Download HW 1-2 - Math with Ms. Winstead

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HW 1-1: Using the Empirical Rule
For each problem set, label the normal curve with the appropriate values, and use the curve to answer
the questions.
1. The mean score on the midterm was an 82 with a standard deviation of 5. Find the probability that a
randomly selected person:
a. scored between 77 and 87
b. scored between 82 and 87
c. scored between 72 and 87
d. scored higher than 92
e. scored less than 77
2. The mean SAT score is 490 with a standard deviation of 100. Find the probability that a randomly selected
student:
a. scored between 390 and 590
b. scored above 790
c. scored less than 490
d. scored between 290 and 490
3. The mean weight of college football players is 200 pounds with a standard deviation of 30. Find the
probability that a randomly selected player:
a. weighs between 170 and 260
b. weighs less than 170
c. weighs over 290
d. weighs less than 140
e. weighs between 140 and 230
4. The average life of a car tire is 28,000 miles with a standard deviation of 3000. Find the probability that a
randomly selected tire will have a life of:
a. between 19,000 and 37,000 miles
b. less than 25,000 miles
c. between 31,000 and 37,000 miles
d. over 22,000 miles
e. below 31,000 miles
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Using Z-Scores to Pick a Winner
The decathlon is an event in track and field that consists of 10 events.
To determine who wins, we have to know whether it is harder to jump an inch higher or run 5 seconds faster.
We have to be able to compare two fundamentally different activities involving different units.
Standard deviations to the rescue! If we knew the mean performance (by world-class athletes) in each event,
and the standard deviation, we could compute how far each performance was from the mean in standard
deviation units (that is, the z-scores).
So consider the three athletes’ performances shown below in a three event competition. Note that each placed
first, second, and third in an event.
Competitor
A
B
C
mean
standard deviation
100 m dash
10.1 sec
9.9 sec
10.3 sec
100 m dash
10 sec
0.2 sec
Event
Shot put
66’
60’
63’
Long Jump
26’
27’
27’ 3”
Shot Put
60’
3’
Long Jump
26’
6”
Who gets the gold medal? Who turned in the most remarkable performance of the competition? Explain your
reasoning using mathematics.
Reiland, NCSU
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HW 1-2: Practice with Normal Distributions
For each question, draw and shade a normal curve, then solve.
1.
The percentage impurity of a chemical can be modelled by a normal distribution with a mean of 5.8
and a standard deviation of 0.5. Obtain the probability that a sample of the chemical has percentage
impurity between 5 and 6.
2.
Melons sold on a market stall have weights that are normally distributed with a mean of 2.18 kg and
a standard deviation of 0.25 kg. For a melon chosen at random, find the probability that its weight
lies between 2 kg and 2.5 kg.
3.
A teacher travels from home to work by car each weekday by one of two routes, X or Y. For route X,
her journey times are normally distributed with a mean of 30.4 minutes and a standard deviation of
3.6 minutes. Calculate the probability that her journey time on a particular day takes between 25
minutes and 35 minutes.
4.
Soup is sold in tins which are filled by a machine. The actual weight of soup delivered to a tin by
the filling machine is always normally distributed about the mean weight with a standard deviation
of 8g. The machine is set originally to deliver a mean weight of 810g.
(a) Determine the probability that the weight of soup in a tin, selected at random, is less than
800g.
(b) Determine the probability that the weight of soup in a tin, selected at random, is between 795
g and 820 g.
5.
The weight, X grams, of a particular variety of orange is normally distributed with mean 205 and
standard deviation 25. A wholesaler decides to grade such oranges by weight. He decides that the
smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the
remainder graded as medium.
Determine, to one decimal place, the maximum weight of an orange graded as:
(i) small
(ii) medium.
6.
Jars of bolognese sauce, sold by a supermarket, are stated to have contents of weight 500 g.
The weights, in grams, of the actual contents of jars in a large batch are normally distributed
with mean 506 and standard deviation 5. Find the weight which is exceeded by the contents of
99.9% of the jars in this batch.
7.
The distance, in kilometers, travelled to work by the employees of a city council may be
modelled by a normal distribution with mean 7.5 and standard deviation 2.5. Find d such that 10%
of the council’s employees travel less than d kilometers to work.
8.
An airline operates a service between Manchester and Paris. The flight time may be modelled
by a normal distribution with mean 85 minutes and standard deviation 8 minutes. In order to gain
publicity for the service, the airline decides to refund fares when a flight time exceeds q minutes.
Find the value of q such that the probability of fares being refunded on a particular flight is 0.001.
