Download Quick Review By Question by Mr ali

Quick Review By Question by Mr ali
PART-1
MULTIPLE CHOICE PART (WORTH 36 POINTS).
Question-1: A pair of dice are tossed once, and the outcome numbers, on
each die, x and y, are recorded. The probability of the event, { x  y }, is:
(a)5/36
(b) 5/18
(c) 5/6
(d)5/9
The answer is: c
...........................................................................................................................
Question-2:A pair of fair dice are tossed once, and the outcome numbers, on
each die, x and y, are recorded. The probability of the event, {x + y = 8}, is:
(a)5/36
(b) 5/18
(c) 5/6
(d)5/9
The answer is: a
……………………………………………………………………………………..
Question-3: The mean of the first n positive even integers: 2, 4,...2n, is:
(a) n/ 2
(b) (n +1)/2
(c) n + 1
(d) (n – 1)/2
The answer is: c
..........................................................................................................................
Question-4: The mean for the geometric distribution, with probability for
success, p = 0.8, is:
(a) 1/2
(b) 3/4
(c) 5/3
(d) 5/4
The answer is: d
……………………………………………………………………………………..
Question-5: On the interval [0, 4], the mean value for the probability density
function, f (x) = 1/4, is:
(a)2.5
(b) 2
(c) 1
(d)3
The answer is: b
..........................................................................................................................
Question-6: On the interval [0, 4], the standard deviation for the probability
density function, f (x) = 1/4, is:
(a) 1/2
(b) 3/4
(c) 4/3
(d) 5/3
The answer is: c
..........................................................................................................................
Question-7:Two flocks of 25 Turkey birds each, have mean weight, 4.3 kg,
with standard deviation of 0.3 kg for flock A, and a mean weight of 4.1 kg
with standard deviation 0.4 kg for flock B. The Z statistic needed to decide
whether there is a difference between flocks, A & B, weight means is:
(a)0
(b) 1
(c) 3
(d)4
The answer is: None (Z = 2).
………………………………………………………………………………...
Question-8: An anti-derivative for the function, f ( x)  3  cos 2 ( x)  sin 2 ( x),
is:
(a)2x
(b)3x  1
(c)3t t2
(d) x
The answer is: a
………………………………………………………………………………...
Question-9: The distance traveled by a particle moving according to the
velocity, v(t) = exp(t)  t + 3/2, in the time interval, [0, 1], is:
(a)e 1
(b)e
(c)e + 1
(d)e + 2
The answer is: b
……………………………………………………………………………………..
Question-10: The second derivative of the function, ln(3x), (x > 0), is:
(a) x
(b)x2
(c) 1/x
(a) x = 0
(b) x = 1
(c) x = 2
(a) (, )
(b) (1, 0)
(d) – 1/x2
The answer is: d
………………………………………………………………………………..
Question-11: The function, f (x) = x exp( x), has a stationary point at:
(d) x = 3
The answer is: b
……………………………………………………………………………………..
Question-12: The function, f (x) = x  x 2, has an absolute maximum at:
(c) (1, 5)
(d) (1/10, 1/20)
The answer is: d
……………………………………………………………………………………..
Question-13: The value of the derivative of the function, y (x) = exp(3x) 
2x, at x = 0, is:
(a)5
(b) 
(c)
(d) 3
1
2
Question - 14: If f ( x)  3x  2 x , then the tangent to the graph of f(x) is a
line parallel to the x-axis, for x equal to:
(a) 1/3
(b) – 1/3
(c) 1/6
(d) – 1/6
……………………………………………………………………………...
Question - 15: The solution of the differential equation,
dy / dx  3 exp( 2 x) , with initial condition y(0) = 1, is:
(a) 3 exp( 2 x)  1 / 2 
(c) 3 exp( 2 x)  1 / 2
3
exp( 2 x)  1 / 2
2
3
(d) exp( 2 x)  1 / 2
2
(b)
…………………………………………………………………….........…..
Question - 16: The average value of the function, f ( x)  sin( 2 x) ,
on the interval [0, π/2]
(a)  2 / 
(b)   / 2
(c)  / 2
(d) 2 / 
.....................................................................................................................
Question - 17: The acceleration of a moving particle traveling according to
2
the trajectory, y (t )  exp( t  ln(t ))  t , when, t = 1, is:
(a) e  2
(b) e + 2
(c) e 1
(d) e +1
.......................................................................................................................
3
2
Question -18: The graph of the function f ( x)  x  3x  1 , is concave
down on the interval :
(a) (1)
(b) (1, )
(c) (1)
(d) (1, )
..…………………………………………………………………………….
