A comparison is made between two bus lines to determine if arrival times of their regular buses from Denver to Durango are off schedule by the same amount of time. For 46randomly selected runs, bus line A was observed to be off schedule an average time of 53 minutes, with standard deviation 15 minutes. For 61 randomly selected runs, bus line B was observed to be off schedule an average of 62 minutes, with standard deviation 13 minutes. Do the data indicate a significant difference in average off-schedule times? Use a 5% level of significance.

a. What are we testing in this problem?

single meansingle proportion     

difference of proportions

difference of means

paired difference

b. What is the level of significance?


c. State the null and alternate hypotheses.

H₀: μ₁ ≤ μ₂; H₁: μ₁ > μ₂

H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂     

H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

H₀: μ₁ ≥ μ₂; H₁: μ₁ < μ₂

d. What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with known population standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown population standard deviations.

The Student's t. We assume that both population distributions are approximately normal with unknown population standard deviations.

The Student's t. We assume that both population distributions are approximately normal with known population standard deviations.

e. What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)


f. Estimate the P-value.

P-value > 0.500

0.250 < P-value < 0.500     

0.100 < P-value < 0.250

0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

g. Sketch the sampling distribution and show the area corresponding to the P-value.

h. Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.     

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

i. Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 to conclude that there is a difference in average off schedule times.

There is insufficient evidence at the 0.05 to conclude that there is a difference in average off schedule times.    

A comparison is made between two bus lines to determine if arrival times of their regular buses from Denver to Durango are off schedule by the same amount of time. For 51 randomly selected runs, bus line A was observed to be off schedule an average time of 53 minutes, with standard deviation 17 minutes. For 61 randomly selected runs, bus line B was observed to be off schedule an average of 60 minutes, with standard deviation 15 minutes. Do the data indicate a significant difference in average off-schedule times? Use a 5% level of significance.

What are we testing in this problem?

difference of proportionssingle proportion    single meandifference of meanspaired difference

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ₁ > μ₂; H₁: μ₁ = μ₂H₀: μ₁ = μ₂; H₁: μ₁ > μ₂    H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.The standard normal. We assume that both population distributions are approximately normal with known standard deviations.    The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.5000.250 < P-value < 0.500    0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 to conclude that there is a difference in average off schedule times.There is insufficient evidence at the 0.05 to conclude that there is a difference in average off schedule times.

Linear programming.

Solve the following two (2) Linear programming problems (#1 and #2) and then answer question 3:

1.. Solve the following LP problem graphically:

Maximize profit =            X + 10Y

Subject to:                        4X + 3Y < /= 36
                                           2X +4Y < / = 40
                                           Y > / = 3
                                           X, Y > / = 0

2. Considering the following LP problem and answer the questions, Part a and Part b:

Maximize profit =            30X₁ + 10X₂

Subject to:                        3X₁ + X₂ < /= 300
                                           X₁ +X₂ < / = 200
                                           X₁ < / = 100
                                           X₂ > / = 50
                                           X₁ – X₂ < / = 0
                                           X₁, X₂ > / = 0

a. Solve graphically
b. Is there more than one optimal solution? Explain

3. How many feasible solutions are there in a LP program/problem? Which ones do we need to examine to find the optimal solution?

Consider the trade relations between the United States and China. Assume the leaders of the countries believe the payoffs to alternative trade policies are as follows. If both countries impose low tariffs, then both countries will gain $60 billion. If both countries impose high tariffs, then both countries will gain $40 billion. If one country imposes high tariffs, the country that has high imposed tariffs will gain $50 billion and the country that imposes low tariffs will receive $20 billion.

Draw the payout matrix describing the payouts given the trade choices by the two countries.

Does the United States have a dominant strategy?

Does China have a dominant strategy?

Using the dominant strategy concept explain your answers. Based on this information, is there a Nash equilibrium for trade policy between the United States and China?

A bag contains four red marbles, two green ones, one lavender one, three yellows, and three orange marbles. HINT [See Example 7.] How many sets of five marbles include at most one of the yellow ones?

Most married couples have two or three personality preferences in common. A random sample of 371 married couples found that 130 had three preferences in common. Another random sample of 573 couples showed that 219 had two personality preferences in common. Let p1 be the population proportion of all married couples who have three personality preferences in common. Let p2 be the population proportion of all married couples who have two personality preferences in common.

(a) Find a 90% confidence interval for p1 – p2. (Round your answers to three decimal places.)

Lower limit:

Upper limit:

Sally Rogers has decided to invest her wealth equally across the following three assets.  

a. What are her expected returns and the risk from her investment in the three​ assets? How do they compare with investing in asset M​ alone?  

b. Could Sally reduce her total risk even more by using assets M and N​ only, assets M and O​ only, or assets N and O​ only? Use a​ 50/50 split between the asset​ pairs, and find the standard deviation of each asset pair.

a. What is the expected return of investing equally in all three assets​ M, N, and​ O?

​(Round to two decimal​ places.)

What is the expected return of investing in asset M​ alone?

​(Round to two decimal​ places.)

What is the standard deviation of the portfolio that invests equally in all three assets​ M, N, and​ O?

​(Round to two decimal​ places.)

What is the standard deviation of asset​ M?

​(Round to two decimal​ places.)

By investing in the portfolio that invests equally in all three assets​ M, N, and O rather than asset M​ alone, Sally can benefit by increasing her return by

and decrease her risk by

​(Round to two decimal​ places.)

b.  What is the expected return of a portfolio of​ 50% asset M and​ 50% asset​ N?

