Mathematics Homework Help

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 SS_E, SS_R, and SS_T.

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.