Problem(3)
Consider a r.v. representing coin throws (Bernoulli Variable with Σ = {0,1} ). Let the true
probability distribution be p(0) = r, p(1) = 1-r.
Someone guesses a different distribution q(0) = s, q(1) = 1-s.
(a) Find expressions for the Kullback–Leibler distances D(p||q) and D(q||p) between the
two distributions in terms of r and s.
(b) Show that in general, D(p||q) ≠ D(q||p) and that equality occurs iff r = s.
(c) Compute D(p||q) and D(q||p) for the case r = 1/2 and s = 1/4.

python or C++

From “The Practice of Statistics in the Life Sciences” by Baldi and Moore. Obesity in adult males is associated with lower levels of sex hormone. A study investigated a possible link between obesity and plasma testosterone concentrations in adolescent males between the ages of 14 and 20 years. Here are the data for 25 obese adolescent males, measured in nanomoles per liter of blood (nmol/l): 0.30 0.24 0.19 0.17 0.18 0.23 0.24 0.06 0.15 0.17 0.18 0.17 0.15 0.12 0.25 0.25 0.25 0.32 0.35 0.37 0.39 0.46 0.49 0.42 0.36

(a) Test whether the average plasma testosterone concentrations in adolescent males between 14 and 20 years of age is 0.25 nmol/l or not. [4 marks]

(b) Suppose we have an interest in knowing whether the average plasma testosterone concentrations in adolescent males between 14 and 20 years of age is greater than 0.25 nmol/l. Test this situation. [4 mark]

(c) Suppose we have an interest in knowing whether the average plasma testosterone concentrations in adolescent males between 14 and 20 years of age is less than 0.25 nmol/l. Test this situation. [4 marks]

From Barbie dolls to runway models, women in Western countries are exposed to unrealistically thin and arguably unhealthy body standards for their gender. A body mass index (BMI) between 18.5 and 24.9 is considered healthy. Using a 3D computer avatar, participants built what they considered the ideal body of an adult of their own gender. Below appears a summary of the results for a sample of 40 female heterosexual Caucasian undergraduate students from British universities that had been recruited by the researchers. Mean BMI was recorded to be 18.85 with standard deviation 1.75. Does the data provide evidence that young Caucasian women in British Universities, on average, aim for an unhealthy ideal body type (corresponding to a BMI less than 18.5)? Use α = 0.10.

(a) Construct a 99% confidence interval for the mean BMI for young Caucasian women in British universities using R. Use at least 2 decimals in your answer. [2 marks]

(b) Would a 99% CI (for the mean BMI) constructed using a smaller sample than the one used for part a) tend to be wider or narrower? Explain. [2 marks]

Consider the following hypotheses tests involving the χ²-distribution.

(a) Determine the p-value for H_o: P(1) = P(2) = P(3) = P(4) = 0.25, with χ² = 10.95. (Give your answer bounds exactly.)
______< p <_____

(b) Determine the p-value for H_o: P(I) = 0.25, P(II) = 0.40, P(III) = 0.35, with χ² = 8.57. (Give your answer bounds exactly.)
_____ < p <______

Consider the null hypotheses for the following multinomial experiments. (Give your answers correct to two decimal places.)

(a) Determine the critical value and critical region that would be used in the classical approach to test H_o: P(1) = P(2) = P(3) = P(4) = 0.25, with α = 0.01.
χ²  ______

(b) Determine the critical value and critical region that would be used in the classical approach to test H_o: P(1) = 0.25, P(2) = 0.40, P(3) = 0.35, with α = 0.025.
χ²  _______

1. The following data is the annual income (in $1,000s of U.S. dollars) taken from nine randomly chosen students in an MPH program: 37 102 34 12 111 56 72 17 33

1. Calculate the sample mean income
2. Calculate the sample median income
3. Calculate the sample standard deviation of these incomes
4. What population could this sample represent?
5. Which would change by a larger amount—the mean or median—if the 34 were replaced by 17, and the 12 replaced by a 31?

X ~ N(50, 13). Suppose that you form random samples of 25 from this distribution. Let X be the random variable of averages. Let ΣX be the random variable of sums.

• Part (a)

  Sketch the distributions of X and X on the same graph.

