Loyola College Testing Of Hypothesis Question Papers

Loyola College M.Sc. Statistics April 2003 Testing Of Hypothesis Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

second SEMESTER – APRIL 2003

ST   2802/  S  817   testing of hypothesis

24.04.2003

1.00 – 4.00                                                                                          Max: 100 Marks

 

 

SECTION – A                      (10 ´ 2 = 20 Marks)

Answer ALL questions.  Each carries TWO marks.

 

  1. Let X be an observation from

Find the power of the test

 

 

for testing H: q =1 against K: q = 2

  1. Give the test function

 

for testing H : q £    vs K: q >  based on an observation drawn from B (3,q),

find the  probability of rejecting H when q =.

  1. When do you say that a family of density functions has MLR property?
  2. What is a similar test?
  3. Define: Confidence set.
  4. Examine the validity of the statement “A MPT is always unbiased”.
  5. Give an example of a family having MLR property but not a member of one

parameter exponential family.

  1. Define UMPUT.
  2. Suppose a test function is of the form

for a family having MLR property in T(x).  Can such a test function satisfy

the condition    bf (q1) = bf (q2) (q1¹q2) ?

  1. Define: Maximal Invariance

 

 

 

 

 

 

 

Section – B                       (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries EIGHT marks

  1. Let = (X1 , X2,….., xn) where Xi’s  are  i i d  with pdf  pq(x) = e-(x–q),x >q, q >0 .

Show that the family of densities pq() has MLR property and hence derive

the UMPT of level a for testing H :  q £ q0 Vs K: q >q0 .

  1. For each q0 ÎW, let A (q0) be the acceptance region of a level – a test for testing.

H(q0): q = q0 and for each sample point x let S(x) denote the set  of parameter

values S(x) = {q|x ÎA(q), qÎW}

  • Show that s(x) is a family of confidence sets for q at confidence level 1-a.
  • If A(q0) is UMP for testing H (q0) at level a against K(q0), then for each q0ÎW,

Show that S(x) minimizes pq(q0ÎS(x)) ” q ÎK(q0) among all level (1-a) families of confidence sets for q .

  1. Solve the problem of minimizing ò f fm+1dm subject to ò f fi d m = ci , i  = 1,2,….,m ,

where f1,f2, …..,fm , fm+1     are (m+1), m integrable functions.

  1. Let the  distribution of X be given by

 

X  0       1            2                  3
Pq (X = x)  q      2q       0.9.-2q           0.1-q

 

Where 0 < q < 0.1.    For testing H: q =0.05 against K: q > 0.05 at

level a =0.05, determine  which  of the following tests (if any) is UMP.

(i)         f (0) = 1, f (1) = f(2) = f(3) = 0

  • f (1) = 5, f(0) = f(2) = f(3) = 0
  • f (3) = 1, f(0) = f(1) = f(2) = 0
  1. Let X be an observation drawn from a population with pdf

pq (x) = q e–qx, x >0, q >0

Derive the UMPT of level a = 0.05 for testing H : q £ 1,  q ³ 2  Vs K : 1< q < 2.

  1. State and prove a necessary condition for all similar tests to have Neyman structure.
  2. Let X and Y be independent Poisson varictes with means l and m

H : l £ m Vs K:  l >m.

  1. Write a descriptive note on invariant tests.

SECTION – C                      (2 ´ 20 = 20 Marks)

Answer any TWO.  Each carries twenty marks.

  1. State and prove Neyman – Pearson lemma.
  2. Derive the UMPUT for testing H: q =q0 Vs K: q ≠ q0 in

pq()  = c(q) eqT(x)      h(x)

  1. Let (X1,X2, ….,Xm) and (Y1,Y2,…,Yn) be samples of sizes m and n respectively

from N(x, s2) and   N (ך,s).  Derive the UMPUT

(unconditional) for testing     (i)  H :ך £ x  Vs  K: ך > x   and

(ii)  H:  ך = x  Vs K:  ך ¹ x

  1. Illustrate, with an example, the steps involved in developing unconditional

UMPUT’s for one-sided testing problems in the multi-parameter

exponential setup

 

