected trip. that is,
Y = M - 20.
a) How does the mean Y relate to the mean of M? Justify your answer. What is the practical importance of μY?
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b) How does the standard deviation of Y relate to the standard deviation of M? Justify our answer. What is the practical
importance of σY?
43. Get on the boat. Based on the analysis of #41, the ferry company decides to increase the cost of a trip to $6. We
can calculate the company's profit Y on a randomly selected trip from the number of cars X. Find the mean and standard
deviation of Y. Show your work.
45. Too cool at the cabin. During the winter months, the temperatures at the Starneses' Colorado cabin can stay well
below freezing for weeks at a time. To prevent the pipes from freezing Mrs. Starnes sets the thermostat at 50° F. She
also buys a digital thermometer that records the indoor temperature each night at midnight. Unfortunately, the
thermometer is programmed to measure the temperature in °C. based on several years' worth of data, the temperature
T in the cabin at midnight follows a Normal Distribution with mean 8.5°C and standard deviation 2.25 °C.
a) Let Y = the temperature in the cabin at midnight on a randomly selected night in °F. F = (9/5)C + 32. Find the mean
and the standard deviation.
b) Find the probability that the midnight temperature in the cabin is below 40 °F. Show your work.
49. Review Independence: In which of the following games of chance would you be willing to assume independence of
X and Y in making a probability model? Explain your answer in each case.
a) In blackjack, you are dealt two cards and examine the total points X on the cards (face cards count 10 points). You can
choose to be dealt another card and compete based on the total points Y on all three cards.
b) In craps, the betting is based on successive rolls of two dice. X is the sum of the faces on the first roll, and Y is the sum
of the faces on the next roll.
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6.2 Combining Random Variables
Example of mean of the Sum: How many total passengers can Pete and Erin expect on a randomly selected day?
If Pete expects µX = 3.75 and Erin expects µY = 3.10 , they will average a total of 3.75 + 3.10 = 6.85 passengers per trip.
Example of Variance of the Sum:
Let S = X + Y as before. Assume that X and Y are independent, which is reasonable since each student was
selected at random.
Here are all the possible combinations of X and Y:
X
12
12
12
13
13
13
14
14
14
15
15
15
16
16
16
17
17
17
18
18
18
P(X)
0.25
0.25
0.25
0.10
0.10
0.10
0.05
0.05
0.05
0.30
0.30
0.30
0.10
0.10
0.10
0.05
0.05
0.05
0.15
0.15
0.15
Y
12
15
18
12
15
18
12
15
18
12
15
18
12
15
18
12
15
18
12
15
18
P(Y)
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
S=X+Y
24
27
30
25
28
31
26
29
32
27
30
33
28
31
34
29
32
35
30
33
36
P(S)=P(X)P(Y)
0.075
0.10
0.075
0.03
0.04
0.03
0.015
0.02
0.015
0.09
0.12
0.09
0.03
0.04
0.03
0.015
0.02
0.015
0.045
0.06
0.045
Find μx = ___________
Find μy = ___________
Find μx+Y = ____________
Find σx = ____________
Find σy = ____________
Find σx+y = ____________
Find σX2 = ____________
Find σY2 = ____________
Find σX+Y2 = ____________
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Example: Mr. Starnes likes between 8.5 and 9 grams of sugar in his hot tea. Suppose the amount of sugar in a randomly
selected packet follows a Normal distribution with mean 2.17 g and standard deviation 0.08 g. If Mr. Starnes selects 4
packets at random, what is the probability his tea will taste right?
Let X = the amount of sugar in a randomly selected packet. Then, T = X1 + X2 + X3 + X4. We want to find P(8.5 ≤ T ≤ 9).
Step 1: Find the mean amount of sugar in tea:
Step 2: Find the Standard Deviation of sugar in tea:
 T2   X2   X2   X2   X2  (0.08)2  (0.08)2  (0.08)2  (0.08)2  0.0256
1
2
3
4
 T  0.0256  0.16
Step 3: Find the z score for the sugar amount in his tea: z  8.5  8.68  1.13
0.16
and
z
9  8.68
 2.00
0.16
Step 4: Draw a normal distribution and shade in the appropriate area under the curve.

