Download economics (hons) * semester*ii

MOCK TEST
SME-II
Maximum Marks : 75
Time : 3 Hours
Instruction : All Questions are compulsory.
Section – I
Attempt any one question out of question Nos. 1 and 2.
1.
2.
1 x 10 = 10
Random samples of size 2 are drawn from the finite population which
consists of the numbers – 5, 6, 7, 8, 9 and 10.
(a)
Calculate the mean and standard deviation of the population.
(b)
List all possible samples of size 2 that can be drawn without
replacement.
(c)
Construct the sampling distribution of the mean for these samples.
(d)
Calculate the mean and standard deviation of the probability
distribution obtained above and illustrate their relation with the
corresponding population values.
If random sample of size m is taken from a population of size N.
(i)
(ii)
How many different equally likely samples can be drawn:
(a)
With replacement;
(b)
Without replacement,
What is the variance of the sample mean if sampling is done:
(a)
With replacement;
(b)
Without replacement,
(iii)
Under what conditions do the answer in (ii) approach each other?
(iv)
Distinguish between simple random sampling and stratified random
sampling. When will you use the latter?
Question No. 3 is compulsory.
3.
5
The following table gives the information on years of education (X) of farmers
and annual yields per acre (Y) on their farms :
X
0
2
4
6
8
10
12
14
16
Y
4
4
6
10
10
8
12
8
6
(i)
Find the regression equation of yield per acre on education and give
an economic interpretation to it.
(ii)
What is the magnitude of the Explained Variation in the dependent
variable? Find the coefficient of correlation from it.
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Attempt any one question out of question Nos. 4 and 5.
4.
1 x 10 = 10
(a)
If two regression lines corresponding to two variables X and Y meet at
a point (2, 3) variance of X = 4, Variance of Y = 1 and correlation
coefficient between X and Y is ½. What is the estimated value of Y for
X = 6?
(b)
Comment on the following:
(i)
Regression coefficients are independent of change of scale and
origin.
(ii)
The two regression lines are mutually perpendicular if X and Y
are independent.
(c)
If all point of a scatter plot lie exactly on the regression line the two
variables are perfectly correlated. Comment.
(d)
Given the regression coefficient of Y on X is 0.80 and that u + 3x = 10
and 2Y + 5v = 25.
Calculate the regression coefficient of v on u.
5.
A random sample of 8 drivers insured with a company and having similar
auto insurance policies was selected. The following table lists their driving
experience (in years) and annual auto insurance premiums (in Rs.)
Driving Experience
Annual Auto Insurance
5
6400
2
8700
12
5000
9
7100
15
4400
6
5600
25
4200
16
6000
(i)
Find the least squares regression line by choosing appropriate
dependent and independent variables.
(ii)
Interpret the meaning of the coefficient of the independent variable
obtained above.
(iii)
Calculate the coefficient of determination and explain what it means.
(iv)
Predict the annual auto insurance premium for a driver with 10 years
of driving experience.
(v)
Construct a 90% Confidence Interval for the population regression
coefficient β .
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Section - II
Question No. 6, 7 and 8 are compulsory
6.
(a)
A population has a density function given by
k  1x k
f x   
 0
0  x 1
otherwise
For n observation x1 , x 2 ,..........x n made from this population, find the
‘Maximum Likelihood Estimator’ of k.
4
(b)


Suppose X ~ N μ, σ 2x and we draw a random sample of size
this population. The following are two
X
(i)
(a)
from
estimators of μ x :
Xi
X
and (ii) X*   i . Check if the above estimators
n
n 1
are unbiased.
7.
n
3
Suppose x1 ............x n be a sample of size n from a normal distribution


N μ, σ 2 . Consider the point estimator of σ 2
s2 
1
x 1  x 2 and S 2  1  x i  x 2

