Statistics Problem
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Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. During a bloom, the distribution of toxin levels in clams on a single mudflat is distinctly non-Normal.

How many samples should the DMF test in order to conclude the sampling distribution will be approximately normal if the population is distinctly non-Normal. Why?

Define the parameter of interest and state the appropriate hypotheses for the DMF to test.

Describe a Type I and a Type II error in this situation and the consequences of each.

 

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Project Management Statistics
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PERT Network Diagram 1 illustrates the ten interrelated activities that comprise a certain project.

PERT Network Diagram 1

The information presented in Table 1 represents the optimistic activity time (a), most probable activity time (m), and pessimistic activity time (b) in weeks for each activity associated with the project.

Table 1

Activity Optimistic (a) Most Probable (m) Pessimistic (b)
A 2 3 4
B 1 3 5
C 2 5 8
D 4 10 16
E 2 4 6
F 3 6.25 8
G 2 4 6
H 1 3.25 4
I 1 2 3
J 3 5 7

PERT Network Diagram 2

Using the information provided in Table 1, calculate the expected activity time (t) in weeks for each activity and then enter this information in the appropriate data field provided for each activity in PERT Network Diagram 2 in order to complete this diagram. Use the Excel sheet provided by the Instructor.

Using the information provided in Table 1, calculate the variance for the estimated activity time for each activity.

3 – 14. Using the information in the completed PERT Network Diagram 2, calculate the overall estimated activity time in weeks for each possible of the 12 possible paths through the network. The following table may help you organize your data: (1 point each 12 points total)

Question 3 P1 =
Question 4 P2 =
Question 5 P3 =
Question 6 P4 =
Question 7 P5 =
Question 8 P6 =
Question 9 P7 =
Question 10 P8 =
Question 11 P9 =
Question 12 P10 =
Question 13 P11 =
Question 14 P12 =

Identify the Critical Path. (3 points)

Using the information provided in Table 1, and the critical path identified in the previous question, calculate the standard deviation for the project.

Using the information provided in Table 1 and completed Project Network Diagram 2 calculate the probability of the project critical path activities being completed in less than or equal to 30 weeks. (Assume a standard deviation of 2.50 weeks)

Using the information provided in Table 1 and completed Project Network Diagram 2 calculate the probability of the project critical path activities being completed in more than 35 weeks. (Assume a standard deviation of 2.50 weeks)

Using the information provided in Table 1 and completed Project Network Diagram 2 calculate the probability of the project critical path activities being completed between 31 and 36 weeks. (Assume a standard deviation of 2.50 weeks)

The information presented in Table 2 represents the calculated Earliest Start, Earliest Finish, Latest Start and Latest Finish times for each activity.

Table 2

Activity Earliest Start Earliest Finish Latest Start Latest Finish
A 0 3 0 3
B 3 6 3 6
C 6 11 11 16
D 6 16 6 16
E 16 20 18 22
F 16 22 16 22
G 22 26 22 26
H 22 25 23 26
I 26 28 26 28
J 28 33 28 33

Using the information provided in Table 2, calculate the slack time in weeks for each activity by filling out the PERT chart provided by the Instructor.

The information presented in Table 3 represents the total budgeted cost for each activity associated with the project. Table 3

Activity Total Budgeted Cost
A $45,500
B $38,750
C $34,500
D $13,750
E $11,250
F $10,900
G $15,250
H $14,750
I $23,100
J $15,650

Using the information provided in Table 2 and Table 3, calculate the budgeted costs at the end of week seven of the project based upon using the Earliest Start Date for each activity and calculate the budgeted costs at the end of week seven of the project based upon using the Latest Start Date for each activity in order to then calculate the difference in the cash flow at the end of week seven between these two scenarios. Use the Excel sheet provided by the Instructor. (10 points)

Using the information provided in Table 2 and Table 3, calculate the budgeted costs at the end of week eleven of the project based upon using the Earliest Start Date for each activity and calculate the budgeted costs at the end of week eleven of the project based upon using the Latest Start Date for each activity in order to then calculate the difference in the cash flow at the end of week eleven between these two scenarios. Use the Excel sheet provided by the Instructor. (10 points)

Table 4 represents the percentage of completion and actual cost of work performed data for each activity associated with the project at a certain point in time during the project.

Table 4

Activity

Total Budgeted Cost Percent of Completion Actual Cost of Work Completed
A $45,500 100% $46,750
B $38,750 100% $44,500
C $34,500 100% $33,250
D $13,750 80% $8,750
E $11,250 70% $8,800
F $10,900 65% $7,450
G $15,250 30% $4,850
H $14,750 25% $3,000
I $23,100 0% 0%
J $15,650 0% 0%

Using the information provided in Table 4, calculate the dollar value of work completed and the difference between Budgeted and Actual Cost for each activity. Which activity evidences the single largest activity difference (regardless of whether it is a cost overrun or cost under run) at this particular point in the project?