Step 5: calculate the probability using correct notation: P(-1.13 ≤ Z ≤ 2.00) = 0.8480
RECALL THAT IS NORMALCDF in calculator but we never write calculator talk on test!
Step 6: Conclude: There is about an 85% chance Mr. Starnes’s tea will taste right.
Practice Problem: Suppose that the weights of a certain variety of apples have weights that are approximately Normally
distributed with a mean of 9 ounces and a standard deviation of 1.5 ounces. If bags of apples are filled by randomly
selecting 12 apples, what is the probability that the sum of the weights of the 12 apples is less than 100 ounces?
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Practice Problem #2: Have you ever been to a fast food restaurant and the lid didn't quite fit?
The diameter C of a randomly selected large drink cup at a fast food restaurant follows a normal distribution with a
mean of 3.96 inches and a standard deviation of 0.01 inches. The diameter L of a randomly selected large lid at this
restaurant follows a Normal distribution with mean 3.98 inches and standard deviation 0.02 inches. For a lid to fit on a
cup, the value of L has to be bigger than the value of C, but not by more than 0.06 inches.
State: __________________________________________________________________________________
Plan: Let ____ = L - C to represent the difference between the lid's diameter and the cup's diameter. Find
P(0<___<0.06)
Do: Find the mean: _______________________________
Find the variance: ______________________________
Find the standard deviation: ________________________
Find the Z scores:
Draw the Area under the curve:
Find the Probability: P(__________) = ________________
Conclude: _________________________________________________________________________________________
__________________________________________________________________________________________________
51. His and her earnings. A study of working couples measures the income X of the husband and the income Y of the
wife in large number of couples in which both parties are employed. Suppose that you knew the mean μx and μY the
variances σx2 and σY2 of both variables in the population.
a) Is it reasonable to take the mean of the total income X + Y to be μx + μY ? Explain your answer.
b) Is it reasonable to take the variance of the total income to be σx2 + σY2? Explain your answer.
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57. Let X = the amount a life insurance company earns on a 5 year term life policy. Calculations reveal that μx = $303.35
and σx = $9707.57
The risk of insuring one person's life is reduced if we insure may people. Suppose that w insure two 21 year old males,
and that their ages at death are independent. If X1 and X2 are the insurer's income from the two insurance policies, the
insurer's average income W on the two policies is W =
𝑋1 +𝑋2
2
= 0.5𝑋1 + 0.5𝑋2
Find the mean and standard deviation of W.
58. If four 21 year old men are insured, the insurer's average income is:
V=
𝑋1 +𝑋2 +𝑋3 +𝑋4
4
= 025𝑋1 + 0.25𝑋2 + 0.25𝑋3 + 0.25𝑋4
When Xi is the income from insuring one man. Assuming that the amount of income earned on individual policies is
independent, find the mean and standard deviation of V.
59. Time and Motion. A time and motion study measures the time required for an assembly line worker to perform a
repetitive task. The data show that the time required to bring a part from a bin to its position on an automotive chassis
varies from car to car according to a Normal Distribution with mean 11 seconds and standard deviation 2 seconds. The
time required to attached the part to the chases follows a Normal distribution with mean 20 seconds and standard
deviation 4 seconds. The study fines the time required for the two steps are independent. A par t that takes a long time
to position, for example, does not take more or less time to attach than other parts.
a) Find mean and standard deviation of the time required for the entire operation of positioning and attaching the part.
b) Management's goal is for the entire process to take less than 30 seconds. Find the probability that this goal will be
met. Show you work.
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63. Swim Team. Hanover High School has the best women's swimming team in the region. The 400 meter freestyle
relay team is undefeated this year. In the 400 meter freestyle relay, each swimmer swims 100 meters. The times, in
seconds, for the four swimmers this season are approximately Normally distributed with means and standard deviations
as shown. Assuming that the swimmer's individual times are independent, find the probability that the total team time
in the 400 meter freestyle relay is less than 220 seconds.
