AQ086-3-2-EMT3
Final Exam
Page 1 of 9
Answer ALL questions.
QUESTION 1 (25 marks)
(a)
The Enormous Turnip is a Russian folktale, which tells the story of a turnip which grew
so large that in order to pull it out, it required the combined forces of a grandfather, a
grandmother, a granddaughter, a dog, a cat and a mouse. It was not told what happened
after the turnip was pulled out, but there are rumors that the turnip was used to make
soup for the whole village. The following shows the number of villagers and the
respective amount of turnip soup that they ate:
Turnip Soup Consumed (ml) Number of Villagers
20 – 29
5
30 – 39
3
40 – 49
8
50 – 59
20
60 – 69
4
(i)
Construct a table with the following column labels:
(A)
Class Midpoint, x
(1 mark)
(B)
fx
(2 marks)
(C)
2
fx
(2 marks)
(ii)
Hence, calculate the values of the followings:
(A)
Mean,
(2 marks)
(B)
Mode,
(2 marks)
(C)
Median,
(2 marks)
(D)
Standard deviation, s.
(2 marks)
(b)
(i)
(ii)
Level 2
A survey is planned to determine the mean annual family medical expenses of
employees of a large company. The management of the company wishes to be 95%
confident that the sample mean is correct to within  $50 of the population mean
annual family medical expenses. A previous study indicates that the standard
deviation is approximately $400.
(A) How large a sample is necessary?
(3 marks)
(B) If management wants to be correct to within  $25 , how many employees
need to be selected?
(2 marks)
Noise levels at various area urban hospitals were measured in decibels. The mean
of the noise levels in 28 corridors was 62.1 decibels, and the standard deviation
was 7.9 decibels. Establish an interval for the population mean that is 95% certain
to include the true mean. Assume that the variable is approximately normally
distributed.
Asia Pacific University of Technology and Innovation
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AQ086-3-2-EMT3
Final Exam
Page 2 of 9
(7 marks)
QUESTION 2 (25 marks)
(a)
(b)
In a lot suppose 50% of the cars there are manufactured in the United States and 15%
of them are compact. 30% of the cars there are manufactured in Europe and 40% of
them are compact. 20% of the cars are manufactured in Japan and 60% of them are
compact.
(i)
Illustrate the manufacturing of the car in a tree diagram.
(3 marks)
(ii)
If a car is picked at random from the lot, find the probability that it is a compact.
(7 marks)
(iii)
Suppose that the car is picked is a compact car, find the probability that it is
from United States.
(3 marks)
A survey was conducted to study the number of students enrolled in different courses
versus their gender. A random sample of data collected revealed the following:
Gender
Male (M) Female (F)
Business & Management (BM)
20
40
Accounting & Finance (AF)
35
25
Computing & Technology (CT)
30
15
Engineering (E)
20
15
Course of Study
From your cross-tabulation find:
P( M  AF )
(i)
(3 marks)
(ii)
P(CT  F )
(3 marks)
(iii)
P( E | M )
(3 marks)
(iv)
P( F | CT )
(3 marks)
QUESTION 3 (25 marks)
(a)
If X ~ Bin (12 , 0.8) , find:
P( X  1) .
(i)
(7 marks)
(ii)
(b)
Level 2
the mean and variance of X.
(4 marks)
Recent crime reports indicate that 3.1 motor vehicle thefts occur each minute in the
United States. Assume that the distribution of thefts per minute can be approximated
Asia Pacific University of Technology and Innovation
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(c)
Page 3 of 9
Final Exam
by Poisson distribution, calculate the probability that there are more than 3 thefts in a
two-minute interval.
(8 marks)
The Mathematics examination mark for students follows normal distribution, with
mean of 45 and standard deviation of 8.
(i)
Find the probability of students who score lower than 50 marks.
(2 marks)
(ii)
Find the probability of students who score between 40 and 60 marks.
(4 marks)
QUESTION 4 (25 marks)
(a)
(b)
You are the manager of a restaurant that delivers pizza to college dormitory rooms. You
have just changed your delivery process in an effort to reduce the mean time between
the order and completion of delivery from the current 25 minutes. A sample of 28 orders
using the new delivery process yields a sample mean of 22.4 minutes and a sample
standard deviation of 6 minutes.
(i)
Conduct hypothesis testing, at significance level of 0.05. Is there evidence that
the population mean delivery time has been reduced below the previous
population mean value of 25 minutes?
(10 marks)
(ii)
If the hypothesis testing is conducted at significance level of 0.01, what would
be the conclusion?
(3 marks)
A sample of 9 clerical staff’s salaries (RM hundred) versus their length of service
(years) with a company is as shown below:
Length of service
Salary
(years)
(RM hundred)
x
y
1
9
2
12
3
14
5
15
5
16
6
20
8
22
9
25
12
30
Computation results:
n = 9 ;  x =51 ;  y = 163 ;
(i)
(ii)
(iii)
Level 2
x
2
= 389 ;
 xy =1111 ;  y
2
=3311
Compute the regression equation in the form yˆ = a + bx . You may use the
computation results given above.
(5 marks)
In the context of this question, interpret the meaning of a and b in part (a).
(2 marks)
Estimate the salary if the length of service was 30 years.
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Page 4 of 9
Final Exam
(1 mark)
(iv)
Compute Pearson’s coefficient of correlation (r).
(2 marks)
(v)
2
Compute the coefficient of determination (R ). Interpret your result.
(2 marks)
Level 2
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Page 5 of 9
Final Exam
FORMULAE AND TABLES
(a)
Coefficient of Variation:
CV =
(b)
(e)
X
100%
s
CV =
Coefficient of Skewness:
Mean − Mode
SK1 =
SD

