PMI Statistics and Probability
NJCTL.org
Sets - KEY
Class Work
1. Draw a sample space for the following situations: Each answer may vary.
a. Rolling a set of two dice and
multiplying the numbers on each
to find an answer.
b. Drawing two marbles out of a
bag, with replacement. Out of 20
marbles, 4 are red, 10 are blue
and the rest are yellow.
c. Spinning a spinner with the
numbers 1 through 8 on it.
2. On the back of this worksheet or on another piece of paper, draw a Venn Diagram that compares the
choices of candy between a group of boys and a group of girls on a field trip. Use the following
information. The boys and their choices are as follows: Logan chose licorice and toffee, Arik chose gum
and Snickers, Jim chose licorice and Snickers, Rob chose toffee and Reese’s and Bill chose gum and
mints. The girls and their choices are as follows: Tess chose mints and Skittles, Brianna chose Skittles
and Mounds, Paige chose Starburst and Snickers, Katie chose toffee and Snickers and Grace chose
Skittles and Starburst.
1
PMI Statistics and Probability
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3. Use the Venn Diagram below to answer the questions.
a. Name the Universe for the problem.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
b. Name all three of the sets involved.
A = {0, 3, 8, 9, 10, 14}
B = {1, 3, 5, 6, 7, 8, 13, 14}
C = {3, 4, 5, 6, 12, 14}
i. A∩B
c. Find:
{3, 8, 14}
ii. BUC
{1, 3, 4, 5, 6, 7, 8, 12, 13, 14}
v. ~(AUB)
vi. A∩B∩C
{2, 4, 11, 12, 15}
iii. A∩(BUC)
iv. ~(BUC)∩A
{3, 8, 14}
{0, 9, 10}
vii. AU(B∩C)
{3, 14}
viii. (AUB)∩~C
{0, 3, 5, 6, 8, 9, 10, 14}
{0, 1, 7, 8, 9, 10, 13}
4. The following Venn Diagram shows the results of a college survey about extra-curricular activities that
men and women do regularly. Use this diagram to list the members of the following sets and then write a
sentence to describe the result.
a. U
{hockey, football, soccer, weights, swimming,
running, walking, aerobics, yoga, eating, sitting}
b. A∩B
{weights, swimming, running}
c. A∩(~B)
{hockey, football, soccer, weights, swimming, running, walking, aerobics, yoga}
d. AUB
e. ~A
{hockey, football, soccer}
{walking, aerobics, yoga, eating, sitting}
f. ~(A∩B)
{walking, aerobics, yoga, hockey, football, soccer, sitting, eating}
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PMI Statistics and Probability
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Sets
Home Work
5. Draw a sample space for the following situations: Each answer may vary.
a. Drawing a card from a normal
deck of cards. Take only the
different suits into consideration.
b. Rolling a single, six-sided die
AND then flipping a coin.
1 – 1/6
2 – 1/6
3 – 1/6
4 – 1/6
5 – 1/6
6 – 1/6
H – 1/2
1/12
1/12
1/12
1/12
1/12
1/12
T – 1/2
1/12
1/12
1/12
1/12
1/12
1/12
c. Flipping a coin AND then
spinning a spinner with the
numbers 1 through 4 on it.
6. On the back of this worksheet or on another piece of paper, draw a Venn Diagram that represents the
following situation. At USA High School, from a survey of 100 people, a student found that 90 had a TV,
40 had a laptop computer and 30 of those same people had both a TV and a laptop.
3
PMI Statistics and Probability
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7. Use the Venn Diagram below to answer the questions.
a. Name the Universe for the problem.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
b. Name all three of the sets involved.