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Loyola College B.Sc. Statistics Nov 2003 Testing Of Hypothesis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST – 5501/STA506 – TESTING OF HYPOTHESIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20 marks)

 

  1. Define the Best critical region for testing a simple hypothesis against an alternative simple hypothesis.
  2. When do you say that a family of probability density functions has a monotone likelihood ratio property?
  3. Explain likelihood ratio criterion.
  4. What is the test statistic used for testing the significance of correlation in a bivariate normal population? What is the distribution of the statistic?
  5. A coin is tossed 100 times. If, the number of heads obtained is 45, can you say the coin is unbiased? justify.
  6. What are non-parametric methods?
  7. 400 oranges are taken from a large consignment and 40 are found to be bad. Find the limits in which the percentage of bad oranges is likely to come at 95% confidence.
  8. Define Type I error and Type II error.
  9. Explain p-value.
  10. Arrange the following sets of x and y values. How many runs you get?

x          40        50        48        60

y          45        51        52        55

 

SECTION-B

Answer any FIVE questions.                                                                           (5×8=40 marks)

 

  1. Let X1, X2, .. Xn be a random sample from N(0,1). Verify whether uniformly most powerful test exists for testing Ho :  q = q1 vs H1 : q ¹ q1 .
  2. Let x have the probability mass function

P [X = x]  =  x = 0, 1,2, ..M.,

It is decided to test Ho : M £ M0 vs H1 : M > M0  based on a sample of size 1 from this distribution.  Derive the uniformly most powerful test.

  1. In a locality 100 persons are randomly selected and asked about their educational attainments. The results one as under

Education

Middle            High school     College

Male                   10                       15                   25

Female                25                        10                  15

Does education depend on sex?

  1. It is decided to study the model

Xi = a + b (ci  – ) + ei

with E(xi) = a + b (ci – ,    V(Xi) = s2.

a random sample of size n = 10 yielded  = 67

= 2.1    = 288 and .

Test H0: b = 0 vs H1 : b ¹ 0 .  Also find 95% confidence interval for b.

  1. Let X have the pdf f ( x ; q) = qx (1-q)1-x x = 0, 1 zero elsewhere. We test H0: q = against H1: q < .   Let X1, X2, X3, X4, X5 be a random simple from this distribution. Obtain the uniformly most powerful test with a =
  2. Explain Sign Test.
  3. Explain how will you construct the confidence interval for ratio of variances, when there are two random samples from 2 independent normal populations. suppose two independent samples of sizes m = 15 and n = 10 yielded

 

 

Find the 98% confidence interval for s |  s.

  1. It is known that the random variable X has a pdf

f (x ; q ) =

0        ,    elsewhere.

It is decided to test H0: q =    Vs H1: q = 4.

If a random sample of size 2 is observed, find probability of Type I error and probability of

type II error.

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

 

  1. a) State and prove Neyman – pearson theorem. (12)
  2. b) Let X have the pdf f( x, q) = 0 < x < q

Let Y1 < Y2 < Y3  < Y4  denote the order statistics of a random sample of size 4 from

this distribution.

We reject H0: q = 1 and accept H1: q ¹ 1  if Y4 £  or Y4 ³ 1.  Find the Power function. (8)

  1. a) Derive likelihood ration test for testify

H0: m1 =m2  Vs       H1: m1 ¹ m2

When the two random samples are drawn from two independent normal populations with mean m1 and m2 and with common unknown variance.                                                 (10)

  1. b) Two independent samples of sizes 8 and 7 items respectively had the following values

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Is the difference between means of samples significant. (Assume common variance)                                                                                                                                             (10)

  1. a) Explain Run test for equality of distributions.                   (12)
  2. b) Apply the Mann-Whitney – Wilcoxon test for the following data to test

H0: Fx =Fy  Vs  H1: Fx = Fy.

