Download b.sc maths stat 2 qp bank 2016 - E

Nehru Arts and Science College
Department of Mathematics
Statistics for Mathematics II
Question Bank
Unit –I
Section A
1. The types of estimators are
a. Point estimator b. Interval estimator c. Estimation of confidence region
d. All the above
2. Estimator is a
a. statistic b. statistics c. Estimate d. None of these
3. An estimator is said to be unbiased estimator for 𝜃 if
a. 𝜃 = 𝜃
b. V (𝜃) = 𝜃
c. E (𝜃) = 𝜃
d. Non
4. Sufficient condition for an estimator tn to be consist end for 𝜃 large n is
a. E ( 𝑡𝑛 ) → 𝜃 b. V ( 𝑡𝑛 ) → 𝜃
c. (a) and (b) d. none
5. If a sufficient estimator exists, then it is a function of the
a. moment estimator b.MLE c. Minimum Chi-square estimator d. Least square
estimator
6. The standard error of the sampling distribution of the mean is
a. the deviation of the sampling distribution of the mean.
b .the standard deviation of the sampling of any statistic.
c. the standard deviation of the sampling distribution of the statistic
d. the standard deviation of the sampling distribution of both mean and variance
7. Crammer – Rao lower bound to variance of unbiased estimator 𝜃 of N ( µ , 𝜃) , when µ is
known is
2
a.𝜃 ⁄𝑛
b. 𝜎⁄
√𝑛
c. 𝜎⁄
√2𝑛
d. 𝜎⁄𝑛
8. If n is the sample size , µ is the population mean and 𝜎 2 is the variance, then the standard
error of the standard deviation is
2
a.𝜃 ⁄𝑛
b. 𝜎⁄
√𝑛
c. 𝜎⁄
√2𝑛
d. 𝜎⁄𝑛
9. An unbiased and consistent estimator of population mean is
a. a sample mean b. standard deviation
c. first sample observation
d. none of these
10. Estimators obtained by the method of moments are
a. consistent
b. efficient
c. unbiased
d. sufficient
Section – B
1. Define Population and Sample.
2. State Neyman’s Factorization theorem.
3. Define (i) un biasedness (ii) consistency (iii) efficiency and (iv) sufficiency
4. Explain sampling distribution and standard error
1
5. Show that sample variance 𝑆 2 = ( 𝑛) ∑ (x - 𝑥̅ ) 2 is not an un biased estimator of population
variance.
6. Show that sample mean is more efficient than the sample median to estimator the
parameter , the mean of a normal population .obtain the efficiency of the sample median.
7. Explain sampling Distribution and standard error.
8. Develop an Unbiased estimator of 𝜎 2 based on a random sample of size n from a
population N ( µ , 𝜎 2 ).
9. Define efficient. Find the efficiency of sample median over sample mean of N ( µ , 𝜎 2 ).
10. Explain un biasedness and consistency in point estimation with example.
Section – C
1. State and prove Neyman factorization theorem
2. State and Establish Rao-Blackwell theorem
3. State and prove Cramer Rao inequality.
4. Define a sufficient statistics. state the factorization theorem on sufficiency.
5. I. State and prove Neyman factorization theorem
II. obtain sufficient for 𝜇 and 𝜎 2 in N (𝜇 , 𝜎 2 ) population.
UNIT –II
SECTION – A
1. Maximum likelihood estimators are always
a. Unbiased b. efficient c. consistent d. sufficient
2. Chi-square distribution is used to find the confidence interval for
𝜎2
c. 𝜎12
a. µ b. Þ
d. 𝜎 2
2
3. Maximum likelihood estimator for 𝜆 of the Poisson distribution is
a.∑ 𝑥
b. 𝑥̅
c. √𝑥̅
d. none of these
4. To find the confidence interval for the ratio of 2 variables we use
a. X2 - statistics b. F – statistics c. t – statistics d. Z – statistics
5. In a confidence interval P { C1 < Z < C2 } = 1 – 1-𝛼 is called as
a. Confidence limit b. confidence coefficient c. limit d. none of these
6. The maximum likelihood estimator are necessarily
a. Unbiased b. sufficient c. most efficient d. unique
7. If T is a sufficient estimator for a parameter 𝜃 and if 𝛹( 𝑇) is one to one function of T ,
then 𝛹( 𝑇) 𝑖𝑠 ______ for 𝛹(𝜃)
a. Efficient b. unbiased estimator c. maximum likelihood estimator d. sufficient
8. In random sampling from normal population N(µ , 𝜎 2 ) the MLE for 𝜎 2 is _____
a. 𝑥̅
1
b. 𝛴 ( xi - µ)2
𝑛
c. 𝛴( xi - µ) d. S2
9. Estimator obtained by the method of moment are____
a. Consist b. efficient c. unbiased d. sufficient
10. In a sufficient estimator exists, then it is a function of the
1. Moment estimator b. MLE c. minimum chi-square estimator d. least
Square estimator
SECTION – B
1. Explain the estimation procedure by the method of moments and indicate the
circumstances under which it is most appropriate.