1
Question - 19: A possible anti-derivative of the function, x  exp( 2 x) ,
is:
2
(b) ln( x)  exp( 2 x) / 2
2
(d) ln( x)  exp( 2 x) / 2
(a) x  2 exp( 2 x)
(c) x  2 exp( 2 x)
…………………………………………………………………………………....
Question - 20: The function, f (x) = |2x3|, is continuous,
but has no derivative value at:
(a) x = 2
(b) x = 3
(c) x= –3/2
(d) x = 3/2
……………………………………………………………………………...
Question - 21: The area under the graph of the function, f ( x)  1  x ,
on the interval [1, 4] equal to:
(a) 3
(b) 23/3
(c) 1
(d) 3/23
…...……………………………………………………………………........
Question - 22: The absolute maximum value of the function,
f ( x)  4  x ,
is equal to:
(a) 0
(b) 4
(c) 4
(d) 2
…………………………………………………………………………………....
Question -23: At the point, x = 0, the value of the second derivative of the
function,
f ( x)  1  x , is:
(a) 
(b) 1/4
(c) non-existent
(d) 0
1
…………………………………………………………………………………....
Question - 24: A committee of 4 persons are chosen at random from a class
containing 7 girls and 9 boys, the probability that the committee will contain
exactly 3 boys, is:
(a)1/3
(b) 21/65
(c) 3/16
(d) 121/130
.......................................................................................................................
Question - 25: On the interval [0, 10], the standard deviation for the
probability density function, f (x) = 1/10, is:
(a)100/3
(b) 25
(c) 25/3
(d) 10 N s.d. = 5 / 3
.......................................................................................................................
Question - 26: A sample of size 100 is to be drawn from a population with
mean 50 and variance 81. The standard error of the sample mean is:
(a) 0.9
(b) 8.1
(c) 0.5
(d) 0.81
.......................................................................................................................
.......................................................................................................................
Question - 27: Consider the list of the grades: 80, 90, 70, 100, 75, 80, 95,
then the triple (mean, median, standard deviation), is:
(a)( 80, 84.3, 11)
(b) ( 84.3, 80, 11) (c) (75, 80, 95)
(d) (75, 84.3, 7)
.......................................................................................................................
Question – 28: A pair of dice are tossed once, and the outcome numbers on
each die, x and y, are recorded. The probability of the event, {x + y = 7 or
9}, is
(a)10/36
(b) 12/36
(c) 17/36
(d) 21/36
.......................................................................................................................
Question – 29: If the probability that Ali will NOT answer any one
question on this CMA04 assignment correctly is 0.7, then his expected grade
out of 40 is:
(a)12
(b) 20
(c) 21
(d) 28
.......................................................................................................................
Question -30: Two persons entered a room, the probability that they were
born at the same day of the week is:
(a)1/7
(b) 6/7
(c) 1/49
(d) 6/49
.......................................................................................................................
Question - 31: The mean for the geometric distribution with probability for
failure, q = 0.2, is:
(a) 1/5
(b) 4/5
(c) 2
(d) 5/4
…………………………………………………………………………………....
Question - 32: Two flocks of 25 Turkey birds each, have mean weight, 4.3
kg, with standard deviation of 0.3 kg for flock A, and mean weight of 4.1 kg
with standard deviation 0.4 kg for flock B. The Z statistic needed to decide
whether there is difference between flocks, A & B, weight mean is,
(a)1.6
(b) 0.8
(c) 2
(d)4
……………………………………………………………………………...
Question - 33: In Question-9, based on the Z statistic obtained, we accept
the null hypothesis, H0, at the significance level of:
(a)10%
(b) 5%
(c) 1%
(d)Reject H0
Question - 34: A pair of fair dice are tossed once, and the product of the
two faces outcome numbers, x and y, is recorded. The probability of the
event, {the product, x y = 49}, is:
(a)1
(b) 1/2
(c) 1/3
(d)0
…………………………………………………………………….........….
Question - 35: On the interval [0, 3], the mean value for the probability
density function, f (x) = 1/3, is:
(a)3
(b) 2
(c) 3/2
(d)1
…………………………………………………………………….........….
Question - 36: The mean of the first 2n positive integers: 1, 2, 3,...n,..2n, is:
(a) 2n+1
(b) (2n+1)/2
(c) n+1
(d) (n+1)/2
......................................................................................................................
Question - 37: The median of the score list: 39, 38, 39, 37, 38, 40, 36, is:
(a)40
(b) 39
(c) 37
(d) 36
The answer is N , The median = 38
……………………………………………………………………………..