​(Round to two decimal​ places.)

What is the expected return of a portfolio of​ 50% asset M and​ 50% asset​ O?

​(Round to two decimal​ places.)

What is the expected return of a portfolio of​ 50% asset N and​ 50% asset​ O?

​(Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset M and​ 50% asset​ N?

​(Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset M and​ 50% asset​ O?

​(Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset N and​ 50% asset​ O?

​(Round to two decimal​ places.)

Could Sally reduce her total risk even more by using assets M and N​ only, assets M and O​ only, or assets N and O​ only?  ​(Select the best​ response.)

A .​Yes, a portfolio of​ 50% of asset M and​ 50% of asset N could reduce the risk to 1.00%.

B. ​Yes, a portfolio of​ 50% of asset M and​ 50% of asset O could reduce the risk to 1.00%.

C. There is not enough information to answer this question.

D. ​No, none of the portfolios using a​ 50-50 split reduce risk.

Benefits of

diversification.

Sally Rogers has decided to invest her wealth equally across the following three assets.

a. What are her expected returns and the risk from her investment in the three assets? How do they compare with investing in asset M alone?

Hint: Find the standard deviations of asset M and of the portfolio equally invested in assets M, N, and O.

b. Could Sally reduce her total risk even more by using assets M and N only, assets M and O only, or assets N and O only? Use a 50/50 split between the asset pairs, and find the standard deviation of each asset pair.

a. What is the expected return of investing equally in all three assets M, N, and O?

________

(Round to two decimal places.)

What is the expected return of investing in asset M alone?

________

(Round to two decimal places.)

What is the standard deviation of the portfolio that invests equally in all three assets M, N, and O?

__________

(Round to two decimal places.)

What is the standard deviation of asset M?

_________

(Round to two decimal places.)

By investing in the portfolio that invests equally in all three assets M, N, and O rather than asset M alone, Sally can benefit by increasing her return by

________ %

and decrease her risk by

_________ %.

(Round to two decimal places.)

b. What is the expected return of a portfolio of 50% asset M and 50% asset N?

___________ %

(Round to two decimal places.)

What is the expected return of a portfolio of 50% asset M and 50% asset O?

______ %

(Round to two decimal places.)

What is the expected return of a portfolio of 50% asset N and 50% asset O?

______ %

(Round to two decimal places.)

What is the standard deviation of a portfolio of 50% asset M and 50% asset N?

_______ %

(Round to two decimal places.)

What is the standard deviation of a portfolio of 50% asset M and 50% asset O?

___________ %

(Round to two decimal places.)

What is the standard deviation of a portfolio of 50% asset N and 50% asset O?

_____%

(Round to two decimal places.)

Could Sally reduce her total risk even more by using assets M and N only, assets M and O only, or assets N and O only? (Select the best response.)

A.No, none of the portfolios using a 50-50 split reduce risk.

B. Yes, a portfolio of 50% of asset M and 50% of asset O could reduce

C. There is not enough information to answer this question.

D. Yes, a portfolio of 50% of asset M and 50% of asset N could reduce the risk to 0.990.99 %.

Salem is a 15-year-old male weighing 46.6 kg. He is known to have asthma and. He accidently fell down and broke his leg while playing football. For that reason he was admitted to the hospital for surgery. Since he was 6 year-old, he presented to the emergency department many times and had 4 hospital admissions for asthma, two of them were to the intensive care unit. He often required a course of oral steroids for one month every few months. However, in the last two years his asthma was well controlled on fluticasone (inhaled steroid) and salbutamol (inhaled β-adrenergic agonist) and he didn’t need any oral steroids. Also, he had not visited the emergency department or been admitted to the hospital for the last two years.

On the day of the accident, the patient had no signs or symptoms of asthma. When he and his father were asked if he had tried aspirin or NSAIDs in the past they said they were not sure. When pain control was discussed, the father wanted to avoid morphine for its addictive effects. Consequently, the doctor prescribed him ibuprofen and planned to give him the first dose in the hospital under close observation soon after the surgery.

The surgery and anaesthesia went fine. One hour later, the patient was given a tablet of ibuprofen 400 mg orally for pain control. 10 minutes after that he began to show symptoms of asthma (shortness of breath and wheezing). For that he used his salbutamol inhaler 8 times. However, his symptoms became worse over the next 20 minutes and he was not able to talk. Soon after that, the patient became cyanosed and needed oxygen by face mask. Salbutamol inhaler was repeated and Hydrocortisone 100mg IV was given. Within 20–30 minutes his condition started to improve. Salbutamol was given every 4 hours and oral Prednisolone 50 mg once daily was initiated (for 6 days). To control the pain of his surgery he was given Morphine 5mg/4h orally and was observed closely overnight for the symptoms of asthma. He was discharged after one week. That week Salem didn’t experience any further asthma symptoms and returned home on his usual inhalers.

1.       What is the most likely explanation (at the biochemical level) for Salem’s symptoms that developed after he was given the oral ibuprofen?

2.       How do you explain the successful relief of the patient’s ibuprofen-induced symptoms after he was treated with hydrocortisone and prednisolone?

3.       In this case study, β-adrenergic agonist and steroids were used to treat and/or prevent asthma symptoms. Mention the other two medication types useful in the treatment of asthma that are mentioned in eicosanoids metabolism chapter of your course. (0.5 mark for each medication type (1 mark total))