X         4.3       5.9       4.9       3.1       5.3       6.4       6.2       3.8       7.5       5.8

Y         5.5       7.9       6.8         9        5.6       6.3       8.5       4.6       7.1                   (8)

  1. a) Explain the procedure of testing equality of two proportions. Also obtain the 95%

confidence interval for the difference in proportions.                                               (10)

  1. b) Explain sequential probability ratio test           (10)

 

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Loyola College B.Sc. Statistics April 2004 Testing Of Hypothesis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5501/STA 506 – TESTING OF HYPOTHESIS

06.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. Define a simple hypothesis and a composite hypothesis.
  2. Let X1, X2,…, Xn be a random sample from N (m, s2).

Write the distributions of   i)            ii) .

  1. Define uniformly most powerful critical region.
  2. Explain Type – I error and Type – II error.
  3. Explain the likelihood ratio principle.
  4. What do you mean by non- parametric methods?
  5. When do we need the randomised test?
  6. Find the number of runs in the sequence.

x yyy   xxx y  x y xxx  yy  xxxxx

  1. Explain the term confidence interval.
  2. What is a p – value?

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

  1. Let X have pdf of the form f (x, q) = q xq-1, 0 < x < 1, zero elsewhere, where q Î {q ½q = 1,2}.  To test Ho: q = 1 vs H1 : q = 2, a random sample of size 2 is chosen.  The critical region is C = { (x1, x2) ½  < x1 x2}.  Find Type I error and Type II error.
  2. Verify whether UMPT exists for testing

Ho: q = q‘ Vs H1: q ¹ q‘

when the random sample X1, X2, …, Xn is from N (q , 1).

  1. Explain Wilcoxon’s Test.
  2. The theory predicts the proportion of beans in the 4 groups A, B, C and D should be

9 : 3 : 3 : 1.  In an experiment among 1600 beans, the number in the 4 groups were 882,

313, 287, 118.  Does the experimental results support the theory?

  1. Explain how will you test for regression coefficients b and a in

yi = a + b (ci – ),    i = 1, 2, … n

  1. Explain the t-test for equality of means of two independent Normal populations.
  2. In a random sample of 500 men from a particular district of Tamil Nadu, 300 are found to be smokers. In one of 1000 men from another district, 550 are smokers.  Do the data indicate that the two districts are significantly different with respect to the prevalence of smoking among men?
  3. Derive the distribution of number of runs.

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

  1. a) State and prove Neyman – Pearson lemma.          (10)
  2. b) Explain monotone likelihood ratio property (MLR) and its use in testing the

hypothesis.                                                                                                                  (10)

 

  1. a) Derive the likelihood ratio test for testing the equality of two variances of two normal

populations N (q1, q3­) and N (q2, q4­), q1, q2 unspecified.                                             (12)

  1. b) Two independent samples of 8 and 7 respectively had the following values of the

variables.

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Do the population variances differ significantly?                                                       (8)

 

  1. a) Explain Man-Whitney – Wilcoxon Test.           (10)
  2. b) Explain Sign-Test (10)

 

  1. a) Test the hypothesis that there is no difference in the quality of the 4 kinds of tyres A,

B, C and D based on the data given below:

                                                                                     Tyre Brand
A B C D
Failed below 40,000 kms 26 23 15 32
Lasted from 40,000 to 60,000 kms 118 93 116 121
Lasted more than 60,000 kms 56 84 69 47
  1. b) Let X1, X2, …, Xn be a random sample from N (q, 100). Find n and c if

Ho­: q = 75 vs H1 :  q = 78.  Given P [ X Î C ½Ho] = .05   and P [X Î C ½H1] = .90.

C = { (x1, x2, …,xn) ½  ≥ c}  is the Best critical region.

 

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Loyola College B.Sc. Statistics April 2006 Testing Of Hypothesis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 19

FIFTH SEMESTER – APRIL 2006

                                                    ST 5501 – TESTING OF HYPOTHESIS

(Also equivalent to STA 506)

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer all the questions.                                               (10 x 2 = 20 Marks)

 

  1. Define a simple and composite hypothesis.
  2. Distinguish between a randomized and non-randomized test.
  3. Provide the form of one-parameter exponential family.
  4. When a test of hypothesis is called uniformly most powerful?
  5. Define likelihood ratio test.
  6. What is a sequential probability ratio test?
  7. Write two uses of chi-square distribution in tests of significance.
  8. Write the 95% confidence interval for the population mean based on large sample.
  9. Define the concept of a quantile of a distribution of a random variable of the continuous type.
  10. Write a note on the sign test.

