Institute of Business Management, Karachi,
Department of CCS&IS (Mathematics & Statistics)
Course: Model & Interference [STA-301, OM-10580]
Friday, July 18, 2021
Summer (Crash) - 2021
Course Supervisor: Ishtiaq Ahmed
Practice Exercise - 1
(Application of Normal Continues Probability Distribution and Applications of z-score)
Q.1
Read Page 257 to 269 from the text book “Statistics for Management” for the concept of Normal Probability
Distribution.
Q.2
Convert the following sample data into standard scores and show that its mean is zero and standard deviation
equal to one.
3
4
0
4
1
3
6
1
5
Q.3
Convert the following sample data into Z-scores (standard scores) and show that its mean is zero and standard
deviation one.
X: 25 14
36
35
30
34
52
26
26
33
44
45
42
Q.4
Use above data of Q.3 and find the follow probabilities with bell-shape illustration:
a)
Pr (X ≤ 20)
b)
Pr (28 ≤ X )
c)
Pr (16 ≤ X ≤ 26)
d)
Pr (19 ≤ X)
e)
Pr (X ≤ 30)
f)
Pr (18 ≤ X ≤ 42)
Q.5
A survey of per capita income indicated that the annual income for people in one state is normally distributed
with a mean of $9,800 and a standard deviation of $1,600. If a person is selected at random, what is the
probability that the person’s annual income is a) greater than $5,000, b) greater than $12,200, c) between
$8,520 and $12,200, d) between $11,400 and $13,000, e) greater than $8,200, f) less than $15,800 and g)
between $9,000 and $13,000? Also show by normal curve.
[Answers: a) 99.87% b) 6.680% c) 72.13% d) 13.59% e) 84.13%, f) 98.78% and g) 66.87%]
Q.6
The weights of newborn babies at a particular hospital have been observed to be normally distributed with a
mean of 7.4 pounds and a standard deviation of 0.4 pound. What is the probability that a baby in this hospital
will weigh a) more than 8 pounds? b) Less than 7 pounds?
[Answers: a) 6.680% b) 15.87%]
Q.7
A student gets 82 marks in a final examination in Accounting; the mean is 75 marks with a standard deviation of
10 marks. In Economics he gets 86 marks in the final examination on which the mean is 80 marks with a
standard deviation of 14 marks. Is this relative standing better in Accounting or Economics? Draw conclusions
with reference to obtain z-scores.
[Student’s relative performance in Accounting was higher than his performance in Economics]
Q.8
The mean temperature in Murree in March is 12oC with a standard deviation of 0.4oC. On March 10 the
temperature is 2.5 standard deviations above the mean. What is the temperature on March 10?
[Answer: Temperature on March 10 was 13oC]
Given a normal distribution with mean = 50 and s.d = 10, find the probability that X assumes a value between
45 and 62. Also show the probability area by normal curve.
[Answer: 57.64%]
Q.9
Page 1 of 3
Model & Inference
Answers:
a) 8.38%
b) 72.21%
c) 17.73%
d) 93.06%
e) 34.83%
f) 72.70%
Summer(Crash)-2021
Q.10
Given a normal distribution with mean = 300 and s.d = 50, find the probability that X assumes a value greater
than 362.
[Answer: 10.75%]
Q.11
Given a normal distribution with mean = 40 and s.d = 6, find the value x that has a) 38% of the area below it and
b) 5% of the area above it.
Answers: a) x = 38.14, b) x = 49.87]
Q.12
A certain types of storage battery lasts on the average 3.0 years, with a standard deviation of 0.5 year.
Assuming that the battery lives are normally distributed, find the probability that a given battery will last less
than 2.3 years.
[Answer: 8.08%]
Q.13
An electrical firm manufactures light bulbs that have a length of life that is normally distributed with mean equal
to 800 hours and a standard deviation of 40 hours. Find the probability that a bulb burns between 778 and 834
hours.
