Download Exercise 1 - Ms. Leslie Math Website

Variability and Statistical Analysis
SENIOR 4 APPLIED MATHEMATICS EXERCISES
Exercise 1: Mean and Standard Deviation
1. For each of the following sets of data calculate the mean, median, mode and
standard deviation.
A
45
45
47
47
55
63
63
65
65
B
15
26
40
49
55
55
75
82
98
C
4
11
27
33
52
60
83
85
140
2. For each of the sets of information given in the following tables use the
calculator to find the mean, median, mode and standard deviation.
Late Prehistoric Arrowhead Measurements (in mm)
Big Goose Creek (52 arrowheads)
Length
Width
16
16
17
17
18
18
18
18
19
20
20
21
21
21
22
22
22
23
23
23
24
24
25
25
25
26
26
26
26
27
27
27
27
27
28
28
28
28
29
30
30
30
30
30
30
31
33
33
34
35
39
40
10
11
11
11
11
11
12
12
12
12
12
12
12
12
13
IS
13
13
13
13
13
13
13
13
13
13
13
13
13
13
14
14
14
14
14
14
14
14
14
14
14
14
14
15
15
15
15
15
15
16
17
18
5
6
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10
11
11
11
12
12
13
Neck Width
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Variability and Statistical Analysis
SENIOR 4 APPLIED MATHEMATICS EXERCISES
3. A class of 30 students received the following marks in a mathematical
examination. Use technology to find the mean, median, mode and standard
deviation of this set of scores.
78
92
62
52
65
59
53
63
68
73
71
63
69
74
73
81
55
71
75
81
84
77
80
75
41
57
91
62
65
49
4. Potatoes are sold at E and S Supermarket in 5 kg bags. Ten bags are
selected at random to provide an estimate of the mean and standard
deviation of all the bags sold at this supermarket. The 10 bags weighed the
following (in kg): 5.4, 5.4, 5.3, 5.2, 5.3, 5.3, 5.1, 5.0, 4.9, and 5.1. Give the
estimates for the mean and standard deviation obtained.
5. Calculate the mean and the standard deviation for the following data, based
on the heights of 100 Senior Years students.
height
153.5 to 160.5
160.5 to 167,5
167.5 to 174.5
174,5 to 181.5
181.5 to 188.5
TOTALS
class
mark
157
164
171
178
185
frequency
5
16
43
27
9
100
6. Calculate the mean and the standard deviation for the following data, based
on the weights of 125 newborn infants, in pounds.
weight
3.5 to 4.5
4.5 to 5.5
5.5 to 6.5
6.5 to 7.5
7.5 to 8.5
8.5 to 9.5
9.5 to 10.5
10.5 to 11.5
TOTALS
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class
mark
4
5
6
7
8
9
10
11
frequency
4
11
19
33
29
17
8
4
125
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Variability and Statistical Analysis
7. Three manufactures of toasters claim that the average life of their toasters,
under normal use, is five years. A consumer’s group decided to test each
company’s claim. It compiles the following list, in years, of toasters
manufactured by each:
Company X:
0.5, 1.6, 2, 3.5, 4, 4.5, 6, 7, 7.9, 8, 10
Company Y:
4, 4, 5, 5, 5, 6, 11, 13, 14, 15, 16
Company Z:
2, 3, 4, 4, 6, 13, 14, 15
a. Which “average” was each company using to support its claim?
b. From which company would you buy a toaster? Why?
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Exercise 2:
1.
Variability and Statistical Analysis
The Normal Distribution and the Approximation to the
Normal Distribution.
Refer to question 1, Ex. 1.
a.
Give the boundaries of one standard deviation above and below the
mean.
(    ,   )
b.
Find the percent of data lying within one standard deviation of the
mean.
(    ,   )
c.
Is this distribution considered normal?
2.
Refer to question #2, Ex. 1.
Find the percent of scores lying within one standard deviation of the mean.
(    ,    ). Do this for each of the sets. Are the distributions
considered “normal”?
3.
A machine packaging candy in 90 gram packages is thought to be faulty.
Ten packages were randomly selected, and the actual masses, in grams
are:
86, 91, 89, 92, 90, 93, 90, 90, 91, 88
If the spread of the masses is too great, the machine is considered to be
faulty. Statistically, we use the standard deviation to judge this. In this
case, if the standard deviation of the set of scores is greater than 1.3, the
machine is considered faulty and will need adjustment or repair. Is the
machine faulty? How did you determine the answer?
4.
Using a graphing calculator, find the mean and standard deviation of the
following frequency distribution. Find the percentage of scores that lie
within 1 standard deviation of the mean. Explain why this is or is not a
normal distribution.
Scores
0
1
2
3
4
5
6
7
8
9
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Frequency
80
60
80
70
60
80
50
60
70
60
Page 4 of 10
Variability and Statistical Analysis
SENIOR 4 APPLIED MATHEMATICS EXERCISES
5.
