2017

Sl. No. of Paper. 2652

Unique Paper Code. 12271303

Name of Paper. Statistical Method for Economics

Name of Course. B.A. (H)

Semester. III

Duration. 03 Hours Maximum Marks. 75

Section –I

Q.1. is compulsory. Do any one out of Q.2. and Q.3.

Q.1.(a) The Height of five players in a football team have a mean of 76 inches, a median of 78
inches and a range of 11 inches.

(i) If the tallest of these players is replaced by a substitute who is two inches taller, would
these three measures changes ? If yes how ?
(ii) If the tallest player is replaced by a substitute who is four inches shorter, which of the
new values (mean, median and range) could your determine ? What would their new
values be ?

(b) Show that for any three events A, B and C with P(C) > 0,

P (A  B / C ) = P (A / C ) + P (B / C ) − P (A  B / C ) (3+2)

Q.2.(a) The marks of 21 students in a 50 marks math’s tests are given below :

18 20 25 28 30 35 36 38 39 40 41 41
41 42 42 43 44 45 45 47 50
Calculate a 10% trimmed mean for the data above.

(b) If A and B are independent events proved that ‘A’ (complement of A) and B’ (complement of
B) would likewise be independent.

(c) An infrastructure company, has launched two projects, one in the Mumbai and other in
Delhi. Suppose that the probability of success of the Mumbai project is 0.8 and the probability of
success of the Delhi project is 0.3 and that the success of anyone project is independent of the
other. If the Delhi project is unsuccessful, what is the probability that the Mumbai one is also
unsuccessful ? Explain your answer. Given that at least one of the two projects is successful,
what is the probability that only the Mumbai project is successful ? (4+3+3)

Q.3.(a) A Factory employs 10 workers in the production department, 8 workers in the packing
department and 7 workers in the delivery department. Out of these workers 5 are to be
randomly selected for a training programme. What is the probability that at least one of the
departments will be unrepresented among the selected workers ?

(b) The marks distribution (in %) of the students of a class were found to be symmetric. Results
revealed that 25% of the students scored 78% or more while one-fourth of these students scored
30% or less. Determine the mean score of these students. Also, find the 10% trimmed mean, if
possible.

(c) Let A be the event that a randomly selected individual likes vanilla flavor, B the event that
randomly selected individual like strawberry flavor and C be the event that a randomly selected
individuals like chocolates flavor. Suppose that;

P(A) = 0.65 P(B) = 0.55 P(C) = 0.70

P(A∪B) = 0.8 P(B∩C) = 0.3 P(A∪B∪C) = 0.9
(i) What is the probability that the individuals likes both the vanilla and the strawberry
flavors ?
(ii) If it is known that the individuals did not like vanilla, what now s the probability
that the individuals liked at least one of the others two flavors ? (4+3+3)

Section – II

Do any two out of Q.4., Q.5., and Q.6.

Q.4.(a) The probability distribution for the number of machines that might break down in a day
has been determined for a machines shop. The probabilities for 0, 1 and 2 breakdowns are
respectively 0.3, 0.6 and 0.1.

(i) Find the means and standard deviations of the number of daily breakdowns.
(ii) Daily repair costs, R found in rupees are calculated as R = 300 + 200 X2, X is the number
of breakdowns. Calculate the expected daily needs repair cost.

(b) A salesman makes initial contact with the potential customers in order to assess whether the
customers would like to purchase the product if offered some lucrative discounts. His survey
suggests that 40% of the customers would purchase the product if offered a discount. If he
contacts 100 customers, then what is the approximate probability that between 45 and 50
customers would actually purchase the product ?

(c) The amount of time, in minute, that a person must wait for a bus is uniformly distributed
between 0 and 15 minutes, inclusive. What is the probability that a person waits fewer than 1.5
minutes? What is his expected waiting time ?

Q.5.(a) The time taken (in hours) by a transport company to deliver consignment of apples from
an orchard in Himachal is a continuous random variable with pdf given by :

1
 25 x for 0  x< 5

2 1
 − x for 5  x  10
 5 25
0 for x< 0 or x> 10



(i) What is the probability that the time taken to deliver the consignment is at the most
03 hours ?
(ii) Compute Expected time taken and the standard deviation of the delivery time.