 

PART – B

Answer any Five questions.                                                       (5 x 8 = 40 Marks)

  1. If X1 X2 …..Xn is a random sample from N (q, 1) ,  q Î R , find a most powerful test for testing H0 : q = 0 against H1 : q = 1.
  2. Let X1 X2 …..X10 denote a random sample of size 10 from a Poisson distribution with mean q. Show that the critical region C defined by  is a best critical region for testing H0: q = 0.1 against H1 : q = 0.5.  For this test determine the significance level   a and the power at  q = 0.5.
  3. X1 X2 …..X25 denote a random sample of size 25 from a normal distribution

N (q, 100).  Find a uniformly most powerful critical region of size a = 0.10 for testing H0: q  = 75 against H1: q > 75.

 

  1. Let Y1 < Y2 < …. < Y5 be the order statistics of a random sample of size n =5 from a distribution with p.d.f

f (x; q) = ½  e– |x– q|,   x Î R, for all real q.  Find the likelihood ratio test for testing Ho : q = q0 against H1 : q ¹ q0.

 

  1. Let X be N (0, q) and let q¢ = 4, q¢¢ = 9, a0 = 0.05 and b0 = 0.10. Show that the SPRT can be based upon the statistic .  Determine c0 (n) and c1 (n).

 

  1. A random sample of 100 units was drawn from a large population. The mean and standard error of the sample are respectively 45 and 8.   Obtain 95% confidence interval for the population mean.

 

  1. For the following data:

X1  :        25        30        45        52        65        75        80        42        50        60

X2 :         60        40        35        50        60        72        63        40        55        62

Test whether the two population variances are equal.  Use 5% significance level.

 

  1. Let X denote the length of time in seconds between two calls entering a college switch board. Let p denote the unique median of this continuous type distribution.  Test H0: p = 6.2 against H1: p < 6.2 based on a random sample of size 20.

 

6.8          5.7       6.9       5.3       4.1       9.8       6.7       7.0       2.1       19.0     18.9

16.9        10.4     44.1     2.9       2.4       4.8       18.9     4.8       7.9

 

PART – C

Answer any Two questions.                                           (2 x 20 = 40 Marks)

 

  1. (a) State and prove Neyman – Pearson theorem.

(b) If  X1 X2 …..Xn  is a random sample from a distribution having p.d.f. of the form f (x; q) = q xq–1,  0 < x < 1, zero elsewhere, show that a best critical region for testing Ho: q = 1 against H1: q = 2 is C = {(x1, x2,…..xn)  :                                     (10+10)

 

  1. Let X1 X2 …..Xn be a random sample from N (q1, q2) and let
    W = {(q1, q2) : –¥ < q1 < ¥, 0 < q2 < ¥} and w = {(q1, q2) : q1 = 0, 0 < q2 < ¥)}.

Perform the likelihood ratio test for testing H0: q1 = 0, q2 > 0            against H1: q1 ¹ 0,
q2 > 0.

  1. (a) Consider the following data:

X:        32        40        50        62        70        35        42        45        19        30

Y:           30        43        55        60        65        23        32        40        20        35

Test whether the two population means are equal at 1% significance level.

 

  • The number of transactions in a teller’s counter during the days of the week is given below:

Mon.             Tue.                 Wed.               Thu.                 Fri.                   Sat

33                 28                    24                    22                    30                    19

Test whether the number of transactions are uniformly distributed or not.  Use 1% level of significance.                                                                                                                    (10+10)

 

  1. Let X and Y denote the weights of ground cinnamon in 120 gram tins packaged by companies A and B respectively. Use Wilcoxon test to test the hypothesis
    H0 : px = py against H0 : px < py.  The weights of n1 = 8 and n2 = 8 tins of cinnamon packages by the companies A and B selected at random yielded the following observations of X and Y respectively.