Question – 38: A mode for the score list: 39, 38, 39, 37, 38, 40, 36, is:
(a)40
(b) 39
(c) 37
(d) 36
The modes are 38 and 39, only 39 is listed as a solution. So b is solution.
Note: the question is asking for A mode, not the mode (s).
......................................................................................................................
Question – 39: The lower quartile of the list: 39, 38, 39, 37, 38, 40, 36, is:
(a)40
(b) 39
(c) 37
(d) 36
...................................................................................................................
Question -40: The number of groups of size 2 chosen out a 7 people is:
(a)1
(b) 35
(c) 42
(d) 21
..……………………………………………………………………………
Question - 41: The mean for the geometric distribution with probability for
success p = 0.6 is:
(a) 1/2
(b) 1
(c) 5/3
(d) 2
…………………………………………………………………………………...
Question - 42: The mean height of a group of 50 students who showed
interest in athletics is 67.5 inches with standard deviation 2.8 inches, and the
mean height of another group of 50 students who did not show any interest
is 68.2 inches with standard deviation 2.5 inches. The Z statistic that would
be needed to test the null hypothesis H0: “Students in second group are taller
than those of the first group ”, is:
(a)1.32
(b) 1.64
(c) 1.96
(d)2.58
……………………………………………………………………………..
Question -43: In Question-9, based on the Z statistic obtained, we accept the
null hypothesis at the significance level of:
(a)1%
(b) 5%
(c) 10%
(d)1, 5 & 10%
Question - 44: The first derivative for f (x) = sin(x)  cos (x), is:
(a) cos (x)  sin (x)
(b) cos(x) + sin(x)
(c) cos(x) + sin(x) + 1
(d) – cos (x) + sin (x)
……………………………………………………………………………...
Question - 45: The acceleration of a moving particle traveling according to
the trajectory, y(t) = exp (1– t) + ln(t), when t = 1, is:
(a) 1
(b)1
(c) 0
(d) 2
…………………………………………………………………….........…..
Question - 46: Pick the correct value of the following definite integral,
4
 [2  cos ( x)  sin
2
2
( x)]dx
1
(a)3
(b) 3
(c) – 3
(d) – 3
………………………………………………………………………...........
Question - 47: For positive t, the solution of the differential equation,
dy/dt = t , with y (2) = 1, is:
(a) exp (t2) 
(b) t 1
(c) t2/2 
(d)exp(t 2)
.......................................................................................................................
Question -48: Any line parallel to the line given by the equation,
y = (3/2) x + w, has a slope equal to:
(a) 0
(b) 1/2
(c) w
(d) 3/2
..…………………………………………………………………………….
Question - 49: A possible anti-derivative of the function, 1+ exp(x), is:
(a)1 exp(x)
(b)exp(x)
(c) x exp(x)
(d)1/exp(x)
…………………………………………………………………………………....
Question - 50: The function, f (x) = |x2|, is continuous, but has no
derivative value at:
(a) x = 2
(b) x = 3
(c) x= –3
(d) x = – 2
……………………………………………………………………………...
Question - 51: The function, f (x) = exp(x) + x , has a stationary point at:
(a) x = 0
(b) x = 1
(c) x = 
(d) x = 2
…...……………………………………………………………………........
Question - 52: The function, f (x) = 4 + x 2 , has an absolute minimum at:
(a) (2, 0)
(b) (4, 0)
(c) (0, 4)
(d) (0 , 2)
…………………………………………………………………………………....
Question -53: At the point x = 10, the value of the second derivative of the
function,
y (x) = x 2  x, is:
(a) 1
(b) 2
(c) 3
(d) 4
1
……………………………………………………………………………
II. ESSAY QUESTIONS PART:(WORTH 64 POINTS)
Please answers as many questions as possible. Each question has 4 subparts and each sub- part is worth 4 points, for a total of 16 points for each
question. In this part, your grade is the total of the best four (4) grades for
answers to four (4) questions, from amongst your answers to the available
six (6) questions. Please show your work, write down your answers neatly
in the space provided directly below each sub-part.
QUESTION OF BLOCK ‘D’
Question–1: Consider the number list: 29, 29, 28, 32, 29, 31, 28, 31, 33, is:
(a) Find the mode and the median for the number list.
The 9 numbers in the list can be rewritten in increasing order as,
28, 28, 29, 29, 29, 31, 31, 32, 33.
Clearly, in this number list, the mode = median = 29.
(b) Find the lower and upper quartiles, for the number list.
