Use the graphical method for linear programming to find the optimal solution for the following problem.

Maximize P = 4x + 5 y

subject to 2x + 4y ≤ 12

                5x + 2y ≤ 10

and      x ≥ 0, y ≥ 0.

graph the feasible region

Jerome is creating a secret passcode for his vault. The vault uses some of the Greek alphabet –possible choices are (?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?}, and can also use odd digits (1, 3, 5, 7, 9). The code will be of length 10 and selected randomly from the possible Greek letters or Roman Numerals mentioned above. a) How many possible secret passcodes can be formed of length 10? b) What is the probability that a randomly selected passcode contains only Greek letters and no numbers? c) What is the probability that the first and last position of a randomly selected passcode contains odd numbers? d) What is the probability that a randomly selected code has no repeats? e) Given that a randomly selected passcode contains only Greek letters, what is the probability that it starts and ends with the letter ??

A restaurant chain that has 3 locations in Portland is trying to determine which of their 3 locations they should keep open on New Year’s Eve. They survey a random sample of customers at each location and ask each whether or not they plan on going out to eat on New Year’s Eve. The results are below. Run a test for independence to decide if the proportion of customers that will go out to eat on New Year’s Eve is dependent on location. Use α=0.05.

Hypotheses:

H,0): The choice to go out on New Year's Eve is _____ restaurant location.

(H,1): The choice to go out on New Year's Eve is _____ restaurant location.

Enter the test statistic - round to 4 decimal places. ______

Enter the P-Value - round to 4 decimal places. ______

Can it be concluded that the choice to go out on New Year's Eve is dependent on restaurant location?

In a test of the effectiveness of garlic for lowering​ cholesterol, 43 subjects were treated with garlic in a processed tablet form. Cholesterol levels were measured before and after the treatment. The changes ​(before−​after) in their levels of LDL cholesterol​ (in mg/dL) have a mean of 5.7and a standard deviation of 17.7Construct a 90% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment. What does the confidence interval suggest about the effectiveness of garlic in reducing LDL​ cholesterol?

An orange juice producer buys only one kind of oranges. The amount of juice squeezed from each of these oranges is approximately normally distributed with a mean of 4.2 ounces and a population standard deviation of 1 ounce. If a sample of 100 oranges is selected:

(a) What is the probability that the average juice squeezed is less than 4.15 ounces?

(b) What is the probability that the average juice squeezed is more than 4.3 ounces?

(c) What is the probability that the average juice squeezed is between 4.15 ounces and 4.3 ounces?

(d) Do we need the Central Limit Theorem to solve (a) and (b)? Why or why not? Explain.

A recent National Science Foundation (NSF) survey indicates that more than 20% of the staff in American research and development laboratories is foreign. Results of the study have been used for pushing legislation aimed at controlling the number of foreign workers in the United States. An organization of foreign-born scientists wants to prove that the NSF survey results do not reflect the true percentage of foreign workers in the U.S. labs. The organization collects a sample of 5,000 laboratory workers in all major labs in the country and finds that 876 are foreign.

a). At 5% level of significance, is there sufficient evidence to conclude that the NSF study overestimated the percentage of foreigners in American laboratories? Your conclusion must be in terms of the P-Value as well as setting up a Rejection Region. Show work.

b). Which statistical distribution should be applied in this situation and why? Explain carefully.

c). What type of error is possible and describe this error in terms of the problem.

d). Based on a 95% confidence level, what is the best case and worst case scenario regarding the percentage of foreigners in American laboratories?

e). Carefully interpret this interval estimation.

f). Using the results of part (d), explain carefully whether or not there is sufficient evidence to conclude that the NSF study overestimated the percentage of foreigners in American laboratories? Explain carefully.

Suppose that a bag of scrabble tiles contains 5 Es, 4 As, 3 Ns and 2 Bs. It is my turn and I draw 4 tiles from the bag without replacement. Assume that my draw is uniformly random. Let C be the event that I got two Es, one A and one N. (a) Compute P(C) by imagining that the tiles are drawn one by one as an ordered sample. (b) Compute P(C) by imagining that the tiles are drawn all at once as an unordered sample.