X:  117.1       121.3        127.8        121.9          117.4       124.5      119.5        115.1

Y:  123.5       125.3        126.5        127.9          122.1       125.6      129.8        117.2

 

 

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Loyola College B.Sc. Statistics Nov 2006 Testing Of Hypothesis Question Paper PDF Download

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                        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 14

FIFTH SEMESTER – NOV 2006

ST 5501 – TESTING OF HYPOTHESIS

(Also equivalent to STA 506)

 

 

Date & Time : 27-10-2006/9.00-12.00   Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions.                                                        10 X 2=20 marks

 

  1. Define Type I and Type II errors.
  2. Give an example each for a simple and composite hypothesis.
  3. When do you say that a critical region is uniformly best?
  4. Define level of significance and power function of a test.
  5. Provide a testing problem for which no uniformly best critical region exists.
  6. When a distribution is said to belong to an exponential family?
  7. When a” Likelihood Ratio Test” is used?
  8. Define Sequential Probability Ratio Test.
  9. Write any two applications of chi-square distribution in testing.
  10. Write a short note on “Sign Test”.

 

 

PART – B

Answer any FIVE questions.                                                        5 X 8=40 marks

 

  1. Let X1, … X10 be a random sample of size 10 from a normal distribution      N (0, 2 ). Find  a best critical region of size  = 0.05 for testing Ho : 2 = 1  against H1: 2 = 2.
  2. Let X1, X2,…. Xn be a random sample from a distribution with the p.d.f.                         f(x;q) = q xq-1   0 < x <  zero elsewhere, where .  Find a sufficient statistic  for q and show that a UMPT of Ho : q = 6 against H1 : q < 6 is based on this statistic.
  3. Let X1,X2, …, Xn be a random sample from the normal distribution N(q,1). Show that the likelihood ratio principle for testing Ho : q = q¢, where q¢ specified, against H1 : q q¢ leads to the inequality
  4. Let X have a Poisson distribution with mean q. Find the sequential probability ratio test for testing Ho : q = 0.02 against H1 : q = 0.07.  Show that this test can be based upon the statistics   If  a0 = 0.20 and =0.10,  find co (n) and    c1 (n).
  5. Let X1,X2,… Xn denote a random sample from a distribution that is N (q,1). where the mean q is unknown. Show that there is no uniformly most powerful test of the simple hypothesis Ho: q = q¢, Where q’ is a fixed number, against the alternative composite hypothesis H1 : ¢.
  6. Let X1, X2, … Xn be a random sample from a Bernoulli distribution with parameter p, where 0<p< 1. Show that the distribution has a  monotone likelihood  ratio in the statistic Y = .

 

 

 

 

 

 

 

  1. The demand for a particular spare part in a factory was found to vary from day-to-day.  In a sample study the following information was obtained.
Day Mon Tue Wed Thu Fri Sat
No of parts demanded 1124 1125 1110 1120 1126 1115

 

Using chi-square test, test the hypothesis that the number of parts demanded does not depend on the day of  the week. Use 1% significance level.

 

  1. The lengths in centimeters of n = 9 fish of a particular species captured off the

New England coast were 32.5, 27.6, 29.3, 30.1, 15.5, 21.7, 22.8, 21.2, 19.0.

Use Wilcoxon test to test Ho : m = 3.7 against the alternative hypothesis

H1 : m > 3.7 at 5% significance level.

 

PART – C

Answer any TWO questions.                                                     2 X 20 =40 marks

 

  1. (a) State and prove Neyman –Pearson theorem.
    • Consider a distribution having a p.d.f. of the form f (x ; q)  = qx (1-q)1-x, x = 0, 1

= 0, otherwise.

Let Ho : q = and  H1: q >. Use the central limit theorem to determine the sample size n of a random sample so that a uniformly most powerful test of H0  against H1 has a power function K(q), with approximately K = 0.05 and K = 0.90.                  (10+10)

  1. Let the independent random variables X and Y have distributions that are N(q1,q3) and N (q2, q3) respectively,  where the means q1, q2 and common variance q3 are unknown.  If X 1, X2, ….,Xn and Y1, Y2, … Ym are independent random samples from these distributions, derive a likelihood ratio test for testing H0 : q1=q2 ,  unspecified and q3 unspecified against all alternatives.
  2. (a) Let X be N (0,q) and let q’ = 4, q” = 9. a0 = 0.05 and =0.10. Show that

the sequential probability ratio test can be based on the statistic

Determine  c0 (n) and c1 (n).