Referring to the list in increasing order above:
The lower quartile [avg of 2nd and 3rd values] = (28+29)/2 = 28.5,
The upper quartile [avg of 7th and 8th values] = (31 + 32)/2 =31.5.
(c) Find the mean for the number list.
The mean = (28 + 28+ 29 + 29 + 29 + 31+ 31+ 32 +33)/9 = 270/9 = 30.
(d) Find the variance and the standard deviation for the number list.
The variance = [2(–2)2 +3(–1)2 + 2(1)2 + (2)2 + (3)2]/8 = 26/8 = 3.25.
The standard deviation = Sqrt(variance) = 3.25  1.8 .
Question–2: Consider the below freezing monthly average temperature list
of a Canadian city:
–29, –29, –28, –32, –29, –31, –29, –31, –33, –30,–31,–32.
(a) Find the mode and the median for the monthly temperatue list.
Clearly, in this list, the mode is –29, while, the median = –30.5.
(b) Find the lower and upper quartiles, for the number list.
The upper quartile = –29, and the lower quartile = – (31 + 32)/2 = –31.5.
(c) Find the mean for the Temperature list.
The mean = – 30.33.
(d) Find the variance and the standard deviation for the number list.
The variance = (1/12)(1/9)[4(4)2 +3(5)2 + 3(2)2 + (8)2 + (1)2] = 2.
The standard deviation = Sqrt(variance) = 2  1.41 .
Question–3: A sample, of 100 pace maker batteries of brand A, showed a
mean life time of 366 days with a standard deviation of 25 days, while a
sample, of 80 pace maker batteries of brand B, showed a mean life time of
360 days, with a standard deviation of 20 days.
(a)Find a, 95% , confidence intervals, for the pace maker population
batteries life means for both brands, A & B.
The Confidence interval,
for Brand A: (366–1.96*25/10, 366+1.96*25/10) = (361.1, 370.9),
for Brand B: (360–1.96*20/9, 360+1.96*20/9) = (355.6, 364.4).
(b) Use this data to find out whether there is a difference between the life
time means of the pace maker batteries populations of brands A and B.
Establish the null, and alternative hypotheses.
Set mean life time of batteries population from brand A =  A ,
Set mean life time of batteries population from brand B =  B .
The null hypothesis: H0 :  A   B .
The alternative hypothesis: H1 :  A   B .
(c)Find the Estimated Standard Error, ESE, for the sampling distribution.
The Estimated Standard Error, ESE, for the sampling distribution is given
by: ESE  252 /100  202 / 80  3.35 .
(d)Find the two sample Test Statistic, Z, and decide whether to accept or
reject the null hypothesis at the, 5%, significance level.
Denote the sample life time mean for brand A = x A .
Denote the sample life time mean for brand B = xB .
Z = [ x A  xB ] / ESE =[366 –360]/ESE = 6/3.354 = 1.79.
So in this case, – 1.96 < Z < 1.96,
and hence, the null hypothesis: H0 is accepted at the, 5% , level.
Question - 4: A sample of a flock of 120 Turkey birds in Farm A, showed a
mean life time of 450 days with a standard deviation of 16 days, while a
sample of 97 Turkey birds in Farm B, showed a mean life time of 400 days
with standard deviation 20 days.
(a) Use the data above to investigate whether there is a difference between
the life time means of Turkey birds Populations in Farms A and B. Establish
the null and alternative hypotheses.
H 0 :  A  B ,
H1 :  A   B .
(b) Find the two sample Test Statistic Z and decide whether to accept or
reject the null hypothesis at the 5% significance level.
(16) 2 (20) 2

 2 .5
120
97
450  400
Z
 20
2 .5
ESE 
Since
Z  1.96 , then H0 is rejected at the 5% significance level
(c) Find a, 99%, confidence intervals, for the life time mean for Turkeys in
Farm A and find a, 90% confidence intervals, for the life time mean for
Turkeys in Farm B.
For Farm A
( X  2.58
SE
n
 16 
 16 
)  (450  2.58
,450  2.58
)
n
 120 
 120 
 (446.23, 453.77)
, X  2.58
SE
, X  1.64
SE
For Farm B
( X  1.64
SE
n
 20 
 20 
)  (400  1.64
, 400  1.64
)
n
 97 
 97 
 (396.67, 403.33)
Question - 5: A sample of 100 Printer Cartridges of brand A (single use),
showed a mean life time efficiency of 42000 print pages with a standard
deviation of 1200 print pages, while a sample, of 100 Printer Cartridges of a
re-filled type brand B ( after multiple refill), showed a mean life time
efficiency of 41500 print pages, with a standard deviation of 1700 print
pages.
