1.The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent table.

Vehicles

per Minute

Observed

Frequency

Vehicles

per Minute

Observed

Frequency

40

14

53

102

41

24

54

96

42

57

55

90

43

111

56

81

44

194

57

73

45

256

58

64

46

296

59

61

47

378

60

59

48

250

61

50

49

185

62

42

50

171

63

29

51

150

64

18

52

110

65

15

In this problem, you’ll be testing whether the above data reflects a Poisson distribution.

a.Derive the maximum likelihood estimator for a Poisson Distribution with parameter l.

b.Using your answer from part 1, if we assume the data does reflect a Poisson Distribution, find the estimator for l.

c.Create an appropriate histogram of the data. Use as many buckets as you think is appropriate.

d.Using the results from parts (2) and (3), and the provided data, run an appropriate test to determine whether the data comes from a Poisson Distribution.

2.Consider the data below.

Hours Studied

1

0

3

1.5

2.75

1

0.5

2

3

1.75

Test Score

76

66

96

84

100

81

85

79

100

81

a.Using Least-Squares regression, find the line of best fit that uses Hours Studied to predict a Test Score.

b.Create a scatterplot of the data and draw your line from part (a).

c.Find SSE, SSR, and SST.

d.Find the Coefficient of Determination, and give a brief interpretation of the answer.

e.Test whether both the intercept and slope parameters are 0. Use α = 0.05

f.We define regression correlation as, where XY is the covariance between Y and X.

It can be determined that, and a correlation is considered Strong if |is above .5. In other words, the sample correlation coefficient is the square root of R2. Find the sample correlation.

g. Run an appropriate test to determine whether this correlation is 0. The test statistic, , has degrees of freedom n-2.

- Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown in the following tabulation. Is there sufficient evidence to conclude that both tests give the same mean impurity level, using α = 0.01? Round numeric answer to 2 decimal places.

Specimen

Test 1

Test 2

1

1.4

1.6

2

1.5

1.9

3

1.7

1.7

4

1.4

1.3

5

1.9

2.2

6

2

2.3

7

1.6

1.9

8

1.5

1.8

4.Consider the data from problem (3). If we could not assume the data came from a paired test, would you reach the same conclusion? Run an appropriate Hypothesis test with α = 0.01.

Sample Solution

built; diminishing the ability of local entrepreneurs to set up competing businesses and increase their wealth. The likelihood of a democratic transition is therefore low, since “democracy is expected to increase redistribution and reduce inequality”[26]; something which is not in the interest of the elite ruling classes. Moreover, economic crises can have a large role to play in mobilising a population against the elites and causing the fall of a non-democratic government. Although the elites do have “the monopoly over large scale violence, […] states in crisis can […] slide […] into even more instability”[7], particularly if a popular revolution is supported by a large proportion of the population, or, as in the case of Syria, the “improving […] economic conditions of the large Syrian refugee communities in neighbouring countries [provide] economic alternatives to joining armed groups”[8]; decreasing the state’s military stronghold over its population. It is certain, however, that the likelihood of the collapse of a non-democratic regime as a result of an economic shock depends on its depth and severity, and the degree to which there is the resulting loss in welfare incentivising the population to mobilise. Furthermore, if the state is able to reallocate resources effectively despite an economic crisis, they may be able to withstand opposition to power; for example, by “[cutting] back outlays on subsidies, enabling it to concentrate more resources on the police, domestic security, and the state’s cultural and media propaganda machinery”[9] (page 165), as was undertaken in Egypt under Mubarak. While wealth and development are undoubtedly significant in causing a shift towards democratic governance, “authoritarian regimes around the world [have shown] that they can reap the benefits of economic development while evading any pressure to relax their political control. [An example is China’s economy, which] has grown explosively over the last 25 years, [even though] its politics have remained essentially stagnant.”[10] Evidence of this undermines the thesis that economic factors are the most significant. A second possible explanation for some non-democratic regimes outliving oth>

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