A = {3, 8, 9, 14, 15}
B = {1, 2, 4, 8, 11, 12, 13}
C = {2, 3, 5, 6, 7, 11}
i. A∩B
c. Find:
{8}
iii. A∩(AUC)
ii. BUC
{1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15}
{8, 9, 14, 15}
vi. A∩(BUC)
v. ~(AUC)
{0,1,4,10,12,13}
iv. ~(A∩C)∩B
{1, 2, 4, 8, 11, 12, 13}
vii. A∩B∩C
{3, 8}
viii. BU(A∩C)
{Ø}
{1, 2, 3, 4, 8, 11, 12, 13}
8. The following Venn Diagram shows activities that male (A) and female (B) teens in Colorado do on the
weekends. Use this diagram to list the members of the following sets and then write a sentence to
describe the result.
a. U
{swimming, running, sailing, skiing,
snowboarding, hiking, fishing, 4-wheeling,
biking, hunting}
b. A∩B
{skiing, snowboarding, hiking, fishing}
c. A∩(~B)
{swimming, running, sailing}
d. AUB
{swimming, running, sailing, skiing,
snowboarding, hiking, fishing, 4-wheeling,
biking, hunting}
e. ~A
{4-wheeling, biking, hiking}
f. ~(A∩B)
{swimming, running, sailing, 4-wheeling, biking, hiking}
Spiral Review
Simplify:
9. √36𝑚8 𝑛6
6m4n3
Simplify:
10.
Factor:
−12𝑥 −6 𝑦 6
11. 24x3 – 36x
18𝑥 −6 𝑦 −7
-2y13
3
12x(2x2 – 3)
4
Simplify:
12.
1
𝑥4
1
𝑥6
x2
PMI Statistics and Probability
NJCTL.org
Independence and Conditional Probability
Class Work
13. Label the events as dependent or independent:
a. Your family decides to take a trip to Disney
World for spring break. Your friend’s family
decides to go to Disneyland.
b. You secretly take out all of the Aces from a deck
of cards and then get your friend to see how many
tries it takes to get an Ace.
Independent
Dependent
14. Decide if the following events are mutually exclusive or overlapping. Then find P(AUB).
a. A = Drawing a red card from a regular deck of cards
B = Drawing a face card from a regular deck of cards
Overlapping
8/13 or 0.62 or 62%
b. A = Rolling an odd number on a six-sided die.
B = Drawing a spade from a regular deck of cards
Mutually exclusive ¾ or o.75 or 75%
15. Find the conditional probability for the following problems:
a. Find the probability that it is raining, given that it
is cold.
b. A bag contains different colored disks that are
numbered from 1 to 10. The probability that the
disk is green is 0.6. The probability that it is green
and odd is 0.3. What is the probability that the disk
is odd, given that it is green?
0.33 or 33%
0.50 or 50%
16. Use the formula
𝑃(𝐵 |𝐴)
𝑃(𝐴)
= 𝑃(𝐵) to mathematically decide if the events are independent.
a. Rolling a 10 on a set of six-sided die and then rolling a 5.
0.084 = 0.084, therefore independent
b. A = taking math during your senior year at high school
B = going to college
0.9 = 0.9, therefore independent
5
PMI Statistics and Probability
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Independence and Conditional Probability
Home Work
17. Label the events as dependent or independent:
a. The cost of a person’s insurance is high.
Looking at the person’s driving record, they have
had a lot of accidents.
b. You drink two 40oz. sodas a day for three
weeks. In that time, you gain 15 pounds.
Dependent
Dependent
18. Decide if the following events are mutually exclusive or overlapping. Then find P(AUB).
a. A bag of 15 marbles has 3 red marbles, 3 blue marbles, 3 yellow marbles, 3 green marbles and 3
black marbles.
A = Drawing a red marble
Mutually exclusive 2/5, 0.4 or 40%
B = Drawing a blue marble
b. Using a regular deck of cards:
A = Drawing an even numbered card
B = Drawing a heart
Overlapping 29/52, 0.558 or 55.8%
19. Find the conditional probability for the following problems:
a. Find the probability that a student is in a band,
given that they take a music class.
