Simulation and Expected Values: Yale Law School says 74% of their students pass the bar exam on their first try.

To simulate passing students, we could assign the random digits as:

00 to 49 = pass first try, 50 to 99 = fail first try

0 to 7 = pass first try, 8 to 9 = fail first try

00 to 73 = pass first try, 74 to 99 = fail first try

0 to 4 = pass first try, 5 to 9 = fail first try

The outcomes for this experiment are __________________, with a probability of ________________, and __________________, with a probability of __________________.

David, a 55 year old high school teacher is a heavy smoker with really poor eating habits. He was admitted to the ER a week ago because he was experiencing symptoms including shortness of breath, chest pain, weakness in his left arm and left leg. He was rushed to the Coronary Care Unit, where clinicians performed Electrocardiogram (ECG) and arteriogram tests. Results showed 80% occlusion of the left main coronary artery.

1. What condition does he have? a) dilated cardiomyopathy, b) atherosclerosis, c) myocardial infarction, d) cardiac arrhythmia, e) endocarditis

2. Which treatment is David most likely to receive for this condition?

3. What are the risk factors for this condition?

4. What are the underlying mechanisms (causes) that result in this condition?

1. A sample of 25 seniors from a large metropolitan area school district had a mean Math SAT score of 450. Suppose we know that the standard deviation of the population of Math SAT scores for seniors in the district is 100. A 90% confidence interval for the population mean SAT score for the population of seniors is used. Which of the following would produce a confidence interval with a smaller margin of error?                                                                                                    (2)

1. Using a sample of 100 seniors
2. Using a confidence level of 95%
3. Using an alpha (α) of 0.01
4. Using a sample of only 10 seniors

2. In a large population of adults, the mean IQ is 112 with a standard deviation of 20. Suppose 200 adults are randomly selected for a market research campaign. What is the sampling distribution of their sample mean IQ? Must show work (explain) to justify choice.                                           (4)

1. exactly normal, mean 112, standard deviation 20
2. approximately normal, mean 112, standard deviation of 0.1
3. approximately normal, mean 112, standard deviation 1.414
4. approximately normal, mean 112, standard deviation 20

3. A final exam in Math 160 has a mean of 73 with a standard deviation of 7.8. If 34 students are randomly selected, find the probability that the mean of their test scores is more than 76.   (6)

4. Given that a piece of data is from a distribution with a mean of 0 and a standard deviation of 1, find the amount of area under the curve between z = – 1.83 and z = 0.                                 (3)

5. A study was conducted to estimate hospital costs for accident victims who wore seat belts. Twenty randomly selected cases to have a distribution that appears to be bell-shaped with a mean of $9004 and a standard deviation of $5629. Construct the 99% confidence interval for the mean of all such costs.                                                                                         

A total of 23 Gossett High School students were admitted to State University. Of those students, 7 were offered athletic scholarships. The school’s guidance counselor looked at each group’s summary statistics of their composite ACT scores, wondering if there was a difference between the groups (those who were not offered scholarships and those who were). The statistics for the 16 students who were not offered scholarships are x̅ = 24.7, s = 2.8 and for the 7 who were, x̅ = 26.5, s = 2.6. Assume that both distributions are approximately normal. Test the counselor’s claim using a 90% Level of Confidence.

We use statcrunch and the p method for homework.

Statistics concepts for engineering management:

The data in the table provides: College GPA, High School GPA, SAT total score, and number of letters of reference.
   a. Generate a model for college GPA as a function of the other three variables.
   b. Is this model useful? Justify your conclusion.
   c. Are any of the variables not useful predictors? Why?

CGPA   HSGPA   SAT   REF
2.04 2.01   1070   5
2.56 3.4   1254   6
3.75 3.68   1466   6
1.1 1.54   706   4
3 3.32   1160   5
0.05 0.33   756   3
1.38 0.36   1058   2
1.5 1.97   1008   7
1.38 2.03   1104   4
4.01 2.05   1200   7
1.5 2.1 896   7
1.29 1.34 848   3
1.9 1.51 958   5
3.11 3.12 1246   6
1.92 2.14 1106   4
0.81 2.6 790   5
1.01 1.9 954   4
3.66 3.06 1500   6
2 1.6    1046   5

I drive to school and am currently looking for a parking spot so I can walk to Jacob's. If I turn into a parking lot to look for a spot to park in that specific lot, the process of looking for a spot takes 1 minute of time whether I find a spot or not.  

Parking lot A is closest to Jacobs. If I get a spot here, it takes me 1 minute to walk into my class at the business school, however there is only a 10% chance I'll find a spot if I look.

Parking lot B is a 4-minute walk; if I pull in to look for a spot, there is a 30% chance I'll find a spot.

Parking lot C is an 8-minute walk; if I pull in to look for a spot, there is a 100% chance I will find a spot.  