Swimmer
Kalin
Megan
Ashley
Lyndie
Mean
55.2
58.0
56.3
54.7
Std. Dev
2.8
3.0
2.8
2.7
TRY ON YOUR OWN PROBLEM (HOMEWORK)
61. Auto Emissions. The amount of nitrogen oxides (NOX) present in the exhaust of a particular type of car varies from
car to car according to a Normal distribution with mean 1.4 grams per mile (g/mi) and standard deviation 0.3 g/mi. Two
randomly selected cars of this type are tested. One has 1.1 g/mi of NOX; the other has 1.9 g/mi. The test station
attendant find this difference in emissions between two similar cars surprising. If the NOX levels for two randomly
chosen cars of this type are independent, find the probability that the difference is at least as large as the value the
attendant observed.
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6.3 Binomial and Geometric Random Variables
Binomial Settings
When the same chance process is repeated several times, we are often interested in whether a particular outcome
does or doesn’t happen on each repetition. In some cases, the number of repeated trials is fixed in advance and we
are interested in the number of times a particular event (called a “success”) occurs. If the trials in these cases are
independent and each success has an equal chance of occurring, we have a binomial setting.
Example: ROLL THE PIGS game: PIG OUT
Land on sides: one has dot up, one has dot down
Toss two pigs 50 times. Success = number of times Pig Out occurs. Failure = number of times it did not.
Keep track of the number of successes.
Total successes = ______/50
__________________________________________________________________________________________________
Binary?
Yes. The outcome was either Pig out or not
Independent? Yes The outcome of one roll does not effect the outcome of the next roll.
Number?
n = 50
Success?
Probability of success = approximately 20%
_________________________________________________________________________________________________
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Example: Determine whether the random variables below have a binomial distribution. Justify your answer.
(a) Roll a fair die 10 times and let X=the number of sixes.
(b) Shoot a basketball 20 times from various distances on the court. Let Y=number of shots made.
(c) Observe the next 100 cars that go by and let C=color.
Binomial Coefficient: Used to determine the number of successes given a binomial event. Much easier than listing out
all possible outcomes, which could take too long!
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Example: Each child of a particular pair of parents has probability 0.25 of having type O blood. Genetics says that
children receive genes from each of their parents independently. If these parents have 5 children, the count X of
children with type O blood is a binomial random variable with n = 5 trials and probability p = 0.25 of a success on each
trial. In this setting, a child with type O blood is a “success” (S) and a child with another blood type is a “failure” (F).
What’s P(X = 2)?
1) Define n = 5 (number of trials) k = 2 (number of successes). Find the number of successful events:
Examine the number of successes from all possible outcomes: It would take forever to list all possible outcomes! This is
why we use the binomial coefficient :)
2) The probability of a success = 0.25 (from problem) The probability of failure = 0.75 (complement of success)
Write it just like above to get full credit on the AP Stat test, however, you can perform the entire calculation in your
calculator:
Go to Distributions: binompdf(n,p,k) Reminder that pdf is for a specific number and cdf is for an interval
DO NOT USE CALCULATOR SPEAK ON THE TEST. WRITE EVERYTHING JUST AS BELOW BUT YOU CAN FIND THE SOLUTION
IN YOUR CALCULATOR 0.2637 all at once!
Binomial Probability Formula is given on the formula sheet!
EXAMPLE 2: Each child of a particular pair of parents has probability 0.25 of having blood type O. Suppose the parents
have 5 children
(a) Find the probability that exactly 3 of the children have type O blood.
(b) Should the parents be surprised if more than 3 of their children have type O blood?
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Conclusion to part b: Since there is only a 1.5% chance that more than 3 children out of 5 would have Type O blood, the
parents should be surprised!
Using your calculator to answer part B. You may have to use 1 - Binomcdf if your calculator does not have the new ROM
1-Binomcdf(5,0.25,3). If it has the new ROM, then just Binomcdf(5,0.25,3)
69. Sowing Seeds. Seed Depot advertises that 85% of its flower seeds will germinate (grow). Supposed that the
company's claim is true. Judy buys a packet with 20 flower seeds from Seed Depot and plants them in her garden. Let X
= the number of seeds that germinate
Binomial or Not?
71. Lefties. Exactly 10% of the students in a school are left-handed. Select students at random from the school, one at a
time, until you find one who is left-handed. Let V = the number of students chosen.
Binomial or Not?