(b) On the statistics examination, grades are normally distributed with the mean grade 72 and
the standard deviation was 9. The top 10% of the students are to receive A’s. What is the
minimum grade a student must get in order to receive an A?

(c) What is the probability of getting a 9 Exactly once in 3 throws with a pairs of fair dice ?
(4+3+3)

Q.6.(a) Suppose that 10% of the probability for a certain distribution that is normal with mean μ
and the variance σ2 is below 60 and that 5% is above 90. What are the values of μ and σ ?

(b) Let X be the time between two successive landings of planes at an airport in a certain city. If
X has an exponential distribution with  = 1, compute the expected tie between two successive
landings. Also calculate the standard deviation of X, what is the value of (1 ≤ X ≤ 2) ?

(c) Let X have the following pdf

Which is a uniform distribution on [0, 1].

ℎ(𝑋) is given by max (𝑋, 1 − 𝑋) =

Find the expected value of ℎ (𝑋) (4+3+3)

Section – III

Do any two out of Q.7., Q.8., and Q.9.

Q.7.(a) Let X and Y have the Joint pmf 𝑝(𝑥. 𝑦) described as follows :

(𝑥, 𝑦) (0, 0) (0, 1) (1, 1) (2,0) (2, 1)

𝑃(𝑥, 𝑦) 1/18 3/18 3/18 6/18 1/18
𝑃(𝑥, 𝑦) is equal to 0 elsewhere :

(i) Find the marginal probability mass function X and Y.

(ii) Find the conditional mean of Y given x = 1.

(b) A production process is composed of three stages – I, II, and III. The time taken at each stage
is normally distributed with mean equal to 15, 30 and 20 seconds. The standards deviation of
time taken at each stage is 1, 2 and 1.5 seconds respectively. Assume time taken by each stage is
independent of time taken by other stages.

(i) What is the probability that it will take more than 1 minute to complete a randomly
chosen process?
(ii) For a randomly chosen process what is the probability that time taken in stage 1 exceeds
17 seconds?

(c) Let X be a normally distributed random variable with means 16 and variance 9. A random
sample of size 𝑛 is chosen from this distribution. Let 𝑋 be the sample mean. The standard
deviation of 𝑋 is found to be 0.3. What is the sample size of 𝑛? (4 + 3 + 3)
𝑥+𝑦
Q.8.(a) If 𝑓(𝑥, 𝑦) = for 𝑥 = 1, 2 and 𝑦 = 1, 2, 3. Find the value of 𝑘 for 𝑓(𝑥, 𝑦) to be valid joint
𝑘
probability mass function. What is the value of E(X) and E(Y)?

(b) The outer diameter of a pipe is normally distributed with average 20 mm and standard
deviation 5 mm. the inner diameter of pipe is normally distributed with average 18 mm and
standard deviation of 4 mm. If X is a sample average of outer diameter of 16 randomly chosen
pipes and Y is sample average from the same sample of 16 pipes,

(i) Describe the distributions of (𝑋̅ − 𝑌̅ )

(ii) Find 𝑃(−1 < (𝑋̅ − 𝑌̅ < 1) (5+5)
Q.9.(a) If two cards are randomly drawn (without replacement) from an ordinary deck of 52
playing cards. Z is the number of aces obtained in the first draw, and W is the total number of
aces obtained in both draws, find

(i) The Joint probability distribution of Z and W,

(ii) The marginal distribution of Z

(b) Let Y denote the engine power of a new car that is launched in three models that differ in
power (Denoted by bhp). Market survey shows that 20% customers want to buy the car with 2
bhp, while 30% buys the car with 5 bhp power. The rest prefer the model with 4 bhp. Derive the
sampling distribution of average engine power using a sample size of 2, if the samples are
obtained through random sampling. (5+5)

Section – IV

Q.10.(a) Varun Publications obtained a random sample of 23 books to determine their average.
The cost of books is assumed to be normally distributed. If the sample means is ₹ 23.56 and the
sample deviation is ₹ 4.65, find

(i) 98% confidences interval for the cost of a book

(ii) How can Varun Publication’s make their result more reliable? Is there any loss
associated with it?

