A new cold medication was tested by giving 125 people the drug and 100 people a placebo. A control group consisted of 115 people who were given no treatment. The number of people in each group who showed improvement is shown in the table below.

a) What is the probability that a randomly selected person in the study either was given the placebo or was in the control group?

b) What is the probability that a randomly selected person either was given the drug or improved?

c) What is the probability that a randomly selected person was given the drug and improved?

d) What is the probability that a randomly selected person who improved was given the drug?

e) What is the probability that a randomly selected person who was given the drug improved?

f) Based on these data, does the drug appear to be effective?

A home-builder company receives a shipment of 1,000 lightbulbs every Monday. The lightbulbs are then used immediately in newly finished homes. During the past 4 weeks, the builder counted that 20 of the 4,000 bulbs received were defective. Explain how you arrive to each of your answers to the questions below.

a). Given the information provided, what is a natural estimate for the probability that a given lightbulb is defective?

b). Assuming that different lightbulbs are defective or not independently of each other, give an exact formula for the probability that the shipment arriving next Monday will contain at most one defective lightbulb.

c). Give an approximate formula for the probability that the next shipment will contains exactly k defective lightbulbs.

d). Explain and justify how you would estimate the probability that during the next 10 calendar year the number of shipments containing exactly 5 defective lightbulbs will be greater than 95 and give an approximate value for this probability.

As of March 31, 2020, there are 113 confirmed cases of the current pandemic in

Kane County, Illinois, and 6 confirmed deaths.

2a. Using these statistics, estimate the probability of a randomly-selected confirmed case of the

virus in Kane County leading to death.

Your probability in 2a does not really apply to all people, but to get an idea of what that number

means, assume your answer to 2a applies to everyone who contracts the virus.

2b. If a family of four people contracts the virus, what is the probability that they all die?

2c. If a family of four people contracts the virus, what is the probability that at least one dies?

2d. Would it be unusual for an infected family of four to all survive the virus?

2e. A study by Pew research shows that 50-64 year olds have a median of 75 facebook friends. If all 75 of those people were infected, what is the probability that at least one dies?

Let X be a random variable representing the compressive strength of concrete cubes from a particular mix. X has a normal distribution with mean and standard deviation 60.14 and 5.02 N/mm2, respectively.

a. The probability that the compressive strength of a cube is between its mean value and c is
0.30. In other words P(μ £ X £ c) = 0.30, where μ is the mean value given above. What is
the value of c?
b. Assume that each cube behaves independently of other cubes and the probability that a
cube will pass a compressive strength test is 0.80. What is the probability that in a sample
of 3 cubes, exactly 1 of them will pass the test?
c. The construction company also performs another test on the cubes. They have found that
the probability that a cube will pass this test is p = 0.80. Assuming that each cube
performs independently of the others, what is the probability that in a sample of 500 the
number of cubes that pass the test is at least 406? Apply any approximation that is
appropriate

Math 1635 Statistics Probability (Chapter 4) Worksheet

1. If there are 20 marbles marked from 1 to 20 in the bag, what is the probability to pick a marble from the bag and the number can be

(a) divided by 2 or 5

(b) divided by 3 or 7

2. When a card is selected from the deck of 52 cards, find the probability of getting

(a) a spade or a face

(b) a queen or black

(c) a club or an 8

3. When 2 dice are rolled, find the probability of getting

(a) A sum of 7

(b) A sum greater than 8.

(c) A sum less than or equal to 5.

4. A bag contains 2 red, 3 green and 5 white balls. A ball is selected at random and its color is noted. Then it is replaced and another ball is selected and its color is noted. Find the probability of:

(a) selecting 2 green balls

(b) selecting red and then green balls

For this assignment, your group will utilize the preliminary data collected in the Topic 2 assignment. Considering the specific requirements of your scenario, complete the following steps using Excel. The accuracy of formulas and calculations will be assessed.

1. Select the appropriate discrete probability distribution. If using a binomial distribution, use the constant probability from the collected data and assume a fixed number of events of 20. If using a Poisson distribution, use the applicable mean from the collected data.
2. Identify the following: the probability of 0 events occurring, the probability of <5 events occurring, and the probability of ≥10 events occurring.
3. Using the mean and standard deviation for the continuous data, identify the applicable values of X for the following: Identify the value of X of 20% of the data, identify the value of X for the top 10% of the data, and 95% of the data lies between two values of X.
   
   How would I Calculate and do #3 and can it be explained step by step so I can better understand ?

67% of adults age 55 or older want to reach their 100th birthday. You randomly select 8 adults age 55 years or older and ask them if they want to reach their 100th birthday. The random variable represents the number of adults ages 55 or older who want to reach their 100th birthday.

a) What is the probability that exactly 5 of them say they want to reach their 100th birthday?

b) What is the probability that at most 5 of them say they want to reach their 100th birthday?

c) What is the probability that more than 4 adults say they want to reach their 100th birthday?

d) What is the probability that at least 5 of them say they want to reach their 100th birthday?

e) What is the probability that less than 4 adults say that they want to reach their 100th birthday?

f) What is the mean and standard deviation of the binomial variable?

Please show your work!

The issues surrounding the levels and structure of executive compensation have gained added prominence in the wake of the financial crisis that erupted in the fall of 2008. Based on the 2006 compensation data obtained from the Securities and Exchange Commission (SEC) website, it was determined that the mean and the standard error of compensation for the 418 highest paid CEOs in publicly traded U.S. companies are $8.63 million and $8.18 million, respectively. An analyst randomly chooses 38 CEO compensations for 2006. [You may find it useful to reference the z table.]

Calculate the expected value and the standard error of the sample mean. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)

d. What is the probability that the sample mean is more than $10 million? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.)

Q2) A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 3%. The probability distribution of the funds is as follows: Show all formulas and provide step by step calculations. Do NOT use Excel formulas!

1. Find the investment proportions in the minimum variance portfolio (MVP) of the two risk asset (5 points)
2. Find the expected return and standard deviation for the minimum variance portfolio (5 points)
3. What portion of your wealth should go to S and B respectively to achieve the tangent portfolio (i.e., the portfolio with the highest Sharpe ratio) (5 points)

Solve using coding in R script:

1) Given a standard normal distribution, find the value of k such that P(Z < k) = 0.0197.

2) Given a normal distribution with mu = E(X) = 32 and sigma^2 = V(X) = 30, find the normal curve area to the left of x = 31. Report your code as well as your final answer (4 decimals).

3) A company pays its employees an average wage of $17.90 an hour with a standard deviation of $1.50. If the wages are approximately normally distributed and paid to the nearest cent, the highest 2.5% of the employees hourly wages is greater than what amount?

4) Review question: Let X be a Binomial Random Variable with 8 trials and a probability of success as 0.37. Suppose you want to find k such that P(X < k) = 0.6625. What is the value of k?