 

 

 

 

(b)  A cigarette manufacturing firm claims that its brand A of the cigarettes

outsells its brand B by 8%. If it is found that 42 out of a sample of 200

smokers  prefer brand A and 18 out of another random sample of 100

smokers prefer brand B  test whether the 8% difference is a valid claim.  Use

5% level of significance.                                                                     (10 + 10)

  1. (a) Below is given the distribution of hair colours for either sex in a University.

Hair colour

Fair Red Medium Dark Jet black
Sex Boys 592 119 849 504 36
Girls 544 97 677 451 14

Test the homogeneity of hair colour for either sex.  Use 5% significance level.

  • Using run test, test for randomness for the following data :

15  77  01  65  69  58  40  81  16  16  20  00  84  22  28  26  46  66  36  86  66  17  43  49  85  40  51  40  10 .                                                                                                    (10+10)

 

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Loyola College B.Sc. Statistics April 2007 Testing Of Hypothesis Question Paper PDF Download

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                LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 17

 

FIFTH SEMESTER – APRIL 2007

ST 5501 – TESTING OF HYPOTHESIS

 

 

 

Date & Time: 28/04/2007 / 1:00 – 4:00 Dept. No.                                              Max. : 100 Marks

 

 

PART – A

Answer ALL the questions.                                                        10 X 2=20 marks

  1. Define a simple and composite statistical hypothesis and give an example each.
  2. What are Type-I and Type-II errors in testing of hypothesis.  Also define the power function.
  3. Define a best critical region (BCR) of size for testing the simple hypothesis against an alternative simple hypothesis.
  4. When do you say that a BCR is uniform?
  5. When do you say that a distribution belongs to an exponential family?
  6. Under what situation likelihood ratio test is used?
  7. Define Sequential Probability Ratio Test (SPRT).
  8. State any two differences between SPRT and other test procedures.
  9. Write the statistic for testing the equality of means when the sample is small.
  10. What is a non-parametric test?

 

PART – B

Answer any FIVE questions.                                                        5 X 8 =40 marks

  1. Let Y have a binomial distribution with parameters n and p. We reject Ho: p = ½ and accept     H1 = p> ½ if y  Find n and c to give a power function K(p)  which is such that K  = 0.10 and K = 0.95 approximately.
  2. State and prove Neyman Pearson theorem.

 

  1. If X1, X2,…Xn is a random sample from normal distribution with mean q and variance 1, find a BCR of size a for testing Ho: q = 0 against H1 : q =1.

 

  1. Let X1, X2, … Xn denote a random sample from a distribution that is N(q, 1). Show that  there is no uniformly most powerful test of the simple hypothesis Ho: q = q¢ where q¢ is a fixed number , against  the alternative composite hypothesis H1 : qq¢.
  2. Let X1, X2, ….Xn be a random sample from a Poisson distribution with parameter q where q>0. Show that the distribution has a monotone likelihood ratio in the statistic Y= .
  3. Let Y1,< Y2 < …. < Y5 be the order statistics of a random sample of size n=5 from a distribution with pdf f(x;q) =   e–|x – q | , -<x<, for all real q.  Find the likelihood ratio test for resting Ho: q = q0 against H1 : q q0.
  4. Ten individuals were chosen at random from a population and their heights     were   found to be in inches 63, 63, 66, 67, 68, 69, 70, 70, 71, 72.  Test the hypothesis   that the height in the population is 66 inches. Use 5%  significance level.

 

  1. Explain the wilcoxon

 

 

PART – C

Answer any TWO questions.                                                              2 X 20 =40 marks

 

  1. (a) Let X1, X2,…, X10 denote a random sample of size 10 from a Poisson

distribution with mean q.  Show that the critical region C defined by

is a best critical region for testing H0 : q = 0.1 against

H1 : q = 0.5.  Determine the significance level a and the power at q = 0.5

for this test.

