printed pages.] No.22.

93SlooS3
[This question paper contains 32 Your Roll

1349
Sr. No. of Question Paper :
2272101203
Unique Paper Code
Economics
Intermediate Statistics for
Name of the Paper

Name of the Course B.A. (H) Economics DSC

II
Semester
Maximum Marks : 90
Duration : 3Hour

Instructions for Candidates

No. on the top immediately on receipt of this question paper.

1. Write your Roll

within each section are to be answered in acontiguous manner on

2. Allquestions
subparts of a
question on a new page, and all
the answer sheet. Start each
other.
question should follow one after the
rounded off to 3 decimal places. The
3. All intermediate calculations should be
provided in statistical tables should not be rounded off. All final calculations
values
should be rounded off to twodecimal places.
calculator is allowed.
4. The use of a simple non-programmable

5. Statistical tables are attached for your reference.

places.
6. In all calculations, figures should be rounded to two decimal

7. Answers may be written either in English or Hindi; but the same mediur
be used throughout the paper.

ZT0.
1349
2

2.

4.

5.

6.

7.

SECTION I
Do any two ques tions.

1. Let Y be the number of contracts

received by a randomly selected
company. Suppose the probability mass function of Y is as followsinfrastructure
:
Y 2 3 4
P(y) 0.2 0.4 0.3 0.1
1349 3

(a) Consider a random sample of two companies, obtain the probability

distribution of S? (sample variance).

(b) Calculate P (S? > 2.7) and P (1.5 < S² <7.9), when a random sample of
size two is selected. (5+5)

Y 1 2 3 4
P(y) 0.2 0.4 0.3 0.1

(a) P(S² > 2.7) 3r P(l.5< S<7.9) Ât TUT ÁâfY, H ATGR aI YG YIgfsch

2. (a) How is systematic sampling different from judgement sampling. If Var(X,)

is variance of X, and Var(X,) is variance of X, then Var (aX, - bX,) =
aVar(X,) + b2Var(X,), where a and b are constants. Is this statement true?
Explain.

(b) The teacher of an economics class of size 35 knows that the time needed to
evaluate a randomly chosen first year paper is a random variable with mean
value of 4 minutes and a standard deviation of 3 minutes. If evaluation times
are independent and the teacher starts evaluation at 5:30 pm and evaluates
continuously, what is the probability that she completes evaluation before
8pm dinner starts? (S+S)

P.T.0.
1349 4

Var(X,) X, I tar Èt Var (aX, - bX,) =a'Var(X,) + b'Var(X,), IE a sit b

(a)

3 If u, =30, L, =40, 4, =50, G', = 15, a', = 25, and o', =5 are mean values
and variance of three independently and normally distributed random variables
X,, X,, and X,, respectively.

(a) Calculate P(24 s X s 39), where X = 0.3X, - X, + 1.7 X,.

(b) Calculate P(X, - 2X, s 3X). Can you find this probability if population is
not normal and sample size is 3? Why/ Why not? (5+5)

(3) P(24 <X< 39), ht V hlGY, GET X=0.3X, -X, + 1.7X, I

1349 5

SECTION II

Attempt any three questions.

4. (a) Suppose a sample of size n is to be drawn from a normal distribution where

true standard deviation is 12.7. How large does n have to be to guarantee
that the width of 97% confidence interval for true average value is 1.2. How
does precision of estimation change if we change the confidence level from
97% to 99%.

(b) Suppose a population is normally distributed with mean and unknown

variance o'. From this population, a sample of size 49 is drawn with
92%
an average value of 3.2 and standard deviation 2.6. Find the
confidence interval for p. Write the interpretation of 92% confidence interval
for u, also write the upper confidence bound for p for the 92% confidence
level. (5+5)

fr 97% fayarH HARIT ti r 1.2 IJUR EH HIPHSH TAG 97% 99%

PT.0.
1349 6

5 (a) Let X,, X,, ---, X, be a random sample of size 5 from the pdf

f(x; 0) = Ox4-! where 0 < x sI

Find the moment estimator of e. If X, = 0.34, X, = 0.27, X, = 0.79,

X, =0.82, X, = 0.19, what will be the moment estimate for .

(b) () If an estimator is unbiased its mean square error (MSE) 1s equal to its
variance, is this statement correct? Answer with the help of relevant

derivation.

(ü) If ø and S(Y -Y)' /n are maximum likelihood estimators (M.L.E.s)

of the mean and variance of a normal distribution, then what will be

the M.L.E. of E(Y)? (5+5)

f(x; 0) =0x1 Ed 0<x<1

AIeet -G GIA Hifsri ue X, =0.34, X, =0.27, X, = 0.79, X, =0.82,

(a)

HTAI 4HI6 (M.L.E,) , E(Y) T M.L.E. I ET?

1349 7

6 (a) Arandom sample of 100items taken from a large batch of articles contains
Sdefective items. Estimate the true average proportion of defective items in
the batch in a way that conveys information about precisionand reliability.
(Assume 95% level of confidence).

(b) Consider a sample of random variables X,, X,, ....,X, where n > 10,
n

E(X) =, Var(X) =² >0 and the estimator Ý n-10 XiX.Calculate

) Bias of ?

(i) Variance of ?

(iii) Mean Square Error of ? (5+5)

? ý= n-10

P.T.O.
1349 8

(a) Apsychologist estimates the mean reaction time for a sample of n 0

respondents.

() Calculate the width of the 90% confidence intervalfor true population

variance o².