0.67 or
67%
20. Use the formula
b. Find the probability that a student gets good
grades, given that they play a sport.
0.6 or 60%
𝑃(𝐵 |𝐴)
𝑃(𝐴)
= 𝑃(𝐵) to mathematically decide if the events are independent.
a. The probability that a person owns the car they drive, i.e. no payments, in USA High School is 40%.
The probability that the person owns their car and knows how to change the oil is 35%. Decide if the
events are independent and then find the probability that a person at USA High knows how to change the
oil on a car, given that they own the car.
0.875 ≠ 0.95, therefore dependent
b.
0.6 ≠ 0.8, therefore dependent
Spiral Review
Factor:
Factor:
Simplify:
Simplify:
21. 125x3 – 27y3
22. 16m2 – 81
23. √24𝑟 4 𝑝8
24.
(5x – 3y)(25x2 + 15xy + 9y2)
(4m – 9)(4m + 9)
2r2p4√6
6 Type equation here.
6
𝑛
3
𝑚
2m
n
PMI Statistics and Probability
NJCTL.org
Measures of Central Tendency
Class Work
For questions 25-27, find the mean, median, mode, range, IQR and the standard deviation of each set of
data. Then, make a quantitative statement about each data set based on the measures of central
tendency. Be sure to include something about the spread of the numbers.
25. Test scores for a physics exam: (in percentages)
96
43
88
97
75
75
76
75
78
75
88.38
88.56
Mean = 77.8
Median = 75.5
Mode = 75
Range = 54
IQR = 13
Standard Deviation = 14.32
**Answers may vary slightly
26. Top ten time results for a Men’s Giant Slalom race: (in seconds)
85.18 85.38 86.48 87.04 87.31 87.40 87.43 88.22
Mean = 87.138
Median = 87.355
Mode = none
Range = 3.38
IQR = 1.74
Standard Deviation = 1.108
**Answers may vary slightly
27. Women’s Spring Board Diving Results: (total points)
414
379.20
362.40
362.20
342.85
332.10
317.80
316.80
345.65
309.40
343.00
295.20
Mean = 343.38
Median = 342.925
Mode = none
Range = 118.8
IQR = 45
Standard Deviation = 31.56
**Answers may vary slightly
28. The table below shows a frequency of test scores on a math test. Find the mean.
86.3
29. Which number would you remove from the list below to get a smaller standard deviation?
78, 78, 90, 99, 77, 86, 85, 85, 34, 88, 76, 87, 92, 92, 72
34
30. A freshmen class of 220 students raised $1200 for a charity. The senior class of 175 students raised
$700. What is the average for both groups?
$950
7
PMI Statistics and Probability
NJCTL.org
Measures of Central Tendency
Home Work
For questions 31-33, find the mean, median, mode, range, IQR and the standard deviation of each set of
data. Then, make a quantitative statement about each data set based on the measures of central
tendency. Be sure to include something about the spread of the numbers.
31. Mortgage rates (in percentages): www.mortgagedaily.com
Year
1971 1976 1981 1986 1991 1996 2001 2006 2011
Rate
7.31% 8.73% 15.58% 9.94% 9.5% 7.92% 7.08% 6.51% 4.84%
Mean = 8.6
Median = 7.92
Mode = none
Range = 10.74
IQR = 2.925
Standard Deviation = 2.87
**Answers may vary slightly
32. Golf scores for the British Open top 14. (4 round total)
273
274
277
277
278
278
279
279
280
280
280
280
280
280
150
13
17
Mean = 278.21
Median = 279
Mode = 280
Range = 7
IQR = 3
Standard Deviation = 2.21
**Answers may vary slightly
33. The number of Facebook friends of 10 students.
45
7
36
12
15
96
20
Mean = 41.1
Median = 18.5
Mode = none
Range = 143
IQR = 32
Standard Deviation = 44.07
**Answers may vary slightly
34. The following is a list of golf scores that a person kept track of over the summer. Find the mean.
Would the standard deviation be large or small?