2. State any four of the optimal properties of the maximum likelihood estimator.
3. Discuss the method of minimum chi-square.
4. Explain how you construct 95% confidence interval for the population mean.
5. From a normal population N(µ , 𝜎 2 ) obtain the maximum likelihood estimator for 𝜎 2
when µ is known
6. Let X1 , X2 , ………, Xn be i.i.d N(µ , 𝜎 2 ) random variable. suppose 𝜎 2 is known the
obtain the 100(1-𝛼 )% confidence interval for µ
7. Explain the method of moment in estimator.
8. State the properties of MLE.
9. A random sample X1 , X2 , ………, Xn is taken from a normal population with mean 0
and variance 𝜎 2 . Examine if ∑𝑛𝑖=1
𝑥𝑖 2
𝑛
is an MVB estimator for 𝜎 2 .
10. Find MLE for the parameter 𝜆 of a Poisson distribution on the basis of a sample of size
n. also find its variance.
SECTION –C
1. Describe the method of maximum likelihood. Find the maximum likelihood estimator for
𝜃 in exponential (𝜃) distribution.
2. If (X1 , X2 , ………, Xm) and (Y1 , Y2 , ………, Yn) be random samples from two
independent N(µ1 , 𝜎 2 ) and N(µ2 , 𝜎 2 ) distributions respectively. Find a 100(1-𝛼 )%
confidence interval for (µ1 - µ2 ).
3. Give one observation from a population with probability density function
2
f (x, 𝜃) = 𝜃 2 (𝜃 − 𝑥)
0 ≤ x ≤ 𝜃 . of 100(1-𝛼 )% confidence interval for 𝜃.
4. Obtain MLE for the parameters µ and 𝜎 2 in N(µ, 𝜎 2 )
5. Explain the method of minimum 𝜒 2 . State its properties.
UNIT – III
SECTION – A
1. Type I error is defined as
a. Reject H0 when H0 is true b. Reject H0 when H1 is true c. accept H0 when H is
true d. accept H0 when H1 is true
2. Testiness H0 : 𝜃 = 10 against H1 : 𝜃 ≠10 leads to
a. One sided left tailed test b. one side right tailed test c. two sided test d. none of
these
3. Systematic sampling means
a. Selection of n continuous b. selection of n units situated at equal stances c. selection
of n large units d. selection of n middle units
4. The sample mean fails to be m.l.e for the unknown parameter 𝜃 in a situation where the
population is
a. Normal (𝜃,1) b. Poisson (𝜃) µ c. both (a) & (b) d. none of these
5. Test for equality of means based on two small samples is said on
a. Normal distribution b. t- distribution c. 𝜒 2 .- distribution d. f- distribution
6. The test of significance of correlation coefficient is unbiased on
a. Normal distribution b. t- distribution c. 𝜒 2 .- distribution d. f- distribution
7. simple random sample can be drawn with the help of
a. random number sample b. lottery method
c. both (a) & (b) d. none
8. Testing H0 : µ = µ0 against H1 : µ > µ0 is
a. one side test b. two sided test
c. one side right test d. one side left test
9. In sampling error refers to
a. Mistakes
b. Bias c. Difference between statistics and parameter d. None
10. When population under investigation is infinite we should use
a. Sample method
census method
b. Census method
c. Sample and census method
d. Neither sample nor
SECTION – B
1.
2.
3.
4.
Explain critical region, one tail test and two tail test.
Explain the t-test for the difference between two means ,giving the assumptions.
Define Type I error and Type II error.
Explain the concepts of (i) simple and composite hypothesis (ii) the power of test (iii)
level of significance
5. Explain the test procedure of testing the difference between two means in small samples.
6. Explain the procedure for test of significance.
7. Explain paired t-test.
8. The mean of a sample of size 10 is 0.742 the standard deviation in the sample is 0.04. test
hypothesis H0 : µ = 0.7000 against H0 : µ ≠ 0.7000.