100%

SK 2 =
3 ( Mean − Median )
SD
Probability
(i)
P( A  B) = P( A) + P( B) − P( A  B)
(ii)
P( A) = 1 − P( A)
(iii)
P( A | B) =
P( A  B)
P( B)
(iii)
P( A | B) =
P( B | A) P( A)
P( B)
(iv)
P( Ai | B) =
P ( Ai ) P ( B | Ai )
P ( A1 ) P ( B | A1 ) + P ( A2 ) P ( B | A2 ) +
(d)
Binomial Distribution
n
n− x
P ( X = x ) =   p x (1 − p ) , x = 0,1,..., n
 x
(e)
Poisson Distribution
P ( X = x) =
e−    x
, x = 0,1,...
x!
(f)
Normal Distribution
X −
Z=

(g)
Confidence Interval
(i)
 s 


, n −1  n 
2
Confidence interval for µ: = X  t 
Z / 2   

e


Sample size: n = 
(ii)
Level 2
+ P( An ) P ( B | An )
2
 pˆ qˆ 


n

2
Confidence interval for p: pˆ  Z  
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Sample size, n =
(h)
Page 6 of 9
Final Exam
(Z / 2 )2  pˆ qˆ
e2
Hypothesis Testing:
Test statistics formula for:
(i)
hypothesis testing for µ when σ known, Z =
X −

n
(ii)
(iii)
(iv)
(i)
X −
s
n
X −
hypothesis testing for µ when σ unknown and n < 30, t =
s
n
pˆ − p
hypothesis testing for p, Z =
pq
n
hypothesis testing for µ when σ unknown and n ≥ 30, Z =
Regression Equation
y = a + bx where b =
(j)
(
)
n  x 2 − ( x )2
and a =
y
x
−b
n
n
Correlation
(i)
Product Moment Correlation Coefficient
n xy − ( x )( y )
r=
n x 2 − ( x )2 n y 2 − ( y )2

(ii)
Level 2
n xy − ( x )( y )


Spearman Rank Correlation Coefficient rs = 1 −
6 d 2
n n2 −1
(
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Page 7 of 9
Final Exam
STANDARD NORMAL CUMULATIVE PROBABILITY TABLE
Cumulative probabilities for NEGATIVE z-values are shown in the following table:
Level 2
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Page 8 of 9
Final Exam
Cumulative probabilities for POSITIVE z-values are shown in the following table:
Level 2
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Final Exam
Page 9 of 9
t-DISTRIBUTION
Level 2
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