  • A	B)
    C)D)

• Part (b)

  Give the distribution of X. (Enter an exact number as an integer, fraction, or decimal.)

  X ~ ____(____,____)

• Part (c)

  Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.)

  P(X < 50) = ________

• Part (d)

  Find the 30th percentile. (Round your answer to two decimal places.)

• _________.

• Part (e)

  Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.)

  P(48 < X < 54) = _________

• Part (f)

  Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.)

  P(17 < X < 48) = ___________

• Part (g)

  Give the distribution of ΣX.
  ΣX ~ ______(_____,_____)

• Part (h)

  Find the minimum value for the upper quartile for ΣX. (Round your answer to two decimal places.)

• ________.

• Part (i)

  Sketch the graph, shade the region, label and scale the horizontal axis for ΣX, and find the probability. (Round your answer to four decimal places.)

  P(1200 < ΣX < 1350) = ___________

The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025.

 for the population of each country by letting 

t = 15

 correspond to 2015. Use the model to predict the population of each country in 2035. (Round your values of b to five decimal places. Round your values of a to one decimal place. Round your population predictions for 2035 to one decimal place.) 

CountryExponential ModelPopulation in 2035 
(in millions)Ay = 

 By = 

 Cy = 

 Dy = 

 Ey = 

(b) You can see that the populations of Country D and Country E are growing at different rates. What constant in the equation 

y = ae^bt

 gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.

The constant b determines the growth rate. The greater the rate of growth, the smaller the value of b.The constant a determines the growth rate. The greater the rate of growth, the greater the value of a.    The constant b determines the growth rate. The greater the rate of growth, the greater the value of b.The constant a determines the growth rate. The greater the rate of growth, the smaller the value of a.

Given: A nutrition store in the mall is selling “Memory Booster” which is a concoction of herbs and minerals that is intended to improve memory performance. To test the hypothesized memory enhancement of the herbal mix, the now famous drug researcher, Dr. Mindfog, obtains a sample of 15 adult volunteers and has each person take the suggested dosage each day for 4 weeks. At the end of the four-week period, each individual takes a standardized memory test. In the general adult population, the standardized memory test is known to have a mean of italic mu = 10, where higher scores demonstrate better performance on the test of memory. The scores for the 15 volunteers are: 12, 11, 13, 15, 8, 11, 13, 15, 10, 9, 10, 16, 15, 11, 14 Which hypothesis test needs to be employed here?

Group of answer choices

z test

one-sample t test

dependent t test

independent t test

Pearson's r

Maryland Home and Community-Based Services (MHCBS) is considering a major expansion that will enable it to attract a different clientele to its organization. Currently, they serve only 34% of the frail elderly seniors and persons with disabilities in the local area. The new chief CEO would like the organization to expand its revenue stream by investing in a senior multipurpose center serving healthy seniors by offering them arts and crafts and health and wellness programs. The center will also contain an Internet café offering nutritious breakfast and lunch options.

The CEO has commissioned a needs assessment, and the study’s results reveal that there are approximately 120 seniors in the local community who are interested in this center and the CEO expects growth of the aging population to be at least 10% each year. Cost growth across all areas of expense is expected to rise by 5% each year. The CEO has presented her proposal and financial information to the Board of Directors, and they have advised her that they are in full support of her strategy only if the program is a benefit to the community and if the organization can recoup its investment in five years. The CEO has asked you if this can be achieved. Based on the information presented in the scenario, calculate the two analyses and explain, in a brief memorandum to the CEO, their implications.

Baseline Information

Monthly Revenue: $125 per senior

Fixed Costs Monthly

Utilities: $590

Health/Wellness Staff: $2,500

Arts/Crafts Staff: $2,000

Supplies: $800

Fitness Equipment Maintenance Contract: $200

Variable Costs

Monthly Breakfast Cost: $25

Monthly Lunch Cost: $15

QUESTIONS

Based on the information above, once the minimum threshold of participants is reached, the initial investment to establish the center is $317,880. The organization anticipates that it will generate $46,920 of net revenues in the first year, $68,166 in the second year, $93,404 in the third year, $123,287in the fourth year, and $158,573 in the fifth year.