(i) Calculate the upper bound on sample size "*n so that the expected
width of the confidence interval calculated in (i) does not exceed the
true population variance o'.

(b) A random sample of 20 workers in a village was found to have a mean

daily income of Rs. 45 and a sample standard deviation of Rs. 8. Based
on the sample data, the government wants to obtain an estimate of
the minimum income earned by workers "w", which covers 99 percent of
workers in the population. Calculate w. (Assume population distribution to
be normal). (5+5)
1349

SECTION III

Attempt any four questions.

8. (a) The chip of a processor requires a special kind of setting. A lot of re-setting
is required if the setting is not optimal. Prior to a demo too many re-setting
operations were required. In a sample of 200 units, 26 chips required re
setting.Atraining workshop explained the process of chips manufacturing in
a dynamic and efficient manner that reduced the task of re-setting. A new
sample of size 200 had only 12 that needed re-setting. Is this sufficient
evidence to conclude at the 0.01 level of significance that the training
workshop havebeen effective in reducing the task of re-setting?

(b) Studying the entry of runners at two busy parks between 6 p.m. and 8p.m.,
it was found that on 40 weekdays there were on the average 247.3 runners
entering the first park while on 30 weekdays there were on the
average 254.1 runners entering the second park. The corresponding
sample standard deviations are s, = 15.2 and s, = 18.7. Test the null
hypothesis 4,-, =0 against the alternative hypothesis , - 4, 0 at the
level of significance a = 0.01. What would be your conclusion using the p
value of the test. (5+5)

PI.O.
1349 10

(a)

fdut s, = 15.2 3r s, = 18.7 vrhTH, -H, =0 hT Tehte4 YRhT

9. (a) A student is timed 20 times in the performance of a task, getting mean

X=7.9 minutes and standard deviation s = 1.2 minutes. If the probability
of a Type Ierror is to be at most 0.05, does this constitute evidence against
the null hypothesis that the average time is less than or equal to 7.5 minutes?
Find the p-value of the test.

(b) Playing 10rounds of golf every week, a golf professional averaged 71.3
with a standard deviation of 2.64. Test the null hypothesis that the consistency
of his game is actually measured by standard deviation o = 2.40, against the
alternative hypothesis that he is less consistent. Use the level of significance
0.05.Assume that the distribution of his score every week, is approximately
normal. (5+5)
1349 11

10. (a) Suppose an automobile company is looking for additives that might increase
gas mileage. As a pilot study, they send thirty cars fueled with a new additive
on a road trip from Bengaluru to Ahmedabad. Without the additive, those
same cars are known to average 25.0 mpg with a standard deviation (o)
of 2.4mpg. Suppose it turns out that the thirty cars average y = 26.3 mpg
with the additive. What should the company conclude at 5% level of
significance?

(b) Using the same information as in part a) suppose the company can commit
two different types of errors. If the additive is effective but the position is
taken that the increase from 25.0 to 26.3 is due solely to chance, the company

will mistakenly pass up a potentially lucrative product. On the other hand,

ifthe additive is not effective but the firm interprets the mileage increase as
"proof' that the additive works, time and money will ultimately be wasted
developing a product that has no intrinsic value. What are these errors known
as? Calculate the probability of both the errors if the rejection region is
given as x 25.718. Also assume that the true population mean (with the
additive) is 25.750. (5+5)

PT.0.
1349 12

11. (a) Arandom sample of 1000 workers from South India shows that their mean
wages are Rs. 47 per hour with a standard deviation of Rs. 28. Arandom
sample of 1500 workers from North India gives a mean wage of Rs. 49 per
hour with a standard deviation of Rs. 40. Test if there is any significant
difference between mean wages across North and South India for the

population of workers at 1 %level of significance.

1349 13

(b) Twodifferent computer processors are compared by measuring the processing

speed for different operations performed by computers using the two
processors. If 12 measurements with the first processor had a standard
deviation of 0.1 GHz and 16 measurements with the second processor had
a standard deviation of 0.15 GHZ, can it be concluded that the processing
speed of the second processor is less uniform? Use a = 0.05 level of
significance. What assumptions must be made as to how the two samples are
obtained? (5+5)

plants to yield
12. (a) In apilot process, almond milk was manufactured inn=8
(in litres) in a week.

26.8 32.5 29.7 24.6 31.5 39.8 26.5 19.9

PT.0.
1349 14

Conduct a test of hypotheses with the intent of showing that the mean
production is less than 36.2.Take level of significance a = 0.0l and assume
a normal population. Based on your conclusion, what error could you have
made? Explain in the context of the problem.

(b) Using the 95% confidence interval, for the mean reading time, following
information is obtained:

Sample Standard 95% Confidence

N Sample Mean
Deviation Interval
15 6.009 1.078 (5.412, 6.606)

() Decide whether or not to reject H,: u = 5.5 hours in favour of

H,: u 5.5 hours at level of significance a = 0.05.

(i) Decide whether or not to reject H,: u = 5.3 hours in favour of

H,: p5.3 hours at level of significance a =0.05.

(iiü) Based on the example what is the relationship between tests for two
sided alternatives and confidence intervals? (5+5)

26.8 32.5 29.7 24.6 31.5 39.8 26.5 19.9

1349 15

Sample Standard95% Confidence

Sample Mean Interval
Deviation
15 6.009 1.078 (5.412, 6.606)

,H8G R a= 0.05 I

P.T.0.