The national survey has conducted a poll examining the financial health of public servants as they approach retirement age. According to responses from the survey of persons 55 years of age and over, 60% of them have stated that they are adequately prepared for retirement. Proposed changes to mandatory retirement laws may mean that persons who would normally be retiring at age 65, may no longer choose to do so, particularly if they feel they are not financially in the position to. Based on the findings of this survey, you want to extrapolate the number of people in your division who are adequately prepared for retirement. To do so, a random and independent sample of employees ages 55 plus where n=10 was conducted.

1. Define the random variable x as the number of persons who feel adequately prepared for retirement. We know that x is a binomial random variable. here define a binominal random variable and interpret what this means in terms of the scenario given in the question.

2. Find the probability that more than 5 people who responded are adequately prepared for retirement. This will help you plan for future hiring. Please use the binomial probability formula and show each step manually while calculating. Secondly, interpret the result you got related to the context of the question. Here I am not only assessing you for getting the right answer/number but also your ability to interpret the answer in a given situation.

Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrinkwrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedure. The distribution of paper currency (per 1000 bars of soap) is given in the following table. Distribution of Paper Currency Prizes Bill Denomination Number of Bills $1 470 $5 240 $10 160 $20 90 $50 39 $100 1 Total 1,000

(a) What is the expected amount of money in a single bar of Jackpot soap? If required, round your answer to two decimal places. Expected value =

(b) What is the standard deviation of the money in a single bar of Jackpot soap? If required, round your answer to two decimal places. Standard deviation =

(c) How many bars of soap would a customer have to buy so that, on average, he or she has purchased four bars containing a $50 or $20 bill? If required, round up your answer to the next whole number. Number of bars of soap =

(d) If a customer buys 7 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger? If required, round your answer to four decimal places. Probability =

In Pennsylvania, 6,165,478 people voted in the election. Trump received 48.18% of the vote and Clinton recieved 47.46%. This doesn't add up to 100% because other candidates received votes. All together these other candidates received 100% - 48.18% - 47.46% = 4.36% of the vote.

Suppose we could select one person at random from the 6+ million voters in PA (note: PA is the common abbreviation for Pennsylvania). We are interested in the chance that we'd choose a Trump, Clinton, or Other voter.

Below is a probability table for the choice:

Suppose we take a simple random sample of ?=1500 n = 1500 voters from the 6+ million voters in PA. What is the expected number of Trump voters? What is the expected number of Clinton voters? To answer these questions, let ?1 T 1 be 1 if the first voter chosen for the sample voted for Trump and 0 if they voted for Clinton or another candidate. Let ?2 T 2 be 1 if the second voter chosen for the sample voted for Trump and 0 if they voted for Clinton or another candidate, and so on. Let's start with some very basic questions. Find: ?(?1000=1) P ( T 1000 = 1 ) ?(?1000=0) P ( T 1000 = 0 ) ?(?17) E ( T 17 )

Problem 3-05 (Algorithmic) Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrinkwrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedure. The distribution of paper currency (per 1000 bars of soap) is given in the following table. Distribution of Paper Currency Prizes Bill Denomination Number of Bills $1 500 $5 250 $10 150 $20 50 $50 49 $100 1 Total 1,000 (a) What is the expected amount of money in a single bar of Jackpot soap? If required, round your answer to two decimal places. Expected value = (b) What is the standard deviation of the money in a single bar of Jackpot soap? If required, round your answer to two decimal places. Standard deviation = (c) How many bars of soap would a customer have to buy so that, on average, he or she has purchased four bars containing a $10 or $20 bill? If required, round up your answer to the next whole number. Number of bars of soap = (d) If a customer buys 9 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger? If required, round your answer to four decimal places. Probability =

Ada Nixon, a student, has just begun a 30-question, multiple-choice exam. For each question, there is exactly one correct answer out of four possible choices. Unfortunately, Ada hasn't prepared well for this exam and has decided to randomly select an answer choice for each question.

(a) (4 points) For a given question, what is the probability Ada picks the correct answer, assuming each answer choice is equally likely to be selected?

(b) Assume the number of questions Ada answers correctly is Binomially distributed.

i. (4 points) What is the average number of questions Ada will answer correctly? Show your work and round your nal answer to 1 decimal place.

ii. (3 points) If Ada answers at least 16 questions correctly, she will receive a passing grade. What is the probability that Ada receives a passing grade? Show your work, including any calculator functions you use, and round your nal answer to 3 decimal places.

(c) (3 points) Suppose Ada comes across a question for which she knows two of the answer choices are certainly wrong, which means the correct answer must be one of the two remaining choices. Assuming Ada will answer every other question using the same random selection procedure as before, will the number of questions she answers correctly remain binomially distributed? State Yes or No and explain why

How would you go about balancing the demands of financial stability, growth, and profitable margins with the demand for the highest quality patient care?

How would you go about balancing the demands of financial stability, growth, and profitable margins with the demand for the highest quality patient care?

Assuming equal concentrations and complete dissociation, rank the following aqueous solutions by their freezing points.

Highest freezing point:​

Lowest Freezing point:​

Should one always select the decision path that has the highest expectation value? Why/Why not? Give an example where one might not.