n 1
n
Show that s 2 is an unbiased estimator of σ 2 while S 2 is a biased
estimator.
6
8.
(b)
To test the durability of a new paint for white centerlines, a highway
department painted strips across heavily travelled roads in 8 different
locations and electronic counters showed that they deteriorated after
being crossed by 142600, 136500, 167800, 1098300, 126400,
133700, 162000 and 149400 cars (All values are to the nearest
hundred). Construct a 95% confidence interval for the average
number of crossings this paint can withstand before it deteriorates.
4
(a)
Using the method of moments, estimate the mean and variance of the
heights of 10 year old children, assuming these conform to a normal
distribution, based on a random sample of 5 such children whose
heights are : 124cm, 122cm, 130cm, 125cm and 132cm.
4
(b)
Dr. Diet has launched a new diet. She argues that it will make
patients lose more than 10 kgs. Over 5 months. A random sample of
25 students follows the diet for 5 months, yielding an average weight
loss of 4kg and a sample variance of 9 kg. Assuming that each
individual’s weight loss is normally distributed with unknown mean
and variance, construct a 95% confidence interval for the mean of the
distribution of each individual’s weight loss.
4
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Section - III
Attempt any one out of 9 & 10
9.
10.
1x5=5
A soft-drink machine at a steak house is regulated so that the amount of drink
dispensed is approximately normally distributed with a mean of 200 milliliters
and a standard deviation of 15 milliliters. The machine is checked periodically
by taking a sample of 9 drinks and computing the average content. If X falls in
the interval 191  X  209 , the machine is thought to be operating satisfactorily;
otherwise, we conclude that μ  200 milliliters.
(i)
Find the probability of committing a type I error when μ  200 milliliters.
(ii)
Find the probability of committing a type II error when μ  215
milliliters.
A fabric manufacturer believes that the proportion of orders for raw material
arriving late is p = 0.6. If a random sample of 10 orders shows that 3 or fewer
arrived late, the hypothesis that p = 0.6 should be rejected in favor of the
alternative p < 0.6. Use the binomial distribution.
(a)
Find the probability of committing a type I error if the true proportion is p
= 0.6.
(b)
Find the probability of committing a type II error for the alternatives p =
0.3.
Attempt any one out of 11 and 12
11.
1 x 10 = 10
In conducting a survey of food prices, two samples of prices of a given food item
were collected. Sample I came from north of the city while sample II came from
south.
If p i is the price recorded from the ith store then the results were:
12.
I
II
n 1  18
n 2  16
P1  0.79
P2  0.83
S12  1.06
S 22  1.27
(i)
Test the hypothesis that there is no difference between the mean price of
the particular food item in the two areas. Assume that the samples are
independent, derived from two normal populations with same variance.
(Use α = 5%)
(ii)
Test the hypothesis that dispersion of prices in south of the city is no
greater than in north of the city. (Use α = 5%)
(i)
The impact of different pay systems on productivity and workers’ level of
satisfaction has always been of interest to labour economists. ‘Fortune’
magazine reported that a sporting goods company experimented with the
effects of two methods of payment on employee morale. Fourteen workers
paid a fixed salary, were given a test measuring morale and scored a
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mean of 79.7 with standard deviation of 8.2. Twelve workers paid on a
commission basis achieved a mean of 72.7 with standard deviation 5.1.
Assuming σ1  σ2 , and using 5% level of significance, what can be
concluded regarding the relative merits of the two pay systems based on
the resulting hypothesis test?
(ii)
A drilling company tests two drill bits by sinking wells to a maximum of
112 feet and recording the number of hours procedure took. The first bit
was used in 12 cases resulting in a mean time of 27.3 hours and
standard deviation 8.7 hours. Ten wells were dug with the second bit in a
mean time of 31.7 hours and standard deviation 8.3 hours.
(a)
Test the H 0 : μ1  μ 2 against H1 : μ1  μ 2 . Set α  0.10 . There is no
evidence to suggest that variances are equal.
(b)
Redo the above test if the drilling company felt drilling times has
equal variances.
Attempt any one out of 13 & 14
13.
14.
1 x 10 = 10
(a)
The manager of a restaurant in a large city claims that waiters working in
all restaurants in his city earn an average of $150 or more in tips per
week. Test this hypothesis if a random sample of 25 waiters selected from
restaurants of this city yielded a mean of $139 in tips per week with a
standard deviation of $28. Assume that the weekly tips for all waiters in
this city have a normal distribution. Use α  0.01 .
(b)
Traditionally 35% of all loans by a National Bank have been to members
of minority groups. During the past year the bank has undertaken efforts
to increase this proportion. Of 150 loans currently outstanding 56 are
identified as having been made to minorities. Has the bank been
successful in its efforts to attract more minority customers. Test the
hypothesis using the 5% level of significance. Also, calculate the P-value
of the test and interpret it.
(a)
In early 1990s Sony Corporation introduced its 32-bit Playstation in the
home videogame market. Management hoped the new product would
increase monthly U.S. sales above the $283 mn. Sony had experienced
the previous decade. A 40-month sample reported a mean of $297 mn.
Assume a standard deviation of $97 mn. Calculate and interpret the Pvalue of the test. What will your conclusion be if the level of significance
is 1%?
(b)
(i)
Define Type I error and Type II error and explain with the help of
diagrams.
(ii)
The print on the packages of 100 watt Electric light bulbs states
that they have an average life of 750 hrs. Assume that the
standard deviation of the lengths of lives of these bulbs is 50 hrs.
A skeptical consumer does not think these bulbs last as long as
the manufacturer claims and she decides to test 64 randomly
selected bulbs. She has set up the decision rule that if the average
life of these bulbs is less than 735 hrs, then she will conclude that
the company has printed too high an average life on the packages.
Approximately what level of significance is the consumer using?
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