. Dean runs The Creamy Bar which specialises in artisan ice cream sold at a local farmer’s market. Prevailing prices in the local market are $8 for a take-home tub of Classic Vanilla and $15 for a tub of Chocolate Almond Fudge.

The local dairy farmer delivers 48 litres of milk every Friday in preparation for market day. Classic Vanilla will need 0.5 litres per tub and Chocolate Almond Fudge requires 3 times as much. Both flavours require 500g of sugar to enhance the taste. There is a total of 20kg of sugar available per market day. For the signature velvety mouthfeel, Dean adds 0.5 litres of heavy cream to Classic Vanilla and double the amount for Chocolate Almond Fudge. He ordered 50 litres of heavy cream from the supplier.

Task 1

Construct a mathematical model for this problem. In doing so, consider the following:

1. What are the decision variables for this problem?
2. Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximised or minimised?
3. What constraints are relevant to this problem? Using the decision variables from part (a), formulate those constraints.

Task 2

Use Excel Solver to obtain a solution to the mathematical problem from Task 1. Your submission should include:

• your Excel spreadsheet
• the Sensitivity Report
• the Answer Report

           Task 3

Use your Excel output to answer the following questions:

1. Describe the linear programming solution to the Dean of The Creamy Bar in terms of:
   • The optimum number of take-home tubs of Classic Vanilla and Chocolate Almond Fudge to prepare each market day.
   • The maximum revenue per market day.
   • Whether all the milk purchased will be fully utilised.
   • Whether all the sugar allocated will be fully utilised.
   • Whether all the heavy cream ordered will be fully used.

    Which of the Solver reports helps you answer these questions?

2. What is the maximum profit per market day if Dean paid $1.2 per litre for milk and cream and $45 for sugar? Note that Dean also draws a $100 salary per market day.

    Which Solver report allows you to answer this question?

3. Due to the popularity of the Chocolate Almond Fudge flavour, Dean is hoping to increase the price to $20 per take-home tub. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the solution and The Creamy Bars’ revenue change, if at all?

4. In preparation for the scorching heat in summer, Dean would like to purchase an extra 10 litres of milk to increase ice cream production. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the solution and The Creamy Bars' revenue change, if at all?

Attach the new Answer Report ONLY, for the scenario in which Dean purchases 58 litres of milk, verifying your calculated maximum revenue per market day.

Task 4

Write a report outlining the solution and discussing your findings from Task 3 (at most two pages, double-spaced, at least 2cm margins, 12pt Times New Roman font or equivalent).

Here are a few points to consider while working through this assignment question:

1. The first step is always to work out the mathematical set up for the problem. This means identifying decision variables, formulating the objective function and then formulating constraints. At this stage, we are not trying to solve the problem or work out interactions among constraints. We simply list all conditions that must be satisfied.

When you complete Task 1, you should have two decision variables, the objective function written in terms of those decision variables, and five constraints, also written in terms of decision variables (some using both decision variables, others just one of them).

2. The second step is to find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The key here is to translate all mathematical expressions from Task 1 into Excel format. Instructions for doing so can be found under Topic 5 in the Excel booklet, as well as in the Linear Programming supplement. In addition, the Lecture notes page in this website gives you access to Excel spreadsheets used to generate Excel output shown in lecture slides for Week 5. It may be worthwhile examining them before attempting Task 2.
3. The final step is interpreting the solution that has been found, which is Task 3.
4. The report in Task 4 is a summary of the results from linear programming and sensitivity analysis in Tasks 2 and 3.

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let α be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. For a two-tailed hypothesis test with level of significance α and null hypothesis H0: μ = k, we reject H0 whenever k falls outside the c = 1 − α confidence interval for μ based on the sample data. When k falls within the c = 1 − α confidence interval, we do not reject H0. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as p, μ1 − μ2, or p1 − p2, which we will study later.) Whenever the value of k given in the null hypothesis falls outside the c = 1 − α confidence interval for the parameter, we reject H0. For example, consider a two-tailed hypothesis test with α = 0.01 and H0: μ = 20 H1: μ ≠ 20

A random sample of size 30 has a sample mean x = 23 from a population with standard deviation σ = 6.