  • Let X have a pdf of the form f (x;q) = , 0 < x < q  zero elsewhere.  Let

Y1< Y2 < Y3 ,< Y4 denote the order statistics of a random sample of size 4

from this distribution. we reject H0 : q = 1 and accept H1 :q1 if either

y4 or Y4  Find the power function K (q),  0 < q, of the test.   (10+10)

  1. Let the random variable X be N(q1,q2). Derive a likelihood radio test for testing                         H0 : q1 = 0, q2 >0 against H1 : q1 0  , q2 >0.

 

  1. (a) Let X be N(0,q) and let q¢ = 4 , q” = 9, a0 = 0.05, and o = 0.10.  Show

that  the sequential probability ratio test can be based upon the statistic

. Determine c0(n) and c1(n) .

  • In a survey of 200 boys, of which 75 were intelligent, 40 has skilled

fathers while 85 of the intelligent boys has unskilled fathers.  Do these

figures support the hypothesis that skilled fathers have intelligent boy?  Use

5% significance level.                                                                       (10+10)

 

  1. (a) An IQ test was administered to 5 persons before and after they were trained.

The results are given below :

 

Candidates I II III IV V
IQ before training 110 120 123 132 125
IQ after training 120 118 125 136 121

 

Test whether there is any improvement in IQ after the training programme. Use 1%

significance level.

  • Let m be the median lung capacity in litres for a male freshman. Use sign test to test at the a = 0.0768  significance level, the null hypothesis  Ho : m = 4.7 against the two sided alternative hypothesis H1 :.m > 4.7.  The observations are : 7.6   4   4.3   5.0   5.7   6.2   4.8   4.7   5.6   5.2   3.7   4.0   5.6   6.8   4.9   3.8   5.6                                                   (10+10)

 

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Loyola College B.Sc. Statistics Nov 2008 Testing Of Hypothesis Question Paper PDF Download

by

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 11

 

FIFTH SEMESTER – November 2008

ST 5501 – TESTING OF HYPOTHESIS

 

 

 

Date : 05-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

Answer ALL the Questions.                                                                               (10 x 2 =20 marks)

 

  1. Define : Best Critical Region
  2. What are randomized tests? Give an example of randomized test.
  3. Give an example of a family of density functions not having MLR Property.
  4. When do you say a density function is a member of one parameter exponential family?
  5. What is the fundamental difference between SPRT and other conventional tests?
  6. Describe: Likelihood ratio criterion.
  7. Mention the statistic used for testing the hypothesis that the population correlation coefficient is zero.
  8. Define Confidence level.
  9. Define the term run in connection with Non-parametric methods
  10. What is Empirical Distribution Function?

 

 

PART – B

 

Answer any FIVE questions.                                                                             (5 x 8 =40 marks)

 

  1. State and prove Neyman Pearson lemma
  2. Show that Binomial densities are members of one parameter exponential family.
  3. Derive the UMPT test of level 0.05 for testing the hypothesis against the alternative based on a single observation drawn from exponential distribution with mean and also obtain its power function
  4. Derive the likelihood ratio test for testing  againstbased on a sample of size 17 drawn from when is unknown.
  5. Write a descriptive note on SPRT
  6. Derive the large sample confidence interval for in the case of exponential distribution with mean
  7. Explain run test for randomness.
  8. Describe Kolmogrov-Smirnov one sample test

 

 

 

PART – C

 

Answer any TWO questions.                                                                             (2 x 20 =40 marks)

 

  1. Derive the MPT of level 0.10 for testing against the alternative in Poisson distribution with mean  based on a sample of size 10 and compute the power of the test.

 

  1. Derive the UMPT for testingagainst based on a sample of size n drawn from and also derive its power function.

 

  1. Derive the likelihood ratio test for testing the equality of two normal population means assuming that the variances are unknown but equal and the sample sizes are different.

 

 

  1. Write short notes on the following:
  • Randomised tests vs Nonrandomised tests
  • Monotone Likelihood Ratio Property
  • Large sample confidence intervals
  • Mann-Whitney U Test

 

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