73. Binomial Setting? A binomial distribution will be approximately correct as a model for one of these two sports
settings and not for the other. Explain why by briefly discussing both settings
a) A NFL kicker has make 80% of his field goal attempts in the past. This season he attempts 20 field goals. The attempts
differ widely in distance, angle, wind, and so on.
b) A NBA player has made 80% of his free-throw attempts in the past. This season he takes 150 free throws. Basketball
free throws are always attempted from 15 feet away with no interference from other players.
75. Blood Types. In the US 44% of adults have type O blood. Suppose we chose 7 US adults at random. Let X = the
number who have type O blood. Use the binomial probability to find P(Y=1). Interpret this answer in context.
77. . Blood Types. Using the information from 75. How surprising would it be to get more than 4 adults with type O
blood in the sample?
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79. Seed Depot advertises that 85% of its flower seeds will germinate (grow). Supposed that the company's claim is
true. Judy buys a packet with 20 flower seeds from Seed Depot and plants them in her garden. Let X = the number of
seeds that germinate.
a) Find the probability that exactly 17 seeds germinate. Show your work.
b) If only 12 seeds actually germinate, should Judy be suspicious that the company's claim is not true? Computer P(X<12)
and use this results to support your answer.
6.3 day 2 Mean and Standard Deviation of a Binomial Distribution
Both Formulas are
given on Formula Sheet
:)
81. Random digit dialing. When an opinion poll calls residential telephone numbers at random, only 20% of the calls
reach a live person. You watch the random digit dialing machine make 15 calls.
Let X = the number of calls that reach a live person.
a) Find and interpret μx
b) find and interpret σx
83. Random digit dialing. Let Y = the number of calls that don't reach a live person.
a) Find the mean of Y. How is it related to the mean of X? Explain why this makes sense.
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b) Find the standard deviation of Y. How is it related to the standard deviation of X? Explain why this makes sense.
85. Aircraft Engines. Engineers define reliability as the probability that an item will perform its function under specific
conditions for a specific period of time. A certain model of aircraft engine is designed so that each engine has
probability 0.999 of performing property for an hour of flight. Company engineers test an SRS of 350 engines of this
model. Let X = the number that operate for an hour without failure.
a) Explain why X is a binomial random variable.
b) Find the mean and standard deviation of X. Interpret each value in context.
c) Two engines failed the test. Are you convinced that this model of engine is less reliable than its supposed to be?
Compute P(X<348) and use the results to justify your answer.
Binomial Distributions in Statistical Sampling
n
1
N
10
87. Airport Security. The TSA is responsible for airport safety. On some flights, TSA officers randomly select passengers
for an extra
security check before boarding. One such flight had 76 passengers - 12 in first class and 64 in coach. Some
passengers were surprised when none of the 10 passengers chosen for screening were seated in first class. Can we use
a binomial distribution to approximate this probability?
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Example: Attitudes Toward Shopping
Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a nationwide random sample of
2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and timeconsuming.” Suppose that exactly 60% of all adult US residents would say “Agree” if asked the same question. Let X =
the number in the sample who agree. Estimate the probability that 1520 or more of the sample agree.
B
I
N
S
Is np > 10?
Is n(1 - p) > 10?
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Geometric Settings - the goal is to repeat a chance behavior until a success occurs.
Geometric Random Variable
NOT GIVEN ON
FORMULA SHEET!
Example: In the board game Monopoly, one way to get out of jail is to roll doubles. How likely is it that someone in jail
would roll doubles on his first, second, or third attempt? If this was the only way to get out of jail, how many turns
would it take, on average?
Let Y = number of attempts it takes to roll doubles one time.
Geometric Distribution?
Binary? Yes, success = doubles, failure = not doubles
Independent? Yes, knowing the outcome of previous rolls does not give additional information about future rolls.
Trials? Yes, we are counting the number of trials needed to get doubles once.
Success? Yes, the probability of success is always 1/6.
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(a) Find the probability that it takes 3 turns to roll doubles to get out of jail.
(b) Find the probability that it takes more than 3 turns to roll doubles and interpret this value in context.
89. No replacement. To use a binomial distribution to approximate the count of success in an SRS, why do we require
that the sample size n be no more than 10% of the population size N?
If the sample size is more than 10% of the population, the amount of change to the make-up of the population is too
much. The probability of success changes too much to be considered constant.