74
82
83
81
71
72
71
85
88
72
71
90
71
77.77
35. Find the IQR for the list in question 34.
13
36. Write a list of 10 scores for a test (in percentages) that would have a large standard deviation.
Answers will vary. Look for a large range.
Spiral Review
Simplify:
Work out:
Multiply:
Simplify:
37.
38. (3x + 1)3
39. (-4r5t6)(-12r-4s-5)
40. √81𝑚6 𝑛4 𝑝16
𝑚
𝑛
𝑝
m
np
27x3 + 27x2 + 9x + 1
48rt6
8
s
5
9m3n2p8
PMI Statistics and Probability
NJCTL.org
Standard Deviation and Normal Distribution
Class Work
41. Create a normal distribution graph for the fast food with the highest calories. Base data is shown
below.
** http://www.statisticbrain.com/fast-food-with-the-highest-calories/
42. The average salary for the NHL (National Hockey League) is 1.3 million. If the graph of professional
sports’ salaries follows a normal distribution with a mean of 1.56 million and a standard deviation of 1.57,
find the z-score of the average NHL salary and then find their percentile pay.
z-score = -0.17, 43rd percentile
43. What percentile is a 4th grader who scored a 92% on a test that has a mean of 87 and a standard
deviation of 2.8.
z-score = 1.79, 96th percentile
44. A factory fills lotion bottles with approximately 20 fl. oz. of lotion. The amount of lotion is normally
distributed and the factory maintains a standard that at least 99% of the lotion bottles will be filled
between 20 and 21 fl. oz. Find the greatest standard deviation that can be allowed. Round to the nearest
hundredth.
σ = 0.17 oz.
45. Out of a class of 243, how many students would fall between 1 and 2 standard deviations above the
mean? Round to the nearest student.
33 students
9
PMI Statistics and Probability
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Standard Deviation and Normal Distribution
Home Work
46. A class at USA College is very hard. On a test, the average score was 63.25% with a standard
deviation of 10.75. If there were 300 students in the class, how many would you expect would get
between a 74 and an 85 on the test? Round to the nearest student.
41 students
47. Create a normal distribution graph to represent the Mean Annual Salaries of Farming, Fishing and
Forestry Occupations. Base data is shown below.
** http://www.statisticbrain.com/farming-and-forestry-occupation-salary-statistics/
48. The average number of photos uploaded on Facebook per day is 250. The data reported has a
standard deviation of 100. Find the number of photographs someone with a z-score of 3.5 uploaded in
one day.
600 photos
49. A manufacturing plant makes nickel plates. They put approximately 2 oz. of nickel in each plate.
The amount of nickel fluctuates slightly, but the manager guarantees that at least 95% of the plates have
between 1.95 and 2.05 oz. of nickel in them. Find the greatest standard deviation that will be allowed to
maintain this standard.
σ = 0.05 ounces
50. A 10th grader scores 72% on a standardized test. The mean of the test was 83% and the standard
deviation is 3.2. What was the 10th graders reported percentile?
<1 percentile
Spiral Review
Simplify:
51.
3𝑚
7
𝑚2
5
15
7
m
Factor:
Multiply:
Simplify:
52. 8x3 + 27
53. (x2 + 3x – 1)(2x2 – x – 2)
54. √81𝑟 4 𝑠 9
(2x + 3)(4x2 – 6x + 9)
24x4 + 5x3 – 7x2 – 5x + 2
10
3
3
3𝑟𝑠 3 √3𝑟
PMI Statistics and Probability
NJCTL.org
Two-Way Frequency Tables
Class Work
Put the following information in a two-way frequency table in the space below and use it to answer the
questions.