9. Explain sampling distribution and standard error
10. Explain chi-square and f- distribution
SECTION – C
1. Explain the t-test for difference of two means.
2. Explain in detail the chi-square test for independence of attributes.
3. State and prove Neyman –Person lemma.
4. A random sample of 10 children have the following birth-weight from a city (in kg)
3.0, 3.5, 4.5, 3.0, 4.25, 3.0, 3.25, 3.5, 4.0, 4.5.test whether the average birth weight of
the children will be 3.5 kg in that city.
5. Explain how you test the equality of variances and state assumptions if any.
UNIT –IV
SECTION –A
1. In SRS with replacement all items are ____
a. Independently and identically distributed random variable b. dependent variables
c. only random variables d. none
2. To test the homogeneity of several variances one has to use
a. T-test b. F test c. Bartlett’s test d. analysis of variance
3. Non-Sampling errors occur in
a. Census Survey b .Sample Survey c. both (a) & (b) d. none
4. In a CRD with t-treatments and n experimental units error degrees of freedom is equal
to_____ a. n-1 b. n-2 c. n-t d. t-n
5. Local control helps to ______ (a) Reduce the number of treatments b. increase the
number of plots c. Reduce the error of variance d. Increase the error
6. For the solution of 𝜃̂ of the likelihood equations the second derivative of L should be
a. Positive b. negative c. Zero d. none
7. A Latin Square design possesses
a. One way Classification b. Two Way Classification
c. Three Way Classification
d. No Way Classification
8. The significance table have frequency in
a. Percentages
b. Proportions c. Frequencies
d. None of these
9. The degree of freedom for X2 in ( 3 X 4) contingency table is
a. 4
b. 6
c.8
d. 2
10. Local control in experimental design is meant to
a. increase the efficiency of the design
c. To form homogeneous blocks
SECTION – B
b. Reduce experimental error
d. all the above
1.
2.
3.
4.
5.
6.
Explain Stratified random sampling procedures.
What is analysis of variance? What purpose does this technique serve?
Explain the method of drawing a systematic sample.
Explain the terms Randomization and Replication.
Explain stratified random sampling given any two merits of stratified random sampling
Explain the one way analysis of variance giving lay out and mathematical model also
given ANOVA table.
7. Derive the formula for V ( ̅̅̅̅
𝑌𝑛) under SRSWOR
8. Explain simple random sampling replacement and without replacement
9. Explain the procedure for test of significance.
10. Explain paired t-test.
SECTION – C
1. Derive an unbiased estimator for population mean in sample random sampling
without replacement. Obtain its standard error
2. Give the complete statistical analysis for two way classification with one observation
per cell.
3. Describe in details the analysis of data obtained from a BD
4. Describe the method of stratified random sample in detail.
5. Describe in detail the analysis of data obtained from a Latin square design.
UNIT – V
SECTION – A
In this unit contains problems of above unit so no one marks from this unit
SECTION – B
1. Prove that the mean of random sample drawn from the Poisson (𝜃) distribution is a
consistent estimator of 𝜃
2. In a sample of 1,000 people, 540 are rice eater and rest and wheat eaters. Test that both
rice and wheat eaters are equally popular at 1% level of significance.
3. What are the properties of‘t’ test?
4. Give the names of the methods of testing of hypothesis.
5. Obtain MLE estimator for the parameter λ of Poisson distribution.
6. The mean and S.D of sample of size 400 are respectively 2.45 cms and 1.16 cms. can you
concludes that the given sample has been from the population with mean 2.65 cms? Test
at 5% level of significance.
7. Difference between SRSWR and SRSWOR.
8. Define CRD. state its advantages
9. Obtain the estimator for 𝜃 in the case of a population with density function
f(x) = ( 1 + 𝜃 ) x 𝜃 ; 𝜃 ≤ x ≤ 1 , 𝜃 > 0 by the method of moments.
10. State Neyman’s factorization lemma. Using this find a sufficient estimator for the
parameter of Bernoulli distributions
SECTION – C
1. Describe the method of systematic sample in detail.
2. Define consistency of an estimator prove if (x1 , x2 , ….. xn) is a random sample from
N(µ , 𝜎 2 ) population, the sample mean consistent estimator for µ.
3. A die is thrown 180 times with the following results
Number
1
2
3
4
5
6
Frequency
25
35
40
22
32
26
Test the hypothesis that the die is an unbiased.
4. Ninety six patients were treated for tuberculosis and the details are given below. Test the
association between the age and death in tuberculosis patients at 5% level of significance.
No. of patients
Age
Dead
Recovered
Young
11
18
Older
23
44
5. Three varieties of coal were analyzed by four chemists and the ash content in the varieties
was found to be as under
Chemists
Varieties
I
II
III
IV
A
8
5
5
7
B
7
6
4
4
C
3
6
5
4
Do the varieties differ significantly in their ash content?