(a) What is the value of c = 1 − α? 2.826 Incorrect: Your answer is incorrect.

Construct a 1 − α confidence interval for μ from the sample data. (Round your answers to two decimal places.)

lower limit

upper limit

What is the value of μ given in the null hypothesis (i.e., what is k)? k = Is this value in the confidence interval?

Yes No Correct: Your answer is correct. Do we reject or fail to reject H0 based on this information? We fail to reject the null hypothesis since μ = 20 is not contained in this interval. We fail to reject the null hypothesis since μ = 20 is contained in this interval. We reject the null hypothesis since μ = 20 is not contained in this interval. We reject the null hypothesis since μ = 20 is contained in this interval. Correct: Your answer is correct.

(b) Using methods of this chapter, find the P-value for the hypothesis test. (Round your answer to four decimal places.)

The lifetime of an electronic component has a Weibull distribution with parameters α​=0.60 and β​=2. Compute the probability the component fails before the expiration of a 4 year warranty.

The maximum flood​ levels, in millions of cubic feet per​ second, for a particular U.S. river have a Weibull distribution with α=5/3 and β=3/2. Find the probability that the maximum flood level for next year will exceed 0.7 million cubic feet per second.

The pages per book in a library are normally distributed with an unknown population mean and standard deviation. A random sample of 41 books is taken and results in a sample mean of 341 pages and sample standard deviation of 22 pages. Find the EBM, margin of error, for a 95% confidence interval estimate for the population mean using the Student's t-distribution.

For this first Pause-Problem, I want you to design three (brief!) studies (they can all be variations on the same idea).

Make sure to note the independent and dependent variables for all

1). One should use an independent two group design

2). One should use a matched OR natural pair design

3). One should use a repeated measures design

A candy company claims that in a large bag​ (over 1,000​ pieces) of Halloween candy half the candies are orange and half the candies are black. You pick candies at random from a bag and discover that of the first 50 you​ eat, 21 are orange.

​a) If it were true that half are orange and half are black​, what is the probability you would have found that at most 21 out of 50 were orange​?

​b) Do you think that half of the candies in the bag are really orange​? Explain.

1. A box contains six 25-watt light bulbs, nine 60-watt light bulbs, and five 100-watt light bulbs. What is the probability a randomly selected a 60 watt light bulb? (PLease explain how did you get your answer)(2 pt) Note: You must provide your answer as a fraction NOT decimal)

2. Cell Phone Provider	Probability
   AT&T	0.271
   Sprint	0.236
   T–Mobile	0.111
   Verizon	0.263

   Does the above table represent the probability model? Explain your answer. (2 pt)

3. The data shows the distance that employees of a certain company travel to work. One of these employees is randomly selected. Determine the probability that the employee travels between 10 and 29 miles to work. (2 pts)

   Distance (miles)	Number of employees
   0 – 9	124
   10 – 19	309
   20 – 29	257
   30 – 39	78
   40 – 49	2

4. In Q3, Is the probability the employee travels between 40 and 49 miles to work considered to be unusual? Explain your answer. (2 pt)

5. The probability that a randomly selected murder resulted from a rifle or shotgun is

   P(rifle or shotgun) = 0.059. Interpret this probability. (2 pt)

   Choose one of the correct answers from below.

   A. If 1000 murders were randomly selected, we would expect about 59 of them to have resulted from a rifle or shotgun.
   B. If any person is randomly selected, there is a 59 % chance the person will be murdered with a rifle or shotgun.
   C. If any person is randomly selected, there is a 5.9% chance the person will be murdered with a rifle or shotgun.
   D. If 1000 murders were randomly selected, exactly 59 of them would have resulted from a rifle or shotgun.

The Economic Policy Institute periodically issues reports on wages of entry level workers. The institute reported that entry level wages for male college graduates were $21.68 per hour and for female college graduates were $18.80 per hour in 2011.† Assume the standard deviation for male graduates is $2.30, and for female graduates it is $2.05. (Round your answers to four decimal places.)