1. The VA has collected the past month’s data from sixteen hospitals. The VA believed the selected hospitals were efficiently run. The objective is to derive a model based on the efficient hospitals to use to comparatively evaluate questionable hospitals. The data are: y = monthly labor spent x1 = monthly X-ray exposures x2 = monthly occupied bed days x3 = average number of days of a patient’s stay SUMMARY OUTPUT Regression Statistics Multiple R 0.9981 R Square 0.9961 Adjusted R2 0.9952 Standard Error 387.1598 Observations 16 ANOVA df SS MS

   		14896.25
   		15896.25
   		16009.25
   		16896.25
   		17986.25

1. The previous multiple regression model is significant to what level?

   		It is not significant
   		0.05
   		0.01
   		0.001
   		0.0000

1. In the VA prolem which is the most significant independent variable?

   		X-ray
   		BedDays
   		Length of stay

How would you interpret a correlation coefficient of -0.95 between the two parameters; age of the patient and the testosterone level? How would your interpretation change if the coefficient was 0.03 or 0.48?

Please provide solutions to the following problems. Please use Excel to solve the problems and submit the Excel spreadsheet.

2. You started a new restaurant. Based on invoices for the first 30 days, you estimated your average grocery bill to be $20,000 with a standard deviation of $2000. You want to start another restaurant in a similar neighborhood and you are planning to prepare a brochure for investors and to work out a deal with a whole sale food distributor. Prepare a 95% confidence interval for the average grocery bill. Would a 99% confidence interval be narrower or wider than the 95% confidence interval?

Stratified Sampling and Cluster Sampling are two approaches used within Probability Sampling Techniques.

Explain using examples, instances where Stratified Sampling would be preferred over Cluster Sampling, and vice versa.

Remember to cite your source using current APA format, and post the url for your paper. Your original post should be no more than 250 words.

A city council suspects a judge of being a "hanging judge" because s/he is perceived as imposing harsher penalties for the same sentence. To investigate this, a random sample of 47 cases is taken from the judge's prior cases that resulted in a guilty verdict for a certain crime. The average jail sentence s/he imposed for the sample is 26 months. The average jail sentence for the same type of crimes is 25 months with a standard deviation of 13 months. What can be concluded with an α of 0.05?

a) What is the appropriate test statistic?
---Select--- na z-test one-sample t-test independent-samples t-test related-samples t-test

b)
Population:
---Select--- cases same type of crimes city council judge's prior cases months
Sample:
---Select--- cases same type of crimes city council judge's prior cases months

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[  ,  ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and select "na" below.
d =  ;   ---Select--- na trivial effect small effect medium effect large effect
r² =  ;   ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

For the same sentence, the judge imposed significantly harsher penalties than other judges.

For the same sentence, the judge imposed significantly milder penalties than other judges.   

For the same sentence, the judge's penalties did not significantly differ from other judges.

Please solve the following questions. Please show all work and all steps.

1a.) Find CDF of the Bernoulli Distribution.

1b.) Explain your reasoning for part 1a above. Do not use a series result.

2a.) For the geometric distribution, show that the geometric pdf is actually a pdf (all probabilities are greater than or equal to 0 and the sum of the pdf from x=1 to x=infinity is equal to 1).

2b.) Find E(X) for 2a above.

2c.) Find V(X) for 2a above.

Suppose a geyser has a mean time between eruptions of 80 minutes. If the interval of time between the eruptions is normally distributed with standard deviation 26 minutes​, answer the following questions. ​(a) What is the probability that a randomly selected time interval between eruptions is longer than 92 ​minutes? The probability that a randomly selected time interval is longer than 92 minutes is approximately 0.3228. ​(Round to four decimal places as​ needed.) ​(b) What is the probability that a random sample of 8 time intervals between eruptions has a mean longer than 92 ​minutes? The probability that the mean of a random sample of 8 time intervals is more than 92 minutes is approximately nothing. ​(Round to four decimal places as​ needed.)

ONLY ANSWER PART B

What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from eastern states, the following information is obtained (units in percentage of hospitals providing at least some charity care):

57.2 56.1 53.1 65.8 59.0 64.7 70.1 64.7 53.5 78.2

Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean percentage x and the sample standard deviation s. (Round your answers to one decimal place.)

x = %

s = %

(b) Find a 90% confidence interval for the population average μ of the percentage of hospitals providing at least some charity care. (Round your answers to one decimal place.)

lower limit %

upper limit %