Which strategy to find a parking spot is best for me (ie. which order should I check the parking lots for spots to park), assuming we are risk-neutral, and simply want to have the earliest expected arrival time to Jacob’s as possible? (Another way to say this is we want the smallest expected value of time spent getting to Hall).

Now, assume I am risk averse (let’s say that in this second case, my class starts in 10 minutes, and there is a large decrease in my utility if I am late for class). Is the best strategy the same as when I am risk-neutral, or has it changed?

Reba Dixon is a fifth-grade school teacher who earned a salary of $38,000 in 2020. She is 45 years old and has been divorced for four years. She receives $1,200 of alimony payments each month from her former husband (divorced on 12/31/2016). Reba also rents out a small apartment building. This year Reba received $50,000 of rental payments from tenants and she incurred $19,500 of expenses associated with the rental.

Reba and her daughter Heather (20 years old at the end of the year) moved to Georgia in January of this year. Reba provides more than one-half of Heather’s support. They had been living in Colorado for the past 15 years, but ever since her divorce, Reba has been wanting to move back to Georgia to be closer to her family. Luckily, last December, a teaching position opened up and Reba and Heather decided to make the move. Reba paid a moving company $2,010 to move their personal belongings, and she and Heather spent two days driving the 1,426 miles to Georgia.

Reba rented a home in Georgia. Heather decided to continue living at home with her mom, but she started attending school full-time in January and throughout the rest of the year at a nearby university. She was awarded a $3,000 partial tuition scholarship this year, and Reba helped out by paying the remaining $500 tuition cost. If possible, Reba thought it would be best to claim the education credit for these expenses.

Reba wasn't sure if she would have enough items to help her benefit from itemizing on her tax return. However, she kept track of several expenses this year that she thought might qualify if she was able to itemize. Reba paid $5,800 in state income taxes and $12,500 in charitable contributions during the year. She also paid the following medical-related expenses for herself and Heather:

Shortly after the move, Reba got distracted while driving and she ran into a street sign. The accident caused $900 in damage to the car and gave her whiplash. Because the repairs were less than her insurance deductible, she paid the entire cost of the repairs. Reba wasn’t able to work for two months after the accident. Fortunately, she received $2,000 from her disability insurance. Her employer, the Central Georgia School District, paid 60 percent of the premiums on the policy as a nontaxable fringe benefit and Reba paid the remaining 40 percent portion.

A few years ago, Reba acquired several investments with her portion of the divorce settlement. This year she reported the following income from her investments: $2,200 of interest income from corporate bonds and $1,500 interest income from City of Denver municipal bonds. Overall, Reba’s stock portfolio appreciated by $12,000, but she did not sell any of her stocks.

Heather reported $6,200 of interest income from corporate bonds she received as gifts from her father over the last several years. This was Heather’s only source of income for the year.

Reba had $10,000 of federal income taxes withheld by her employer. Heather made $1,000 of estimated tax payments during the year. Reba did not make any estimated payments. Reba had qualifying insurance for purposes of the Affordable Care Act (ACA).

a. Determine Reba’s federal income taxes due or taxes payable for the current year. Use Tax Rate Schedule for reference. (Do not round intermediate values. Round your final answers to the nearest whole dollar amount. Leave no answer blank. Enter zero if applicable.)

2020 Tax Rate Schedules

Individuals

Schedule X-Single

Schedule Y-1-Married Filing Jointly or Qualifying Widow(er)

Schedule Z-Head of Household

Schedule Y-2-Married Filing Separately

Bob is a recent law school graduate who intends to take the state bar exam. According to the National Conference on Bar Examiners, about 55% of all people who take the state bar exam pass. Let n = 1, 2, 3, ... represent the number of times a person takes the bar exam until the first pass.

(a) Write out a formula for the probability distribution of the random variable n. (Use p and n in your answer.)
P(n) =

(b) What is the probability that Bob first passes the bar exam on the second try (n = 2)? (Use 3 decimal places.)


(c) What is the probability that Bob needs three attempts to pass the bar exam? (Use 3 decimal places.)


(d) What is the probability that Bob needs more than three attempts to pass the bar exam? (Use 3 decimal places.)


(e) What is the expected number of attempts at the state bar exam Bob must make for his (first) pass? Hint: Use μfor the geometric distribution and round.

Twenty five high school students complete a preparation program for taking the SAT test. Here are the SAT scores from the 25 students who completed the SAT prep program: 434 694 457 534 720 400 484 478 610 641 425 636 454 514 563 370 499 640 501 625 612 471 598 509 531 The mean of these scores is 536.00. We know that the population average for SAT scores is 500 with a standard deviation of 100. The question is, are these students’ SAT scores significantly greater than a population mean of 500 with a population standard deviation of 100 ? Note that the the maker of the SAT prep program claims that it will increase (and not decrease) your SAT score. So, you would be justified in conducting a one-directional test. (alpha = .05).

Choose between

A - The prep program didn't result in significant improvement in SAT scores

B- The prep program resulted in significant improvement in SAT scores