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HW 1-3: Biased versus Unbiased Questions and Sampling Methods
Tell whether the question is potentially biased. Explain your answer. If the question is potentially biased,
rewrite it so that it is not.
1.
Do you think the city should risk an increase in pollution by allowing expansion of the Northern
Industrial Park?
2.
In a survey about Americans’ interest in soccer, the first 25 people admitted to a high school soccer
game were asked, “How interested are you in the world’s most popular sport, soccer?”
3.
Don’t you agree that the school needs a new baseball field more than a new science lab?
4.
You want to determine whether to serve hamburgers or pizza at a soccer team party.
a) Write a survey question that would likely produce biased results.
b) Write a survey question that would likely produce unbiased results.
5.
You want to find students’ opinions on the current attendance policy. Give two ways that your sample
for the survey might be selected. The first must be an example of a biased sample and the second must
be an example of an unbiased sample. Thoroughly explain your answers.
6.
Two toothpaste manufacturers each claim that 4 out of every 5 dentists use their brand exclusively. Both
manufacturers can support their claims with survey results. Explain how this is possible.
For each situation, identify the sampling technique used. (simple random, cluster, stratified, convenience,
voluntary response, or systematic)
7.
Every fifth person boarding a plane is searched thoroughly.
8.
At a local community College, five math classes are randomly selected out of 20 and all of the students
from each class are interviewed.
9.
A researcher randomly selects and interviews fifty male and fifty female teachers.
10.
A researcher for an airline interviews all of the passengers on five randomly selected flights.
11.
Based on 12,500 responses from 42,000 surveys sent to its alumni, a major university estimated that the
annual salary of its alumni was 92,500.
12.
A community college student interviews the first 100 students to enter the building to determine the
percentage of students that own a car.
13.
The names of 70 contestants are written on 70 cards, The cards are placed in a bag, and three names are
picked from the bag.
14.
Calling randomly generated telephone numbers, a study asked 855 U.S. adults which medical conditions
could be prevented by their diet.
Reiland, NCSU
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HW 1-4: Margin of Error, Population Parameters and Sample Statistics Practice
A survey of a sample population gathers information from a few people and then the results are used to reflect
the opinions of a larger population. The reason that researchers and pollsters use sample population is that it is
cheaper and easier to poll a few people rather than everybody. One key to successful surveys of sample
populations is finding the appropriate size for the sample that will give accurate results without spending too
much time or money.
Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter Olympics
in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed.
This result can be used to predict how many of all 31 million American teens favor ski jumping.
How?
To determine how accurately the results of surveying 900 American teens truly reflect the results of surveying
all 31 million American teens, a margin of error should be given. When pollsters report the margin or error for
their surveys, they are stating their confidence mathematically in the data they have collected. The margin of
1
error can be calculated by using the formula 𝑛, where n is the number in a sample size.
√
1
1
For the above sample, the margin of error would be
= 30 = 0.03, 𝑜𝑟 3%,. Since the actual statistic could
√900
be larger or smaller than the true amount, the margin of error can be expressed as ± 3%.
1. Find the margin of error for a survey of 100 American teens.
2. Compare that to the margin of error for a survey of 90,000 teens.
3. Draw a conclusion about the margin of error based on the size of the sample. Why do you think this is so?
4. If you want to cut your margin of error in half, what would you have to do to the sample size? Why?
5. Find the sample size needed to create a margin of error of 2%.
6. For each statement, identify whether the numbers underlined are statistics or parameters.
a. Of all U.S. kindergarten teachers, 32% say that knowing the alphabet is an essential skill.
b. Of the 800 U.S. kindergarten teachers polled, 34% say that knowing the alphabet is an essential skill.
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7. Of the U.S. adult population, 36% has an allergy. A sample of 1200 randomly selected adults resulted in
33.2% reporting an allergy.
a. Describe the population.
b. What is the sample?
c. Describe the variable.
d. Identify the statistic and give its value.
e. Identify the parameter and give its value.
8. In your own words, explain why the parameter is fixed and the statistic varies.
9. Suppose a 12 year old asked you to explain the difference between a sample and a population, how would
you explain it to him/her? How might you explain why you would want to take a sample, rather than surveying
every member of the population?
10. In your own words, explain the difference between a statistic and a parameter.
11. Determine whether the numerical value is a parameter or a statistic (and explain):
a) A recent survey by the alumni of a major university indicated that the average salary of 10,000 of its 300,000
graduates was 125,000.
b) The average salary of all assembly-line employees at a certain car manufacturer is $33,000.
c) The average late fee for 360 credit card holders was found to be $56.75.
