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.

3. 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