(b) What is Mean Square Error (MSE) of an estimator  ? Show that MSE ( ) = Variance ( )
+ (Bias ( ))
2

(c) Consider a random sample 𝑋1 , 𝑋2 , 𝑋3 … . . 𝑋𝑛 from a population from a probability

distribution function

𝑓(𝑥; 𝜃) = 3(1 + 𝑥 ∗ 𝜃). Where −2 < 𝑥 < 2

(i) Show that θ = (sample mean/16) is an unbiased estimator for θ.

(ii) If a sample size is 3 and the sample is (–1, 1, 2) give point estimator for θ.

Q.11.(a) For each of the following confidences intervals drawn from normally distributed
populations, find the confidences level, width and mention the distribution associated with the
statistic used :

𝜎 𝜎
(i) (𝑋 − 1.4 − , 𝑋 + 2.05 − )
√49 √49

𝑠 𝑠
(ii) (𝑋 − 2.069 − , 𝑋 + 2.807 − ) where 𝑛 = 24
√𝑛 √𝑛
(b) Consider a random sample (𝑥1 , 𝑥2 , 𝑥3 … . . 𝑥𝑛 ) from a probability mass function

𝑝(𝑥; 𝜃) = 2𝑥/𝜃(𝜃 + 1) where 𝑥 = 1, 2, 3….θ.

(i) Find an estimator for θ using methods of moments.

(ii) If sample size of 3 and sample is (1, 3, 5) give point estimate for theta.

(c) Let, (𝑥1 , 𝑥2 … . . 𝑥𝑛 ) be a random sample from a population with mean 𝑝 and a standard
ni=1 (x i − X )2
deviation σ. Show that S 2 = is unbiased estimator of population variances,
n −1
denoted by σ2. (4+3+3)

Q.12.(a) A sample of 50 houses shows a smoke level of 645.16 ppm and standards deviation of
164.43 ppm.

(i) Find the 92% confidence level for true smoke levels.
(ii) Suppose that the population standard deviations is 175 ppm, what must be sample size
if we want size of error to be 25 ppm only at 95% confidence level ?

(b) Suppose that true average marks of Section A and Section B of Economics (Hons) are
equal in college to μ. The variance in marks for Section A is σ2, whereas it is 4σ2 for section
B. let 𝐴 denote average marks from a sample of size of M from a section A, while 𝐵 denote
average marks for sample size of N from section B. Let the estimator for μ be
 =  A + (1 − )B

(i) Under what circumstances  is unbiased.

(ii) What value of α will minimize variance of this estimator ?

(c) Consider a random sample (x1, x2, x3, xn) from an probability distribution function

f (x ;  ) =  x ( +1) , where 0  x  1. Find a maximum likelihood estimator for θ. (4+3+3)

2018
Sl. No. of Paper. 7497

Unique Paper Code. 12271303

Name of Paper. Statistical Method for Economics

Name of Course. B.A. (H)

Semester. III

Duration. 03 Hours Maximum Marks. 75

Attempt all Sections.

Section –I

Q.1. is Compulsory. Attempt any one from Q.2. and Q.3.

Q.1.(a) Suppose that P(A) = 1/8 and P(B) = ¼, then what is P(A∩B) and P(AB) if : (2+2)

(i) A and B are mutually exclusive events.

(ii) A and B are independent events.

(b) Descriptive Statistics of a data set are given as follows :

Mean = 535, Median = 500, Mode = 500, Standard Deviation = 96, Minimum = 220, Maximum
= 925, 5th Percentile = 400, 10th Percentile = 430, 90th Percentile = 640, 95th Percentile = 720,
What can you conclude about the skewness of the histogram ?

Q.2.(a) A crime is committed by one of two suspect, A and B. initially, there is equal
evidence against both of them. In further investigation at the crime scene, it is found that the
guilty party had a blood type found in 10% of the population. Suspect A does not match this
blood type, whereas the blood type of Suspect B is unknown. Given this new information,
what is the probability that A is the guilty party ?