12. Identify each of the following as a parameter or a statistic. If you need to make an assumption about who or
what the population is, explain your assumption.
a. The proportion of voters who voted for President Bush in the 2004 election.
b. The proportion of voters surveyed by CNN who voted for John Kerry in the 2004 election.
c. The proportion of voters among our school's faculty who voted for Ralph Nader in the 2004 election.
d. The average number of points scored in a Super Bowl game.
Reiland, NCSU
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HW 1-5: Simulation Practice with Random Number Table
Random digit table
Line
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
19223
73676
45467
52711
95592
68417
82739
60940
36009
38448
81486
59636
62568
45149
61041
14459
38167
73190
95857
35476
71487
13873
54580
71035
96746
95034
47150
71709
38889
94007
35013
57890
72024
19365
48789
69487
88804
70206
32992
77684
26056
98532
32533
07118
55972
09984
81598
81507
09001
12149
05756
99400
77558
93074
69971
15529
20807
17868
15412
18338
60513
04634
40325
75730
94322
31424
62183
04470
87664
39421
29077
95052
27102
43367
37823
28713
01927
00095
60227
91481
72765
47511
24943
39638
24697
09297
71197
03699
66280
24709
80371
70632
29669
92099
65850
14863
90908
56027
49497
71868
96409
27754
32863
40011
60779
85089
81676
61790
85453
39364
00412
19352
71080
03819
73698
65103
23417
84407
58806
04266
61683
73592
55892
72719
18442
12531
42648
29485
85848
53791
57067
55300
90656
46816
42006
71238
73089
22553
56202
14526
62253
26185
90785
66979
35435
47052
75186
33063
96758
35119
42544
82425
82226
48767
17297
50211
94383
87964
83485
76688
27649
84898
11486
02938
31893
50490
41448
65956
98624
43742
62224
87136
41842
27611
62103
82853
36290
90056
52573
59335
47487
14893
18883
41979
08708
39950
45785
11776
70915
32592
61181
75532
86382
84826
11937
51025
95761
81868
91596
39244
1. A club contains 33 students and 10 faculty members. The students are:
Aisen
Albrecht
Bhagava
Bonini
Chakra
Ding
DuFour
Dwivedi
Gartner
Hollensburg
Huang
Joseph
The faculty members are:
Beck
Burse
Brown
Diamente
Kittaka
Kuhn
Lee
Lipp
Lundberg
Marshall
May
MacDonald
Neukam
Patel
Pham
Ranaweera
Rokop
Sommer
Stuy
Terrell
Thomas
Thompson
Laumeyer
Mitchener
Nesbitt
Sohalski
Twining
Wernke
Thyen
Wang
Yerant
The club can send four students and two faculty members to a convention and decides to choose those who will
go by random selection.
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a. Use the random digit table above to choose a sample of 4 students. Start on line 109. List the numbers and
the names below.
b. Use the random digit table to choose a sample of 2 faculty members. Start on line 112. List the numbers and
the names below.
2. You are a marketing executive for a clothing company. Choose a SRS of 10 of the 440 retail
outlets in New York that sell your company's products. Describe how you would label the retail outlets and
select your sample, starting on line 102.
3. Your school will send a delegation of 35 seniors to a student life convention. 200 girls and 150 boys are
eligible to be chosen. If a sample of 20 girls and a separate sample of 15 boys are each selected randomly, it
gives each senior the same chance to be chosen to attend the convention. Beginning at line 105 in the random
digits table below, select the first four senior girls to be in the sample. Explain your procedures clearly.
4. Five boxes, each containing 24 cartons of strawberries, are delivered in a shipment to a grocery store. The
produce manager always selects a few cartons randomly to inspect. He knows better than to just look at some of
the cartons on the top or only in one box, because sometimes the rotten ones are on the bottom. Today he
wishes to select a total of 6 cartons to inspect. He has the boxes arranged in order and has a set way to count the
cartons inside each box. Explain the process used to make the random selection using a random digit table.
5. Five of the employees at the Stellar Boutique are going to be selected to go to training in Las Vegas for four
days. Everyone wants to go of course, so the owner has decided to make the selection randomly. She has
decided to send two managers and three sales representatives. The employees' names are listed in the table
below.
Managers
Angela
Barbara
Elise
Gigi
Malena
Rosie
Tammie
Veronica
Sales Representatives
Alfie
Betty
Carrie
Cathy
Darcy
Fred
Heidu
Sales Representatives
Irma
Joe
Katarina
Lynn
Marcie
Nancy
Orville
Sales Representatives
Ray
Sandy
Shirley
Suzi
Tawny
Wendy
Zoe
Explain the process she can follow to use a random digit table to select the employees who will get to go to the
training. Select the managers first, then select the sales representatives using the random number table starting
on line 116.