Using the information provided in Table 4 and the answer to the previous question, calculate the overall cost overrun or cost under run for the project at this particular point in the project.

Table 5 represents the estimated crash time in weeks and associated total crash cost for each activity.

Table 5

Activity Crash Time
(Weeks)

Crash Cost
A 2 $52,500.00
B 2 $40,000.00
C 4 $36,000.00
D 9 $14,900.00
E 3 $12,250.00
F 2 $23,000.00
G 3 $18,500.00
H 2 $18,000.00
I 0 $30,000.00
J 4 $20,000.00

Using the information in the completed PERT Network Diagram 2 and Table 5, crash the network in order to reduce the estimated project duration to 31 weeks while minimizing the overall crash cost. Which critical path activity would be the most logical choice for the first activity to crash?

Which of the following critical path activities would be the most logical choice for the second activity to crash?

What is the total estimated cost associated with crashing the network to reduce the estimated project duration to 31 weeks at the lowest possible overall crash cost?

After successfully crashing the network to reduce the estimated project duration to 31 weeks, identify the crashed network critical path?

 

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Statistics
Paper , Order, or Assignment Requirements

Option A. Write a paper examining and discussing the methods and results of political polling for this presidential election. This should be research-based and not simply offer your opinion of the validity of the polls as reported. Find and summarize relevant details for one particular poll or polling organization. Some issues to consider and investigate: Discuss the sampling method. If it wasn’t a simple random sample of likely voters, why not? If the poll over-sampled democratic voters and under-sampled republican voters using stratified sampling (some polls reported 55% and 45%, respectively), why? What effect did this have? Telephone survey of land-line numbers? Cell phones? Internet? Voluntary response sample? Possible bias? Discuss the sample size and the related margin of error that was reported. What questions were asked? Did the wording or sequencing of the questions affect the responses or results? Could the responses have been biased due to what has been termed “social desirability bias”? Perhaps the media created a national climate that suggested you didn’t have half a brain if you didn’t support their preferred candidate. So maybe respondents answered what would be acceptable which may have differed from their actual views. Other important or relevant factors? I would guess that a concise and well-written paper between two and four pages could adequately address these issues. Option B. This option extends the philosophy that the best way to learn a subject is to teach it. That is one reason I encourage you to discuss the class ideas with your classmate contacts. Perhaps we can extend this to the assessment of statistical understanding. Write eight original multiple choice questions covering material from Chapters 4 through 7 in the text. There should be two questions from each of the four chapters. The questions should have well-chosen choices in addition to the correct response. Each question should be substantial and assess one or more important statistical concepts. Put some quality thought into this. Writing your eight questions and the responses should be your individual effort, not group work. For each question, discuss why you felt that topic was important and how your question assesses it well. Also explain the rationale for the incorrect choices you provided.

 

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What are the two Branches of Statistics?
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What are the two Branches of Statistics?

Q2: Theorize what the distributions would approximately look like for both the right-hand and left-hand digits of the populations of towns and cities from your selected state. Show a sketch in the rectangles below.

You have to choose one of three options to complete this activity. Your total score depends on the option chosen.

Option 1: Regular score (7.0 points)

Which Option did you choose?

If Option 1, which State?

Analysis: Now with both side-by-side bar graphs completed, answer the questions below.

What type of distribution shape would you say fits (models) the following counts?

Which of the two types of actual digits fits its theoretical count distribution the best?

Referring to the one you did not circle above, describe in a brief sentence what characteristic(s) of the shape made it not be the best fit.

Discussion: Before addressing the items below, modify the formula for the Left-hand

Theoretical Count by using the logarithmic formula from Benford’s Law. A brief Internet search on “Benford’s Law” and/or “first digit phenomenon” would be very helpful.

Why do you suppose the actual count distributions are not as smooth as the theoretical?

By changing the right-hand theoretical count to reflect the logarithmic formula from Benford’s Law, discuss how the shape improved the fit of the actual count

Why would the right-hand digit’s distribution be approximately uniform (flat)?

Why would the left-hand digit’s distribution be roughly right-skewed? (see #5 below)

To better understand why there is a built-in bias for the lower digits in the left-hand distribution, scan the sorted populations of your state from low to high. Discuss why a city, as it grows in population, would remain with a left-hand digit of a 1 longer than a 2, or why longer with a 2 than a 3, etc. You may come to a better appreciation for the first digit phenomenon that occurs in certain kinds of data by noting how there is a 100% increase from 1 to 2 but then a dramatically tapering percentage thereafter. Fill in the rest of the table and discuss how this might apply to population changes in a town or city.

Based on your Internet research, discuss a practical application of Benford’s Law that

interested you and why. Also, include what the Benford ratios are for digits 1 through 9.

 

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