(b) Differentiate between descriptive statistics and inferential statistics. Identify, which of
the following statements (A or B) is inferential in nature : (3+3)

(i) In a random sample of 300 people in Delhi, 240 read at least one newspaper daily.
 A : Eighty percent of people sampled read at least one newspaper daily.
B : Eighty percent of all the people in Delhi read at least one newspaper daily.
(ii) In a random sample of 100 students in University of Delhi, 60 students are non-
residents of Delhi.
A : Sixty percent of students in University of Delhi are from outside of Delhi.
B : Sixty percent of sampled students of Delhi University were from outside Delhi .
(3+1+1)

Q.3.(a) Consider an experiment of tossing three unbiased coins. Find :

(i) The Probability of three heads given a head on the first coin.

(ii) The probability of three tails given at least one tail.

(b) A company has 20 employees in human resources department, 10 employees in sales

department and 8 in accounts department. The manager wants to select 5 employees for regular
feedback exercise and every employee has an equal chance of selection :

(i) What is the probability that all the selected workers are from the same department ?

(ii) What is the probability that at least one of the departments will be presented in the sample
of workers ?

Section – B

Attempt any two from Q.4., Q.5, and Q.6.

Q.4.(a) consider the cumulative distribution function of a discrete random variable 𝑋

𝐹(𝑦) = 0 for 𝑥 < 1

1
= 3 for 1 < 𝑥 < 4
1
= 2 for 4 < 𝑥 < 6
5
= 6 for 6 < 𝑥 < 10
= 1 for 𝑥 > 10

Find :

(i) 𝑃(𝑋 = 10)

(ii) 𝑃(2 < 𝑋 < 6)
(iii) The probability distribution of 𝑋. (2+2+2)

(b) Suppose that the p.d.f of a continuous variable 𝑌 is as follows :

 1
𝑓(𝑦) = (9 − 𝑦 2 ) for −3 < 𝑦 < 3
36

Otherwise

Find :

(i) 𝑃(−1 < 𝑦 < 2)

(ii) 𝑃(𝑦 > 1).

Q.5.(a) Suppose that 20% of people don not wear belts while driving on highways and are
required to be fined.

In a random check of 200 vehicles over a month, let 𝑋 be the number of vehicles fined due to not
wearing the seat belts. What is the probability that 𝑋 is :

(i) At least 40
(ii) Between 35 and 50. (3+3)

(b) A random variable 𝑋 has 𝐸(𝑋) = 10 and 𝑉(𝑋) = 4, compute :

(i) 𝐸(𝑋 2 + 4𝑋)

(ii) 𝑉(4𝑋 + 10). (2+2)

Q.6.(a) On his tour, a night watchman has to open a door in the dark he has 20 keys, only one of
which fits the lock. He makes use of two different methods to open the door :

Method A : He carefully tries the keys one by one to avoid using the same key twice.

Method B : He tries the keys at random.

Define the random variable 𝑋𝐴 and 𝑋𝐵 as the number of necessary trials to open the door when
using method A and B respectively. Work out the probability distribution of 𝑋𝐴 and 𝑋𝐵 . (3+3)

(b) Calculate 𝐸(𝑋) of the following probability distribution function :

2
𝑓(𝑥) = for 0 < 𝑥 < 1
3

1
= 3 for 2 < 𝑥 < 3

= 0 elsewhere. (4)

Section – C

Attempt any two from Q.7., Q.8. and Q.9.

 1
Q.7.(a) If 𝑋 and 𝑌 have the Joint probability mass function as 𝑝(−1, 0) = 0, 𝑝(−1, 1) = 4 ,
1
𝑝(0, 0) = 6 , 𝑝(0, 1) = 0, 𝑝(1,0) = 1/ 12 and 𝑝(1, 1) = 1/2; show that :

(i) Cov(𝑋, 𝑌) = 0

(ii) The two random variable are not independent. (3+3)

(b) Differentiate between a parameter and a statics. Which of the following are statistics and
why?