Reiland, NCSU
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7. Use a random digit table, starting on line 112, to select an SRS of five of the fifty U.S. States. Explain your
process thoroughly and report the five states that you chose. Repeat this a second time, but begin on a different
line on the random digit table. Compare your lists to another classmate's lists. Did you end up with any of the
same states in your samples?
Alabama
California
Florida
Illinois
Kentucky
Massachusetts
Missouri
New Hampshire
North Carolina
Oregon
South Dakota
Vermont
Wisconsin
Alaska
Colorado
Georgia
Indiana
Louisiana
Michigan
Montana
New Jersey
North Dakota
Pennsylvania
Tennessee
Virginia
Wyoming
Arizona
Connecticut
Hawaii
Iowa
Maine
Minnesota
Nebraska
New Mexico
Ohio
Rhode Island
Texas
Washington
Arkansas
Delaware
Idaho
Kansas
Maryland
Mississippi
Nevada
New York
Oklahoma
South Carolina
Utah
West Virginia
8. Washington High School has had some recent problems with students using steroids. The district decides that
it will randomly test student athletes for steroids and other drugs. The boy's hockey team is to be tested. There
are 13 players on the varsity team and 21 players on the junior varsity team. Use a table of random digits
starting at line 122, to choose a stratified random sample of 3 varsity players and 5 junior varsity players to be
tested. Remember to clearly describe your process.
10. At the start of this season, Major League Baseball fans were asked which American League Central team
would be most likely to win the division this year. The table below gives the results of the poll.
Most Likely to Win AL Central
Team
Chicago
Cleveland
Detroit
Kansas City
Minnesota
Probability
0.14
0.23
0.33
0.02
0.28
Using the random digit table, starting on line 117, simulate the results when asking 10 fans who they think will
win the AL Central.
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HW 1-6: Expected Value and Fair Games
1. The outcomes of a game of chance are determined by flipping three coins. The payouts for each trial are
below.
Option #1
Outcome
All 3 coins are heads
exactly 2 coins are heads
any other result
Payout
Win $60
Win $5
lose $20
Option #2
Outcome
All 3 coins are heads
exactly 2 coins are heads
any other result
Payout
Win $45
Win $20
lose $25
Option #3
Outcome
All 3 coins are heads
exactly 2 coins are heads
any other result
Payout
Win $30
Win $10
lose $15
You must play this game 40 times. Choose the option you think will win the most money. State your option and
explain why you are choosing this option below.
I am choosing option # __________
Reason:
2. You roll a 6-sided die and win points or lose points as follows:
Die shows a 1, 2 or 3:
Die shows a 4 or 5:
Die shows a 6:
+10 points
- 13 points
- 1 point
a. How many points can you expect to end up with on average (expected value)?
b. Is this a fair game? Why or why not?
3. Two thousand tickets were sold for $25 each to benefit the Lake Zurich Baseball Association. The grand
prize was a $5000 family trip to Florida. 2nd prize was $2500 cash. 3rd prize was $1000 cash. Find the
expected value.
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4. If the sum of two rolled dice is 8 or more, you win $2; if not you lose $2.
a. Is this a fair game? Why or why not?
b. To have a fair game, the $2 winnings should instead be what amount?
5. An 18-year old student must decide whether to spend $160 for one year’s car collision damage insurance.
The insurance carries a $100 deductible, which means that when the student files a damage claim, the student
must pay $100 of the damage amount, with the insurance company paying the rest (up to the value of the car).
Because the car is only worth $1500, the student consults with an insurance agent who presented him with the
following table of damage amounts and probabilities based on the driving records for 18-year olds in the region.
Event
Payoff
Probability
Accident costing
$1500
$1400
0.05
Accident costing
$1000
$900
0.02
Accident costing
$500
$400
0.03
No Accident
$0
0.90
a. What is the expected value of this insurance? What does this mean?
b. Should the student buy the insurance?
6. Throw a die. If you win $2 when the number is even and lose $1 when the number is odd, what is the
expected value? If you pay $1 to play the game, will you win in the long run?
7. A company has a choice of three marketing strategies. The first will cost $150,000 and has a 40% chance of
$1,500,000 in profits and a 60% chance of $500,000 in profits. The second strategy will cost $50,000 and has a
20% chance of $1,000,000 in profits and an 80% chance of $600,000 in profits. The third strategy will cost
$80,000 and has a 50% chance of $1,000,000 in profits and a 50% chance of $400,000 in profits. Which is the
best strategy?
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