55. At the humane society, during a summer, a total of 625 dogs and cats were brought in to see the
veterinarian. Out of those animals, 145 dogs and 15 cats were positive for heartworm. 455 of the
animals were dogs, the rest were cats.
Positive for heartworm
Negative for heartworm
Total
Dogs
0.232
0.496
0.728
Cats
0.024
0.248
0.272
Total
0.256
0.744
1
a. Find the probability that a dog came in tested positive for heartworm.
0.318 or 31.8%
b. What percentage of cats did not have heartworm?
91%
c. What is the probability that an animal who tested positive for heartworm was a dog?
91%
d. Find the probability that a cat came in and tested positive for heartworm.
0.089 or 8.9%
56. At USA University, 300 students were interviewed about the sports that they played. Out of the 300,
170 were girls, the rest were boys. 121 of the girls and 103 of the boys played a sport. The rest did not
play any sports.
Plays a sport
Does not play a sport
Total
Girls
0.403
0.163
0.567
Boys
0.343
0.09
0.433
Total
0.746
0.253
1
a. What percentage of girls played a sport?
71%
b. Find the probability that a person who played a sport was a boy.
0.46 or 46%
c. Of the boys, what percentage did not play a sport?
21%
d. Find the probability that a person who did not play a sport was a girl.
0.637 or 64%
11
PMI Statistics and Probability
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Two-Way Frequency Tables
Home Work
Put the following information in a two-way frequency table in the space below and use it to answer the
questions.
57. At USA College, 1000 students were surveyed about social websites. Half of the college students
surveyed were girls. Out of the girls, 460 reported having an account on a social website. Of the boys,
320 stated that they had an account.
Account on social website
No account on social website
Total
Girls
0.46
0.04
0.5
Boys
0.32
0.18
0.5
Total
0.78
0.22
1
a. What is the probability that a person who has a social account is a girl?
b. Find the percentage of boys that do not have an account.
0.59 or 59%
36%
c. Of the girls, what is the probability that they do not have a social account?
0.08 or 8%
d. Given that a person has a social account, what is the probability that they are a boy?
0.41 or 41%
58. A group of students at a university studied leukemia in dogs and cats. 2000 pets participated in their
study. 750 were dogs and the rest were cats. 57 of the dogs and 423 cats had leukemia.
Positive for leukemia
Negative for leukemia
Total
Dogs
0.0285
0.3465
0.375
Cats
0.2115
0.4135
0.625
Total
0.24
0.76
1
a. Did dogs have a greater percentage of leukemia or the cats? Prove your answer mathematically.
The cats are more likely to be diagnosed with leukemia. Make sure students use data from their table to justify answers.
b. Of the animals that had leukemia, what percentage were cats?
88%
c. Given that the animal was a dog, what is the probability that it would have leukemia?
0.076 or 7.6%
d. Find the probability that a dog would not have leukemia.
0.924 or 92.4%
Spiral Review
Simplify: (Be careful!)
59.
3
𝑥
1
−3
𝑥
3
1 − 3𝑥
Work out:
60.
3
∙
4
∙
25
15 10 24
1
12
Multiply:
Simplify:
61. (3xy5z6)4
62. √64𝑚6 𝑛20
81x4y20z24
12
4
4
2𝑚𝑛5 √4𝑚2
PMI Statistics and Probability
NJCTL.org
Sampling and Experiments
Class Work
63. It was reported that 63% of prison inmates cannot read. A local mayor said that the stat is too low
and that it is more like 70%. Develop a simulation with 10 trials and calculate a margin of error for this
situation. Does the mayor’s 70% fall within this margin? Prove or disprove his allegation.
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent 63% of the prison population that cannot read as well as having the
design of the simulation being able to randomly generate results. Make sure students did the
simulation 10 times and generated the mean and margin of error correctly. Correct answers
should include a comparison of their results to the 63% as well as a determination of whether the
number falls within their margin of error.