(𝑋𝑖 −𝜇 )
(i) ∑ 𝜎

(𝑋𝑖 −𝑋)
(ii) ∑ 𝑛

𝑋
(iii) ∑ 𝑛𝑖

max(𝑋𝑖 −𝜇)−min(𝑋𝑖− 𝜇 )
(iv) (2+2)
𝑛

Q.8.(a) If the joint probability distribution of 𝑋 and 𝑌 is given by :

𝑓(𝑥, 𝑦) = 𝑐(𝑥 2 + 𝑦 2 )

For, 𝑥 = −1, 0, 1, 3 and 𝑦 = −1, 2 ,3

(i) Find the value of 𝑐

(ii) Calculate 𝑃(𝑋 + 𝑌 > 2)
(iii) Find the conditional distribution of 𝑌 for 𝑥 = 1. (2+2+2)

(b) Consider the population with mean 82 and the standard deviation 12 :

(i) If a random sample of size 64 is selected, what is the probability that the sample mean
will lie between 80.8 and 83.2?
(ii) With a random sample of size 100, what is the probability that the sample mean will lie
between 80.8 and 83.2?
(iii) How does the increase in sample size affect the probability ? (2+1+1)

Q.9.(a) A firm sells commodities 𝑋 and 𝑌 at prices ₹ 10 and ₹ 2 respectively and a fixed cost 𝐶 is
incurred in the process. If the expected sales and variance of 𝑋 are 100 and 25 and for 𝑌 are 200
and 64 respectively. Find the expected value and standard deviation of the revenue of firm
given that the covariance of sales of 𝑋 and 𝑌 is −2. (3)
(b) Two independent experiments are run in which two different types of paint are compared.
Eighteen specimens are painted using type A, and the drying time, in hours, is recorded for
each. The same is dome with type B. the population is normally distributed with standard
deviations for both known to be 1.0. Assuming that the mean drying time is equal for the two
types of paint, find 𝑃(𝑋𝐴 − 𝑋𝐵 > 1.0), where 𝑋𝐴 and 𝑋𝐵 are average drying times for samples of
two paints.

(c) Suppose that 𝑋 and 𝑌 have a discrete joint distribution for which the joint probability mass
function is as follows :

1
𝑓(𝑥, 𝑦) = 30 (𝑥 + 𝑦) for 𝑥 = 0, 1, 2 and 𝑦 = 0, 1, 2, 3 = 0

Otherwise

Determine the marginal probability mass function for X and Y. (4)

Section – D

Attempt ant two from Q.10, Q.11. and Q.12

Q.10.(a) How does the increase in the confidence level affect the precision of estimates? Explain.
(2)

(b) Let, 𝑋11 , 𝑋12 , … … 𝑋|𝑛| and 𝑋21 , 𝑋22 … … . 𝑋2𝑛2 be two random samples from the population the
following a binomial distribution. The parameter to be estimated is 𝑝, defined as the proportion
of success in the two samples. Which of the following is a better point estimate in terms of
efficiency and lesser variance ?

𝑋1+ 𝑋2
(i) 𝑛1 + 𝑛2

𝑋 𝑋
( 𝑛1 )+(𝑛2
1 2
(ii) (5)
2

(c) Let 𝑋 be a uniformly distributed random variable over the interval [0, θ]. Find the moment
estimator of θ. (3)

Q.11.(a) Suppose a random variable 𝑌 has the following pdf :

1
𝑓𝑌 (𝑦: 𝜃) = 𝜃𝑘 𝜃 (𝑦)𝜃+1 𝑦 > 𝑘; 𝜃 > 1

Where 𝑘 is known. Find the maximum likelihood estimator for 𝜃 if the information has been
collected from a random sample 20 individuals. (5)
(b) Consider a normal distribution with the value of 𝜎 known :

(i) What is the confidence level for the interval + 2.81𝜎/√𝑛 ?

(ii) What is the confidence level for the interval + 1.44 𝜎/√𝑛 ?

(iii) What is the value of 𝑧𝛼 /2 for the confidence interval represented by the confidence level of
99.7% ? (2+2+1)

Q.12. (a) A sample survey at a supermarket showed that 204 of 300 shoppers regularly use
cents-off coupons :

(i) Construct a 99% confidence interval for the corresponding true proportion.

(ii) Would a 90% confidence interval calculated from the same sample have been narrower or
wider than the one calculated in part (i) ? Explain the reasoning. (3+2)

(b) A director of a firm wants to study absenteeism among the employees. The number of days
that an employee is absent in a month follows a normal distribution with mean of 11.2 days an
standard deviation of 4.5 days.

(i) What sample size is needed so that the 95% confidence interval for the true mean
absenteeism has a width of less than 3 days ?

(ii) “The calculated confidence interval implies that the true population means lies in it with
probability 95%”. Is the statement correct? Why or why not ? (3+2)