Problem: A coin is tossed three times in succession. List the set of equally likely outcomes (HHH, HHT etc).

Find the probability of getting exactly one tail. (a)

Find the probability of getting exactly 2 heads. (b)

2. If one person is RANDOMLY selected from the population described in the table below find the probability that that person is divorced. (a)

Find the probability that the person is a male. (b)

Find the probability that the person is both a Male and a Widower. (c)

Given that a person is a male what is the probability that he is married? (d)

3. In a given population of 100 students 15 are familiar with the statistics software SAS, 35 are familiar with SPSS and 5 students are familiar with both. Use the following venn diagram and formulas as an aid to answering the problems.

For any randomly chosen individual in this population

1. P(A) = probability of knowing SAS =
2. P(B) = probability of knowing SPSS =
3. P(A intersection B) = probability of knowing both SAS and SPSS =
4. P(A union B) = probability of knowing SAS or SPSS =
5. P(A|B) = probability of knowing SAS given that the student knows SPSS =
6. P(B|A) = probability of knowing SPSS given that the student knows SAS =

4. Answer the following questions using the probability tree diagram provided.

If students randomly chose answers in a series of 3 multiple choice questions with options a through e.

1. What is the probability that they will get all 3 correct?
2. What is the probability that they will get all 3 incorrect?
3. What is the probability that they will get two or more correct?

5. Optional and a bit more challenging. Lets assume that in the problem above the percentage correct for the questions were as follows (assume each question is an independent event and students who answered the first question correctly had no advantage over students who did not:

What is the probability that a randomly chosen student will have gotten all three correct?

What is the probability that a randomly chosen student will have gotten all three incorrect?

What is the probability that a randomly chosen student will get the last two questions correct given that he got the first question incorrect? (Assume the likelihood of correctly answering any question is independent of performance on the other questions).

5. A simple screening test is developed by OFFRXUSE Drug Company to detect sensitivity to nickel.

A sample group of 100 subjects known to be sensitive to nickle are tested. Eighty of these subjects test positive.

1. Calculate the sensitivity of this test
   Sensitivity =
   The test was correct in identify positive subjects _ __ of the time
   A control group of 100 subjects known to not sensitive to nickle are tested. Ninety of these subjects test negative.
2. Calculate the specificity of this test
   Specificity =
   The test correctly identified negative subjects _____ of the time.
   6. The prevalence of nickel allergy in a particular population is known to be 20 percent. A thousand people in this population are to be screened using this test for nickel sensitivity. For the purposes of this calculation fill in the expected values for the missing totals below:
   1. Complete the 2x2 table for the outcomes of this test

Of the 200 nickel sensitive subjects 80% (the sensitivity of the test) will test positive. That means that the remaining 20% will test (false negative) Fill in the column Sensitive to Nickel for this ideal population.

Similarly of the 800 subjects not sensitive to nickel 90% (the specificity of the test) 720 will test negative and 80 will test positive. Fill in the Not Sensitive Column with the expected result

b. What is the positive predictive value of this test

PPV = The number actually positive who are sensitive / the total number that test positive

= TP / (TP + FP) = _

b. What is the negative predictive value of this test

NPV = The number actually are not sensitive / the total number that test negative.

= TN / (TN+ FN) = _

7. Given then same sensitivity and specificity for this diagnostic test for nickel sensitivity (80% specificity and 90% sensitivity) what would be positive and negative predictive values of this test in a population that had a 1% prevalence of nickel sensitivity.

N=1000 prevalence = 1% sensitivity=80% specificity=90%

b. What is the positive predictive value of this test

PPV = The number actually positive who test positive / the total number that test positive

= TP / (TP + FP) =

c. What is the negative predictive value of this test

PPV = The number actually negative who test negative / the total number that test positive

= TN / (TN+ FN) = __

The positive predictive value is now (lower or higher )

What happened?

Can you think of diagnostic tests for diseases with a low prevalence that have low Positive Predictive Value?

1. Assume that the heights of adult Caucasian men have a mean of 68 inches and a standard deviation of 4 inches. Find the probability that an randomly chosen individual chosen from this population has a mean height greater than 72 inches. 
(b) Less than 60 inches. (c) Between 60 and 72 inches.
2. Assume that the heights of adult Caucasian women have a mean of 63 inches and a standard deviation of 2.5 inches What percentage of women in this population have a mean height (a) greater than 63 inches, (b) greater than 68 inches (c) less than 68 inches.
3. The gestation time for humans has a mean of 266 days and a standard deviation of 25 days. What percentage of pregnancies would you expect to be (a) greater than 291 days, (b) less than 241 days, (c) between 241 days and 291 days.
4. Assume that the heights of adult US women have a mean of 63 inches and a standard deviation of 2.5 inches. What percentage of women would you expect to have a mean height (a) less than 63 inches, (b) less than 60.5 inches, (c) greater than 60.5 inches.
5. The body temperatures of adults have a mean of 98.6 ° F and a standard deviation of 0.60 ° F. Find the probability that a randomly selected individual has a mean body temperature is greater than 98.9 ° F.
6. Assume that the salaries of elementary school teachers in the United States have a mean of $32,000 and a standard deviation of $3000. Find the percentage of teachers you would expect to have a salary (a) greater than $36,500, (b) less than 30,500,
7. Assume that blood pressure readings have a mean of 120 and a standard deviation of 8. Find the probability that the blood pressure of a randomly selected individual will be greater than 130. 
 Less than 110. Less than 110 or greater than 130.
8. A study indicates that 4% of American teenagers have tattoos. You randomly sample 30 teenagers. What is the likelihood that exactly 3 will have a tattoo? (b) 3 or less will have tattoos, (c) 4 or more will have tattoos.
9. An electronic defibrillator has a probability of having a defect of .002. In a lot of 100 defibrillators what is the probability that 1 or more are defective?
10. Suppose that 48% of individuals 85 years and older Dementia. You go to a nursing home that has been feeding residents a diet consisting of natural, organic, and healthy foods including free range chickens, omega 3 eggs, flax seed, etc. You find that only 29 of 100 residents who are 85-years and older have Alzheimer’s disease? What is the likelihood of this outcome?

Price: $32.27

Solution: The downloadable solution consists of 13 pages, 1927 words and 33 charts.
Deliverable: Word Document