Using a graphing calculator, find the mean and standard deviation of the
following frequency distribution. Find the percentage of scores that lie
within 1 and 2 standard deviations of the mean. Explain why this is or is
not a normal distribution.
Scores
5
6
7
8
9
10
11
6.
Frequency
9
11
22
38
19
13
8
The administration of a medical clinic wishes to find out how long people
have to wait to be seen by a doctor. The waiting times (in minutes) for 20
patients were recorded as follows:
5.5
7.9
4.5
9.5
7.9
4.2
5.8
1.5
12.0
15.0
4.8
3.3
1.5
6.5
10.6
12.7
8.8
11.0
20.0
7.8
Find the mean, the median and the standard deviation for this data.
Is this data considered a normal distribution for one standard deviation?
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Variability and Statistical Analysis
SENIOR 4 APPLIED MATHEMATICS EXERCISES
Exercise 3: Z-scores
1,
The average mark in Sarah’s English class is 60%, with a standard
deviation of 16. The average mark in her math class is 58%, with a
standard deviation of 10. If Sarah scores 72% in English, and 68% in
math, how many standard deviations is each of her grades above
average? What does this tell us about her performance in the two
subjects?
2.
Peter’s goal is to maintain his marks at least 2.5 z-scores above the mean
in all of his subject. Determine the minimum marks he must obtain in each
subject.
Subject
Mean
Chemistry
English
Math
66
62
68
Standard
Deviation
7
12
8
Physics
73
4
Minimum
Mark
3.
Two students applying for a scholarship have the following scores.
Johanna has a mark of 82% in the College Entrance Survey Test, which
has a mean of 75% and standard deviation of 8%. Harvey has a mark of
78% in the Scholarship Aptitude Exam, which has a mean of 70% and
standard deviation of 5%. Use standardized scores to compare the marks.
4.
A college conducts an entrance exam consisting of an English language
skills test and a mathematics test. Alycia scored 210 out of 300 on the
language test and 540 out of 600 on the mathematics test. Compare
Alycia’s performance on the two parts of the test in the following cases.
English Test
Mean
Case 1
Case 2
Case 3
5.
80%
60%
60%
Mathematics Test
Standard
Deviation
5%
10%
5%
Mean
83%
83%
90%
Standard
Deviation
7%
3.5%
3.5%
The IdleWyld College of Advanced Technology advertises that the
minimum requirement in mathematics for entry to courses in the Computer
Engineering Department is 90% in an Entrance Test in which the mean
mark is known to be 78% with standard deviation of 8%. Liem has taken
the Joint Examining Boards test (which is acceptable to the College that
has a mean of 73% with standard deviation of 10%. What is the minimum
acceptable mark in this test?
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Variability and Statistical Analysis
Exercise 4: Normal distribution and Z-Scores
Use the graphing calculator to solve the following problems.
1. In a normal distribution, find the probability that a score picked at random lies
in each interval described by the following z-scores.
a) z< 0
d) z < 3
g) z > 2.5
b) z<1
e) z < 3.5
h) z > -2
c) z < 2.5
f) z > 0
2. In a normal distribution, find the probability that a score picked at random lies
in each interval described by the following z-scores.
a) –1 < z <1
d) -1.645< z < 1.645
g) -0.5 < z < 1.2
b) –1.96 < z < 1.96
e) 1 < z < 2
c) –2 < z < 2
f) 0.5 < z <2.7
3. Find the z-score, p. where p > 0, so that the area between z = 0 and z = p is
given.
a) Area = 0.1700
b) Area = 0.4500
c) Area = 0.4750
d) Area = 0.49400
4. Find the z-score, p and -p so that the area between z = -p and z = p is given.
a) Area = 0.9000
e) Area = 0.7500
b) Area = 0.5000
d) Area = 0.9500
5. In a normal distribution, find the probability that a score picked at random lies
in each interval described by the following 2-scores.
a) z >1.96 or z< -1.96
b) z > 2.5 or z < -2.5
c) z > 1 or z < -1
d) z > 0.675 or z < -0.675
6. According to Nielsen Media Research, people watch television an average of
6.98 hours per day. Assume that those times are normally distributed with a
standard deviation of 3.80 hours. Find percentage of viewers who watch
television more than 8.0 hours per day
7. In a court of law a woman testified that she gave birth 300 days after
conception. Normal pregnancies last 268 days with a standard deviation of 15
days. Is it reasonable to believe that she could be telling the truth? Explain
your reasoning.
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Variability and Statistical Analysis
8.
The weights of men aged 18 to 74 are normally distributed with a mean of
173 pounds and a standard deviation of 30 pounds (based on a national
health survey). Find the percentage of the population that has weights
between 190 and 225 pounds. In a group of 400 men aged 18 to 74 years,
how many are expected to weigh between 190 and 225 pounds?