1. Perform the break-even analysis to determine how many seniors would need to have full monthly membership and pay for breakfast and lunch for UMUC Home and Community-Based Services to cover its monthly expenses.
2. Calculate the payback period to determine how long it will take for the organization to recover its initial investment of establishing the senior multipurpose center.
3. Based on the information presented in the scenario, calculate the two analyses and explain, in a brief memorandum to the CEO, their implications.
   1. Provide an excel spreadsheet for the specified budget periods.
   2. Provide calculations demonstrating your computations of how you arrived to the answers of the questions
   3. Provide a narrative explaining your calculations, so that if you do not have the correct answer I can at least review how you arrived at your conclusions and potentially render partial credit.

Suppose a poll of 20 voters is taken in a large city. The purpose is to determine x, the number who favor a certain candidate for mayor. Suppose that 40% of all the city’s voters favor the candidate.

a. Find the mean and standard deviation of x.

b. What is the probability that x <=10.

c. Find the probability that x > 12.

d. Find the probability that x = 11

e. Graph the probability distribution of x.

A woman has 10 keys, of which exactly one will open her car door.

1. If she tries the keys at random, discarding those that do not work, what is the probability that she will open the car door on her fourth try?

2. If she tries the keys at random and does not discard previously tried keys, what is the probability that she will open her car door on her fourth try?

DOVERCOURT a non-profit organization has paid a researcher to use SPSS and analyze some data they have collected from the month of October 2016. These data include the following variables as presented in the table below.

Number of adults members who participate in the Dovercourt activities

• Age is in average years for a specific hour
• The number of participants represents a monthly average.

Question:

Enter the data in the SPSS spreadsheet and then briefly answer the following question, taking about half a page for question (including output).

• Use a box plot to describe the number of total participants. Explain what it shows.

Cholesterol is a type of fat found in the blood. It is measured as a concentration: the number of milligrams of cholesterol found per deciliter of blood (mg/dL). A high level of total cholesterol in the bloodstream increases risk for heart disease. For this problem, assume cholesterol in men and women follows a normal distribution, and that “adult man” and “adult woman” refers to a man/woman in the U.S. over age 20. For adult men, total cholesterol has a mean of 188 mg/dL and a standard deviation of 43 mg/dL. For adult women, total cholesterol has a mean of 193 mg/dL and a standard deviation of 42 mg/dL. The CDC defines “high cholesterol” as having total cholesterol of 240 mg/dL or higher, “borderline high” as having a total cholesterol of more than 200 but less than 240, and “healthy” as having total cholesterol of 200 or less. A study published in 2017 indicated that about 11.3% of adult men and 13.2% of adult women have high cholesterol.

1) A researcher measures the total cholesterol of a randomly selected group of 36 adult women, and counts the number of them who have high cholesterol. (Assume that 13.2% of adult women have high cholesterol.)

a. What is the probability that exactly 4 of these 36 women have high cholesterol?

b. What is the probability that 8 or less of these 36 women have high cholesterol?

2) A doctor recommends drastic lifestyle changes for all adults who are in the top 5% of total cholesterol levels.

a. What total cholesterol level is the cutoff for the top 5% of women? (Round to 1 decimal place.)

b. What total cholesterol level is the cutoff for the top 5% of men? (Round to 1 decimal place.)

Write it in P program. This is only one question with multiple parts, so please answer all the parts of the question, it is not much. Please also include a screenshot of your R program.

1. Generator 500 random numbers from the Binomial distribution with size=100, p=0.5 (define it X).
2. Standardize the vector X using the following equation (define it Z):

Z=(X-E(X))/sd(X)

Where E(X)= mean of B(n,p), which is np, sd(X) = standard deviation of B(n,p), which is sqrt(np(1-p)).

3. Make a histogram of Z and discuss the shape of the histogram.
4. Convince your answer from 3 using the method we have learned in the class.
5. Do the same thing for (n, p) = (100, 0.1), (100, 0.3), (100, 0.7), (100, 0.9), and draw 4 histograms of Z, and discuss about the shape of the histogram.

1. A survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 54% rented a car during the past 12 months for personal reasons, and 30% rented a car during the past 12 month for both business and personal reasons.

(a) (3 points) What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?

(b) (1 points) What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?