• October 2nd, 2017

Homework Assignment #1 – Statistics and Probability

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Scenario: The head of the chemistry department at a local university has tasked a student teaching assistant to prepare a variety of statistics pertaining to the final course grades earned by students who have completed the university’s CHEM 103 course during the past ten years. These statistics will be used as a basis for predicting the potential performance of future students who enroll in the course. A total of 1,125 students have completed the course during the past ten years. Rather than trying to have the teaching assistant collect and analyze data for all 1,125 students, the department head has agreed to allow the teaching assistant to prepare the necessary statistics using a small random sample consisting of 25 students who completed the course during the past ten years. The following table summarizes the final course grades earned by the 25 randomly selected students.

Student No. Course Grade Student No. Course Grade
1 83.17 14 86.76
2 82.31 15 88.69
3 90.18 16 86.89
4 93.63 17 88.57
5 92.38 18 96.93
6 96.06 19 86.48
7 83.75 20 92.15
8 85.17 21 92.32
9 86.67 22 78.70
10 94.27 23 92.67
11 83.11 24 74.56
12 82.16 25 78.48
13 89.48

Data: For the purposes of this assignment, assume that: (1) the small sample consisting of 25 students is truly representative of the population of 1,125 students from which it was drawn; and (2) the 1,125 students who have completed the course during the past ten years constitute a truly representative sample of all future students who will eventually enroll in the course. Based upon these assumptions, use the sample data provided in the preceding table to answer the following questions. This HW is worth a total of 100 total points. The points for each question may vary.

Question 1. What is the predicted range for the mean grade for an average future student enrolling in the CHEM 103 course? (3 points)

Question 2. What is the predicted mean grade for an average future student enrolling in the CHEM 103 course? (3 points)

Question 3. What is the predicted median grade for an average future student enrolling in the CHEM 103 course? (3 points)

Question 4. What is the predicted standard error of the mean grade for an average future student enrolling in the CHEM 103 course? (3 points)

Question 5. Assuming the level of confidence for the interval estimate is not specified, what is the predicted interval estimate for the mean grade for an average future student enrolling in the CHEM 103 course? (4 points)

Question 6. Assuming that a 99% level of confidence for the interval estimate is desired, using the sample data, what is the predicted interval estimate for the mean grade for an average future student enrolling in the CHEM 103 course? (4 points)

Question 7. Using the sample data, calculate the predicted variance for the grades for average future students enrolling in the CHEM 103 course? (4 points)

Question 8. Using the sample data, calculate the predicted standard deviation for the grades for average future students enrolling in the CHEM 103 course? (4 points)

Data: Using the previously provided sample data for the final course grades earned by the 25 randomly selected students, determine the frequency, relative frequency and cumulative frequency for each of the following eleven grade classes.

Grade Classes Grade Classes
0.00-10.00 60.00-70.00
10.00-20.00 70.00-80.00
20.00-30.00 80.00-90.00
30.00-40.00 90.00-100.00
40.00-50.00 100.00 or more
50.00-60.00

Question 9. Assuming that student grades would theoretically be symmetrically distributed around the 70.00 – 80.00 grade class, does the histogram indicate that the distribution of actual student grades is skewed (either positively skewed or negatively skewed)? (3 points)

Question 10. Which grade class evidences the highest frequency of actual student grades? (3 points)

Question 11. What is the relative frequency for the 70.00 – 80.00 grade class? (3 points)

Question 12. What is the cumulative frequency for the 80.00 – 90.00 grade class? (3 points)

Data: The following table summarizes the grades for ten group projects for two groups of high school students. Use the information in the table below to answer the following question.

Project No. Group 1 Grades Group 2 Grades
1 96.11 99.22
2 87.65 96.44
3 74.56 73.61
4 92.01 85.07
5 98.69 68.51
6 90.25 95.58
7 78.35 96.97
8 88.89 76.35
9 94.94 89.35
10 77.22 86.99

Question 13. What is the mean project grade for Group 1? (3 points)

Question 14. What is the mean project grade for Group 2? (3 points)

Question 15. What is the standard deviation for the mean project grade for Group 1? (3 points)

Question 16. What is the standard deviation for the mean project grade for Group 2? (3 points)

Question 17. Which group of students exhibits the least degree of central tendency about the mean value for their project grades? (3 points)

Data: A standard deck of playing cards consists of fifty-two cards. The cards in each deck consist of four suits, namely spades (♠), clubs (♣), diamonds (♦) and hearts (♥). Each suit consists of thirteen cards, namely ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2. In the game of poker, a royal flush consists of the ace, king, queen, jack, and 10 of the same suit (e.g., ace of spades, king of spades, queen of spades, jack of spades and 10 of spades).

Question 18. What is the probability of randomly selecting five cards from a randomly shuffled deck of playing cards that constitute a royal flush, assuming the order in which the cards are selected is irrelevant (i.e., not important) and the suit is irrelevant (i.e., not important)? (4 points)

Question 19. What is the probability of randomly selecting five cards from a randomly shuffled deck of playing cards that constitute a royal flush, assuming the cards must be selected in a specific order, namely ace, king, queen, jack and 10, and the suit is irrelevant (i.e., not important)? (4 points)

Data: 45 students in two sections of a college Physics 101 course recently took a mid-term exam. 12 students earned an A, 9 students earned a B, 8 students earned a C, 8 students earned a D and 8 students earned an F on the exam. The students were queried regarding the number of hours they had devoted to studying for the exam, 10 of the students who earned an A, 6 of the students who earned a B, 6 of the students who earned a C, 4 of the students who earned a D, and 2 of the students who earned an F reported that they had devoted more than 8 hours to studying for the exam. The remaining students reported that they had devoted no more than 8 hours to studying for the exam.

Question 20. What is the probability of a randomly selected student having earned an A on the exam? (3 points)

Question 21. What is the probability of a randomly selected student having earned a B on the exam? (3 points)

Question 22. What is the probability of a randomly selected student having devoted no more than 8 hours to studying for the exam? (3 points)

Question 23. What is the probability of a randomly selected student having earned an A on the exam given they devoted no more than 8 hours to studying for the exam? (3 points)

Question 24. What is the probability of a randomly selected student having earned an F on the exam given they devoted more than 8 hours to studying for the exam? (3 points)

Question 25. What is the probability of a randomly selected student having earned an A or a B on the exam given they devoted more than 8 hours to studying for the exam? (3 points)

Data: Scores for a certain exam follow a normal distribution with a mean of 82.54 and a standard deviation of 3.15. Answer questions 26 and 27 using the preceding information.

Question 26. What is the standard Z-score associated with a score of 88.12? (3 points)

Question 27. What is the probability that a randomly selected student’s score will fall between a standard Z-score of -1.65 and a standard Z-score of 1.88? (3 points)

Data: The mean time required to complete a certain type of construction project is 61 weeks with a standard deviation of 4.15 weeks. Answer questions 28-31 using the preceding information and modeling this situation as a normal distribution.

Question 28. What is the probability of the completing the project in no more than 58 weeks? (3 points)

Question 29. What is the probability of the completing the project in more than 62 weeks? (3 points)

Question 30. What is the probability of completing the project between 58 weeks and 65 weeks? (3 points)

Question 31. What is the probability of completing the project within plus or minus one standard deviation of the mean? (3 points)

 

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