9.
Scores on a university entrance biology aptitude test have a mean score of
8.0 and a standard deviation of 2.6. If 600 prospective students wrote the
test, how many would be expected to score between 6.0 and 7.0?
10. The cholesterol levels in men aged 18 to 74 are normally distributed with a
mean of 178,1 and a standard deviation of 40.7. All units are in mg/1OO mL
of blood. What is the probability that the cholesterol level of a randomly
selected man (18 to 74 years) is between 100 and 200?
11. The heights of six-year-old girls are normally distributed with a mean of
117.8cm and a standard deviation of 5.52 cm. Find the probability that the
height of a randomly selected six-year-old girl will be between 117.8 cm and
120.56 cm.
12. The mean pulse rate of males aged 18 to 25 years is 72 beats per minute,
and the standard deviation is 97. If the military states that a rate of 100 or
higher is unsuitable for army recruits, what portion of the male population (18
to 25) would be unsuitable for military service on these grounds?
13. Two students have obtained the following marks in comparable math
courses. Juana has 82% in an exam with a mean score of 78% and a
standard deviation of 5%. Hans scored 73% in an exam with a mean of62%
and standard deviation of 8%. Use z-scores to compare their results.
Comment on these results.
14. The principal and staff of a school decide to use z-scores to translate test
marks into letter grades. They agree to the following:
F -D -C -B -A–
less than –1
from –1 and less than –0.5
from –0.5 and less than 0.5
from 0.5 and less than 1
1 and above.
The test scores for one class of 36 students were:
23, 34, 36, 39, 42, 44, 48, 50, 52, 54, 54, 55, 62, 62, 63, 64, 64,
65, 66, 67, 70, 71, 71, 75, 80, 81, 83, 85, 87, 88, 89, 94, 96, 98
100, 100
Find the zones of scores for the letter grades.
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Variability and Statistical Analysis
Exercise 5: Confidence Intervals
(For each of the following find the mean and the standard deviation)
  np
  npq
1.
The survey of all the students in a school indicated that 32% of the
students are left-handed. Find an interval for the number of left-handed
students expected to be enrolled in a class of 24 students 95% of the time.
2.
Records at a garden center indicate that 75% of a particular kind of tulip
bulb will grow successfully. A customer buys 20 such bulbs. Find the
95% interval for the number of bulbs from the 20 that will grow
successfully.
3.
Fred takes a duck hunting holiday. He has a probability of 0.4 of hitting
any duck that he shoots. He plans to take 30 shots. Find a confidence
interval for the number of ducks he could expect to shoot that would be
true 19 times out of 20.
4.
If 15% of all computer chips manufactured are faulty, find a 90%
confidence interval for the mean number of faulty chips in batches of 100
chips.
5.
A long term study shows that 20% of all cars have oil leaks. A sample of
140 cars is tested for oil leaks. Would it be reasonable to expect 40 cars
to have oil leaks? Explain.
6.
A shoe store owner knows from past experience that 75% of all customers
pay with credit cards. In one day the store sold shoes to 120 customers.
How many customers would you expect to pay by credit card 19 times out
of 20?
7.
A recruiter for the armed forces finds that 55 out of 200 candidates tested
are unfit for service. Find a 95% confidence interval for the actual
percentage of candidates who are unfit for service.
8.
The highway patrol discover that, at one location, there is a 60%
probability that a motorist will exceed the speed limit. Assume that 3500
cars drive this road daily. Determine a 90% confidence interval for the
mean number of motorists exceeding the speed limit daily
9.
In a sample of 100 students, 72 favoured a change in school hours. Find
a 95% confidence interval for the percentage of all students in favour of a
change in school hours.
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SENIOR 4 APPLIED MATHEMATICS EXERCISES
Variability and Statistical Analysis
Exercise 6: Opinion Polls
(For each of the following find the mean and the standard deviation)
  np
  npq
1. There are two candidates in an election, A and B. A poll of 400 voters
selected at random finds that 208 intend to vote for Candidate A. Give a 95%
confidence interval for the percentage of voters favourable to A at the time of
the poll. (Assume that there are no undecided voters.)
2. A public opinion survey has shown that 80 persons in a random sample of 400
persons answered “Yes” to a certain question. Within what limits can it be
asserted that the true population percentage of “Yes” voters would lie?
3. In a random sample of 100 people in a particular district, 70 people were in
favour of declaring March 2lst a public holiday. Find a 95% confidence limit for
the percentage of all people who would be in favour of this public holiday.
4. A pollster interviewed 1000 people selected randomly and found that 650 of
them preferred Brand X to Brand Y. Find a 95% confidence interval for the
percentage of the whole population that would prefer Brand X.
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