(a) Use the data above to investigate whether there is a difference between
the life time efficiency means of Printer Cartridges Populations of brands A
and B(single use vs. multiple use). Establish the null and alternative
hypotheses.
H 0 :  A  B ,
H1 :  A   B .
(b) Find the two sample Test Statistic Z and decide whether to accept or
reject the null hypothesis at the 5% significance level.
2
2
(1200)
(1700)
ESE 

 208.087
100
100
42000  41500
Z
 2 .4
208.087
Since
Z  1.96 , then H0 is rejected at the 5% significance level
(c) Find a, 90%, confidence intervals, for the Printer Cartridges population
life efficiency means for both brands, A & B. Ignoring the inconvenience of
multiple refill, which brand would you recommend for the university, or a
large publishing firm to use?.
For brand A
( X  1.96SE , X  1.96SE )  (42000  1.96(1200), 42000  1.96(1200))
 (40032,43968)
F
or brand B
( X  1.96SE , X  1.96SE )  (41500  1.96(1700), 41500  1.96(1700))
 (38712,44288)
The length of the confidence interval for brand A = 3936,
the length of the confidence interval for brand B = 5576.
So, we will recommend brand B, because it’s confidence interval is longer
than brand A
Question - 6: In an experiment, a pair of fair dices with faces numbered, 1
through 6, are tossed, and the absolute difference , Z  x  y , is recorded at
each toss.
(a) Find the size of the sample space S (the set of different values of Z), and
the median of S. Find the values of Z with the highest frequency (that is find
the mode(s) of the sample space S).
S={0, 1, 2, 3, 4, 5},
Sample size = 6
Median = 2.5
Mode = 1
(b) List all outcomes in the event A = {Z is an odd number} and find the
probability, P (A). For the events B = {Z is a square number}, and
C = {Z is a prime number} find P (B), P ( C), and P(C  B).
A={1, 3, 5},
B={0,1,4},
C={2, 3, 5},
P ( A) 
10  6  2 1
 ,
36
2
P( B) 
P(C ) 
6  10  4 20 5


36
36 9
862 4

36
9
P(C  B)  P(C )  P( B)  P(C  B)

4 5
 0 1
9 9
(c) Find the expected value, and the variance, for the difference, Z, in this
experiment.
6
10
8
, P (1)  , P (2)  ,
36
36
36
6
4
2
P (3)  , P (4)  , P (5) 
36
36
36
5
6
10
8
6
4
2
E ( Z )   iP(i )  0( )  1( )  2( )  3( )  4( )  5( )
i 0
36
36
36
36
36
36
2
 (5  8  9  8  5)  1.94
36
P ( 0) 
n
Var ( Z )   i 2 P(i )   2 
0
10  4(8)  9(6)  16(4)  25(2)
 (1.94) 2  2.06
36
Question - 7: In an experiment, two fair dices with faces numbered, 1
through 6, are tossed, and the sum, Z = x + y, is recorded at each toss.
(a) Find the size of the sample space S (the set of different values of Z), and
the median of S. Find the values of Z with the highest frequency (that is find
the modes of the sample space S).
S  2,3,4,5,6,7,8,9,10,11,12, the size of the sample space = 11
median = 7.
mode = 7.
(b) Find the expected value for the sum, Z, in this experiment.
P(2)=1/36; P(3)=2/36; P(4)=3/36; P(5)=4/36; P(6)=5/36; P(7)=6/36;
P(8)=5/36; P(9)=4/36; P(10)=3/36; P(11)=2/36; P(12)=1/36;
12
E(Z) =
 iP (i)  (7 / 36)(2)(1  2  3  4  5  3)  (7 / 36)(2)(18)  7.
i 2
(c) List all outcomes in the event A = {Z is a square number}. Find the
probability, P ({Z is not a square number}). Consider the events B = {Z is
odd}, and C = {Z is an odd square number}. Find P (B), P ( C), and
P(A  B).
A  4,9 , B  {3,5,7,9,11}, C  {9} = A  B
P(A) = (3+4)/36= 7/36.
P ({Z is not a square number}) = P(Not in A) = 29/36.
P(B)= (2+4+6+4+2)/36=1/2.
P(C)=P({9})=4/36=1/9.
P( A  B)  P( A)  P( B)  P( A  B) =
P( A)  P( B)  P(C ) = (7+184)/36=21/36= 7/12.
QUESTION OF BLOCK ‘C’