64. A 3rd grader guesses 4 answers on a True/False test. Design a simulation to find the probability that
the young guesser would get two out of the 4 answers correct. Run your simulation at least 10 times.
Answers will vary. Look for thorough explanation that includes assigns appropriate values to True
and False. Make sure students did the simulation 10 times and generated the mean and margin of
error correctly. Correct answers should include probability.
65. Soggy Burger is giving out 6 different game cards with each burger purchased. After you collect all 6
cards, you can redeem them for a free burger. Design a simulation and run it 10 times. Come up with a
mean number from your data to estimate the number of burgers that you must purchase in order to get a
free one.
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent different game cards as well as having the design of the simulation being
able to randomly generate results. Make sure students did the simulation 10 times and generated
the mean and IQR correctly. Correct answers should include a comparison of their results as well as a
determination of a range of possibilities.
66. The number of cancer cures in dogs was 15% better using treatment A over treatment B. Create and
run a simulation to decide if the number 0 falls in your margin of error. What does the number 0
represent in this situation?
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent the percentage of the difference between the trials as well as having the
design of the simulation being able to randomly generate results. Make sure students did the
simulation 10 times and generated the margin of error correctly. Correct answers should include a
comparison of their results as well as a determination of whether the number falls within their
margin of error.
13
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Sampling and Experiments
Home Work
67. You are bored in class and start flipping a coin with your friend. Heads you win, tails your friend
wins. She wins 5 times in a row. Is she somehow cheating? Use a simulation to prove or disprove a
claim that she is cheating.
Answers will vary. Students’ answers will depend on their simulations. Look for a thorough
explanation that includes references to their data.
68. Your friend is terrified of flying and claims that 50% of fatalities happen when the plane is descending
or landing. In reality, you know that the odds of a plane crashing are 1 in 29.4 million. Despite that, your
friend continues to be terrified. Finally, you decide to look up information about plane crashes on the
internet and find the actual stat of crashing while descending or landing is 41%. Prove or disprove your
friend’s claim of 50% using a simulation.
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent 41% of crashes as well as having the design of the simulation being able
to randomly generate results. Make sure students did the simulation 10 times and generated
the mean and margin of error correctly. Correct answers should include a comparison of their
results to the 50% as well as a determination of whether the number falls within their margin of
error.
69. Two different skin creams are being tested on how much they help a person’s skin. Cream A
reported that 62% of people had better skin using the cream and Cream B reported 51% of people having
improved skin. Is Cream A really better than Cream B? Design a simulation using the difference of the
successes of the creams to see if the number of zero (no difference) falls within your margin of error.
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent the percentages for each skin cream as well as having the design of the
simulation being able to randomly generate results. Make sure students did the simulation 10 times
and generated the mean and margin of error correctly. Correct answers should include a comparison
of their results as well as a determination of whether the number falls within their margin of error.
70. A student at your school claims that 2/3 of the school plays video games every night. You feel that
this stat is way too high. You ask your homeroom of 36 students who plays video games at night. 17 of
those students admitted playing every night. Is this a large enough sample to prove the original claim to
be incorrect? Develop a simulation to prove or disprove the claim that 2/3 or the school plays video
games each night.
Answers will vary. Look for thorough explanation that includes assigning numbers which
appropriately represent 17 of the 36 students that played video games as well as having the
design of the simulation being able to randomly generate results. Make sure students did the
simulation 10 times and generated the mean and margin of error correctly. Correct answers
should include a comparison of their results to the 67% as well as a determination of whether the
number falls within their margin of error.
Spiral Review
Simplify: (Be careful!)
2
71. 1
𝑥
+5
2𝑥
1 + 5𝑥
Work out:
1
8 35
72.
∙ ∙
16 15
7
24
4
Multiply:
73. (-10xy2z6)3
-1000x3y6z18
14
Simplify:
3
74. √64𝑚6 𝑛20
3
4𝑚2 𝑛6 √4𝑛2