1. What is the probability that a sample of 60 male graduates will provide a sample mean within $0.50 of the population mean, $21.68?

2. What is the probability that a sample of 60 female graduates will provide a sample mean within $0.50 of the population mean, $18.80?

3. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $0.50 of the population mean? Why?

4. What is the probability that a sample of 130 female graduates will provide a sample mean less than the population mean by more than $0.30?

The length of western rattlesnakes are normally distributed with a mean of 60 inches and a standard deviation of 4 inches. Enter answers as a decimal rounded to 4 decimal places with a 0 to the left of the decimal point.

A) Suppose a rattlesnake is found on a mountain trail: a. What is the probability that the rattlesnakes' length will be equal to or less than 54.2 inches?

B) What is the probability its' length will be equal to or greater than 54.2 inches?

C) What is the probability that the rattlesnakes' length will be between 54.2 inches and 65.8 inches?

D) Suppose a nest of 16 rattlesnakes are found on the mountain trail:

What is the probability that the average length of the rattlesnakes will be 60.85 inches or more?

ONE SAMPLE T-TEST

Assignment 1: Change in Mood Following a Romantic Movie Clip

A researcher asks participants to watch a short romantic movie clip. The movie clip depicts a romantic scene ending with two long-lost lovers embracing in a kiss. After the short romantic movie clip, participants are asked to indicate how the romantic movie clip has affected their mood on a bipolar scale ranging from -3 (much worse mood) to +3 (much better mood), with 0 indicating no change in mood. The results are given below. It was assumed that the average participant would give a rating of 0 if there were no change in mood. Test whether or not participants reported a significant change in mood at a .05 level of significance using a two­tailed test.

SCORES: -3, -3, -3, -3, -3, -3, -3, -3, -3,

-2, -2, -2, -2, -2, -2, -2, -2, -2,

2,

-1, -1, -1, -1, -1, -1, -1

0, 0, 0, 0, 0, 0, 0, 0, 0, 0

With regard to the SPSS exercise, answer the following questions:

Based on the SPSS output, state the following values:

Sample size, Sample mean, Sample standard deviation, Estimated standard error                                                 

Based on the table shown in SPSS, state the following values associated with the test statistic:

Mean difference, t obtained (t), Degrees of freedom (df), Significance (2-tailed)

2,006 persons are cross classified by disease and exposure.

Our outcome of interest is having the disease.

Use the following 2 x 2 table to answer questions below.

a. What percent of people have disease?

b. What percent of the exposed persons have disease?

c. What is the odds of disease among the exposed?

d. What is the odds of disease among the unexposed?

e. What is the odds ratio of disease among the exposed and unexposed. Interpret the odds ratio and comment whether the exposure is deleterious or protective.

Traditional definition of independence says that events A and B are
independent if and only if P(A n B)=P(A)×P(B). Show that P(A n B)=P(A)×P(B) if
and only if
a. P(A n B’) = P(A) × P(B’)b. P(A’ n B’) = P(A’) × P(B’)
You may use these without proof in your solutions:
o P(A n B’) = P(A) – P(A n B). [This can be proven by first showing that
(A ∩ B ′ ) ∪̇ (A ∩ B) = A and using the Addition Rule.]
o P(A n B) = 1 – P(A’ n B’). [This can be proven by noting that under de
Morgan’s Law, (A n B)’ = A’ n B’.]

A friend of mine is giving a dinner party. His current wine supply includes 12 bottles of zinfandel, 8 of merlot, and 10 of cabernet (he only drinks red wine), all from different wineries.

(a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?
ways

(b) If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?
ways

(c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?
ways

(d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (Round your answer to three decimal places.)


(e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety? (Round your answer to three decimal places.)

Assume the binomial distribution is appropriate. If N = 10 and P = 0.30, the probability of getting at least 8 P events is _________.

Two players (player A and player B) are playing a game against each other repeatedly until one is bankrupt. When a player wins a game they take $1 from the other player. All plays of the game are independent and identical. Suppose player A starts with $6 and player B starts with $6. If player A wins a game with probability 0.5, what is the probability the game ends (someone loses all their money) on exactly the 10th play of the game?