[Answer: 51.11%]
Q.14
On an examination the average grade was 74 and the standard deviation was 7. If 12% of the class are given
A’s, and the grades are curved to follow a normal distribution, what is the lowest possible of grade A and
highest possible of grade B?
[The lowest grade A is 83 and the grade B is 82]
Q.15
Given a normal distribution with mean = 40 and s.d = 6, find
a)
the area below 32;
b)
the area above 27;
c)
the area between 42 and 51;
d)
the X value that has 13% of the area above it;
e)
the X value that has 45% of the area below it.
Answers:
a) 9.18%
b) 98.50%
c) 33.71%
d) x = 46.78
e) x = 39.22
Q.16
Given a normal distribution with mean = 200 and Variance = 100, find
a)
the area below 214
b)
the area above 179;
c)
the area between 188 and 206;
d)
the X value that has 80% of the area below it;
e)
the two X values containing the middle 75% of the area.
Answers:
a) 91.92%
b) 98.21%
c) 61.06%
d) x = 208.4
e) x1 = 188.5
x2 = 211.5
Q.17
Different typing skills are required secretaries depending on whether one is working in a law office, an
accounting firm, or for a research mathematical group at a major university. In order to evaluate candidates for
these positions, an employment agency administers three distinct standardized typing samples. A time penalty
has been incorporated into the scoring of each sample based on the number of typing errors. The mean and
standard deviation for each test together with the score achieved by a recent applicant are given in the following
Table:
Sample
Applicant’s Score
Mean
Standard Deviation
Law
141 seconds
180 sec
30 sec
Accounting
07 minutes
10 min
02 min
Scientific
33 minutes
26 min
05 min
Required:
For what type of position does this applicant seem to be best suited?
[Answer: Accounting seem to be best suited]
Q.18
Page 2 of 3
In preparation for a spring clearance sale at a women’s apparel shop, 2000 brochures were placed in
preaddressed envelopes, sealed, and stamped. This process took an average of 7 seconds per envelope with a
standard deviation of 1.8 seconds. Assuming this population of times to be approximately bell-shaped.
Model & Inference
Summer(Crash)-2021
Required:
Find the number of envelopes that took
a)
from 5.2 to 8.8 seconds to prepare;
b)
from 3.4 to 10.6 seconds to prepare;
c)
from 1.6 to 12.4 seconds to prepare.
Answers:
a) 1,365 or 1,366 envelopes
b) 1,908 or 1,909 envelopes
c) 1,994 or 1,995 envelopes
Q.19
Mathematics achievement test scores for 1000 students were found to have a mean and standard deviation of
500 and 100, respectively. If the distribution of sores is bell-shaped, approximately how many of the scores
Answers:
a)
fall in the interval 400 – 600?
a) 680 scores
b)
exceed 700
b) 25 scores
c)
fall below 200
c) 1 or 2 scores
Q.20
A pharmaceutical store delivers an average of 40 prescription-filled orders a day, with a standard deviation of 8
orders. Assuming a bell- shaped population for daily number of prescriptions delivered, what percentage of the
time will the driver make
Answers:
a)
between 24 and 56 deliveries?
a) 95.44%
b)
less than 32 deliveries?
b) 15.87%
Q.21
If a set of grades on a Marketing course examination are approximately normally distributed with a mean of 76
and a standard deviation of 7.9, find
Answers:
a) 66
b) 88
c) 78
Required:
i)
the lowest passing grade, if the lowest 10% of the students are given F’s;
ii)
the highest B, if the top 5% of the students are given A’s;
iii)
the lowest B if the top 10% of the students are given A’s and the next 25% are given B’s.
Q.22
In a mathematics examination the average grade was 82 and the standard deviation was 5. All students with
grads from 88 to 94 received a grade of B. If the grades are approximately normally distributed and 8 students
received a B grade, how many students took the examination? [Ans.: 75]
- : Good Luck : -
Page 3 of 3
Model & Inference
Summer(Crash)-2021