NATIONAL INSTITUTE OF TECHNOLOGY CALICUT
Department of Mathematics
Winter Semester 2024-25
MA2012E MATHEMATICS IV
Tutorial Sheet V
1. The kilometer per liter (km/l) figure for a new engine are recorded for fixed speeds between 50 and 112 km/
h
56
104
64
88
112
96
84
68
80
100 60
14.7 13.2 14.5 13.2 12.8 13.4 13.3 14.5 13.8 13
72
14.6 14.3
Speed (km/hr)
Mileage (km/l)
Determine the best fit straight line for the mileage results and predict the mileage for a speed of 90 km/
hr and 110 km/hr.
2. The following data pertain to the number of jobs per day and the CPU time required.
No. of jobs :X
CPU Time: Y
1
2
2
5
3
4
4
9
5
10
Obtain a least square fit of a line to the observations on CPU time. Also estimate the mean CPU time when
X = 3.5.
3. The production rate data shown is the result of using a new technique developed by a method analyst. The
output in units per hour is measured after each 25 units to determine when the worker has reached the
standard of 40 units/hour. The data for eight new employees are given below:
Total No. of units produced
25
50
75
100
125
150
175
200
Units per hour produced
16
28
28
29
30
32
40
42
Fit a second degree polynomial and an exponential curve using the method of least squares. Which one
is the best fit?
4. The following data pertains to the cosmic ray doses measured at various altitudes
Altitude (feet)
X
50 450
780
1200 4400 4800
5300
Dose rate (unit/year)
Y
28 30
32
36
69
51
58
(i) Fit a curve of the form y = e a x+b
(ii) Estimate the mean doses at an altitude of 3000 ft.
5. The following data pertains to the demand for a product (in thousands of units) and its price (in cents)
charged in 5 different market areas
Price
X
20
16
10
11
14
Demand
Y
22
41
120
89
56
Fit a power function ∝ x β and use it to estimate the demand when the price of the product is 12 cent.
6. A correlation study between decreasing chemical potency and time exposed to artificial room light resulted
in r = 0.06 for a sample of size 24. Is there a significant correlation present?
7. Percentage of moisture in sausage after smoke curing for a certain time is given as follows:
Time (h)
115
125
185
200
75
80
150
175
90
100
Percentage of moisture
21
19
11
10
50
42
13
12
26
24
(a)
(b)
Is there is any significant correlation between the moisture content and smoke curing
times? (1- ∝ = 0.95)
Construct a 95% confidence interval for the correlation coefficient.
8. Associate with a computer job there are 2 random variables: CPU time required (Y) and the number of disk
I/O operations (X). Given the following data.
Time (Sec)
Y
40
38
42
50
60
30
20
25
40
39
Number
X
398
390
410
502
590
305
210
252
398
392
(i) Find r
(ii) Obtain 90% CI for ρ
(iii) Test for the significance of ρ.
9. A safety engineer is testing 4 different types of smoke alarm systems. After installing 5 of each type in a
smoke chamber, he introduced smoke to uniform level, electrically connected the alarms, and observed the
reaction time in seconds. Is there a significant difference in the reaction time of the 4 types?
Alarm type
Observations
1
2
3
4
1
5.2
7.4
3.9
12.3
2
6.3
8.1
6.4
9.4
3
4.9
5.9
7.9
7.8
4
3.2
6.5
9.2
10.8
5
6.8
4.9
4.1
8.5
10. The following are the weight loses of certain machine part in (milligrams) due to friction, when three
different lubricants were used under controlled conditions
Lubricant
Lubricant
Lubricant
A
B
C
:
:
:
12.2
10.9
12.7
11.8
5.7
19.9
13.1
13.5
13.6
11.0
9.4
11.7
3.9
11.4
18.3
4.1
15.7
14.3
10.3
10.8
22.8
8.4
14.0
20.4
Test at the 0.01 level of significance whether the differences among the sample mean can be attributed
to chance.
11. To find the best arrangement of instruments on a control panel of an airplane, 3 different arrangements were
tested by simulating an emergency condition and observing the reaction time required to correct the
condition. The reaction times (in tenths seconds) of 28 pilots (randomly assigned to the different
arrangements) were as follows.
Arrangements I
14 13 9 15 11 13 14 11
Arrangements II
10 12 9 7
11 8
12
9 10 13 9 10
Arrangements III
11 5
9 10 6
8
8
7
Test at the level of significance ∝ = 0.01, whether we can reject the null hypothesis that the differences
among the arrangement have no effect.
