Question

Following U(0,1) were generated by a random number generation method:

251

 

597

 

449

 

464

769

 

729

 

998

 

447

153

 

012

 

919

 

279

575

 

922

 

470

 

413

390

 

691

 

372

 

494

335

 

817

 

851

 

972

066

 

064

 

643

 

983

104

 

539

 

509

 

432

200

 

419

 

913

 

368

019

 

305

 

445

 

574

 Apply chi-square test to test the fit of the distribution

16 Feb 2018
Answer :
Word Count : 549
To apply the chi-square goodness-of-fit test, we will follow these steps: ### Step 1: Understand the Data The given numbers seem to be uniform (0,1) random values scaled up by 1000 (i.e., they range from 0 to 999). We will divide this range into 10 equal bins: \[ [0-99], [100-199], \dots, [900-999] \] Since the numbers are uniformly distributed, we expect each bin to contain an equal number of values. ### Step 2: Compute Expected and Observed Frequencies - Expected Frequency: If the numbers follow \(U(0,1)\), each of the 10 bins should contain roughly \( \frac{\text{Total Numbers}}{10} \) values. - Observed Frequency: Count how many numbers fall into each bin. ### Step 3: Compute the Chi-Square Statistic The chi-square statistic is calculated as: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where: - \( O_i \) = observed count in bin \( i \) - \( E_i \) = expected count in bin \( i \) ### Step 4: Compare with Chi-Square Distribution - Degrees of Freedom (df) = \(
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Related Questions

The following data represent the number of units of production per day turned out by 5 different workers using four different types of machines:

Workers

Machine Type

A

B

C

D

1

2

3

4

5

54

56

44

53

48

48

50

46

48

52

57

62

54

56

59

46

53

42

43

49

 

a) Test whether the mean productivity is the same for the different machine types.

b) Test whether the 5 workers differ with respect to mean productivity.

Below are given the weights of 5 students:

S. No

1         2         3        4            5

Weights (Kg)

56       49        66        64        55

a) Calculate Population Mean, Population mean square and Population Variance.

b) Enumerate all possible samples of size 2 without replacement and show that

i) Sample mean (\overline{y})is unbiased for Population mean \overline{Y}

ii) Sample mean square (s2) is unbiased for the Population Mean Square (S2)

iii) Var(\overline{y})\frac{N-n}{N},\frac{S^{2}}{n}

Select a sample of 3 cluster and calculate the sample cluster mean \overline{Y}c

Below is given the plan and yields of 22 factorial experiment involving 2 factors A and B each at two levels 0 and 1:

Block I

(1)

117

b

106

ab

129

a

124

Block II

ab

124

(1)

120

b

117

a

124

Block III

(1)

111

a

127

b

114

ab

126

Block IV

ab

125

a

131

b

112

(1)

108

Block V

ab

95

b

97

(1)

73

a

128

Block VI

a

158

(1)

81

ab

125

b

117

Analyse the design. Does treatment effects A differ from treatment effect B significantly?

Suppose from a total of 20 bearing trees of Guava in a village, 5 clusters of size 4 trees each were selected and Yield (in Kgs) is a given in the following table:

Cluster

1st Tree

2nd Tree

3rd Tree

4th Tree

1

2

3

4

5

5

11

36

7

2

4

1

10

15

22

1

4

19

12

8

15

7

11

10

6

i) ) Y_{i}^{5}(i = 1,2...,...,.5) cluster totals 

ii) ) \overline{Y}_{i}^{5}(i = 1,2...,..., -5) cluster Means 

iii) \overline{Y}_{C} Population Mean per Cluster and 

iv) \overline{Y}, i.e., Population Mean Per Unit  
  

The height of the plants are given in the table given below for an experiment involving six treatments in four randomised blocks. The treatments are indicated by numbers with brackets:

Blocks

Heights of Plants (cms)

I

(1)

24.7

(3)

27.7

(2)

20.6

(4)

16.2

(5)

16.2

(6)

24.9

II

(3)

22.7

(2)

28.8

(1)

27.3

(4)

15

(6)

22.5

(5)

17

III

(6)

26.3

(4)

19.6

(1)

38.5

(3)

36.8

(2)

39.5

(5)

15.4

IV

(5)

17.7

(2)

31

(1)

28.5

(4)

14.1

(3)

34.9

(6)

22.6

Analysis the data using suitable design and draw the conclusion.

Given the following population, allocate a sample of size n = 10 by optimum allocation:

Stratum I :

2       4       4       6       6        8

Stratum II :

8       8      12     16     16       18

Stratum III :

16    16     18      22      22     26
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