12. Three different interactive systems are to be compared with respect to their response time to an editing
request owing to chance fluctuations, among the other transactions in process. It was decided to take 10 sets
of samples at randomly chosen times for each system and record the response time as follows:
Session
System response time (Sec.)
A
B
C
1
0.96
0.82
0.75
2
1.03
0.68
0.56
3
0.77
1.08
0.63
4
0.88
0.76
0.69
5
1.66
0.83
0.73
6
0.99
0.74
0.75
7
0.72
0.77
0.60
8
0.86
0.85
0.63
9
0.97
0.79
0.59
10
0.90
0.71
0.61
Test at 1% level of significance, whether there is a significant difference in the responsiveness of the 3
systems.
13. The lifetime of a tube ( X 1 ) and filament diameter ( X 2 ) are distributed as a bivariate normal with following
2
2
parameters: µ1 = 2000 hrs, µ 2 = 0.10 inch, σ12 = 2500, hrs σ22 = 0.01 inch , ρ = 0.87. The
quality control manager wishes to determine the life of each tube by measuring the filament diameter. If a
filament diameter is 0.095, what is the probability that the tube will last 1900 hours?
14. A college professor has noticed that grades on each of quizzes have a bivariate normal distribution with
following
parameters:
µ1 = 75, µ 2 = 83 σ12 = 25, σ22 = 16, ρ = 0.8.
If a student receives a grade of 80 on the first quiz, what is the probability that he will do better on the
second one? How is the answer affected by making ρ = − 0.8?
15. The following tables gives the results of measurements of train resistance:
V
20
40
60
80
100
120
R
5.5
9.1
14.9
22.8
33.3
46.0
Here, V is the velocity in Kms/hour and R is the resistance in Kg/ quintal. Fit a relation of the form
R = a + bV + cV 2 to the data. Estimate R when V=90.
16. As head of a department of a consumer’s research organization, you have responsibility for testing and
comparing lifetimes of light bulbs for four brands. Your test data are as shown below, each entry
representing the life time of a bulb, measured in hundreds of hours:
Brand
A
B
C
D
20
19
21
25
23
21
24
20
22
23
20
20
Can we infer that the mean life-times of the four brands are equal?
17. Given the following data on the number of hours that 10 students studied for a test and their scores
in the test:
Hours studied x:
4
9
10
14
4
7
12
22
1
17
Test score
y:
31
58
65
73
37
44
60
91
21
84
(i) Test the null hypothesis that the slope of the regression line β = 3 against the alternative
hypothesis
β > 3 at 0.01 level of significance. (ii) Obtain a 95% confidence interval for β.
18. The frequency of chirping of a cricket is thought to be related to temperature. This suggests the
possibility that temperature can be estimated from the chirp frequency. Let the following data on
the number of chirps per second, x by a cricket and the temperature, y in Fahrenheit is shown
below.
x:
20
16
20
18
17
16
15
17
15
16
y:
89
72
93
84
81
75
70
82
69
83
Find the fitted regression line that approximates the regression of temperature on the number of
chirps per second by the cricket. Further, find a 98% confidence interval for the slope
coefficient β .
19. In a controlled study to examine the relationship between humidity ( X ) and the extent of solvent
evaporation ( Y ), a random sample of 25 paired observations on X, the relative humidity (%) and
Y, the solvent evaporation (%) is collected and summarized as follows: n = 25,
25
x = 1314.90 ,
∑ i
i=1
25
25
x 2 = 76309.53,
∑ i
i=1
25
25
i=1
i=1
y = 235.70 . 90 ,
y 2 = 2286.07 and
∑ i
∑ i
x y = 11824.44 .
∑ i i
i=1
Estimate the regression line of Y on X and test the significance of the slope coefficient β at 5% level
of significance.
20. Test the null hypothesis β = 3 against the alternative hypothesis β > 3 at the 0.01 level of
significance, using the following data:
n = 10,
∑
x = 100,
∑
x 2 = 1376,
∑
y = 564,
∑
x y = 6945, Sxx = 376, Sxy = 1305, Syy = 4752.4,
∑
^ = 21.69 .
y 2 = 36562, β^ = 3.471, α
Construct a 95% confidence interval for the slope of the regression line.
21. The following data show the advertising expenses (expressed as a percentage of total expenses)
and the net operating profits (expressed as a percentage of total sales) in a random sample of 6
drugstores;
Advertising 1.5
expenses
1.0
2.8
0.4
1.3
2.0
Net
operating
profits
2.8
5.4
1.9
2.9
4.3
3.6
(a) Fit a least squares line which will enable us to predict net operating profits in terms of
advertising expenses.
(b) Test H0 : β = 2 against H1 : β > 2 at the 0.01 level of significance.