PLEASE INCLUDE ALL MATHEMATICAL WORK WITH YOUR ANSWERS.

1. Compute the following values given the availability of six low-frequency, four mid-frequency, and eight high-frequency tones.

   1. How many different sequences of the eight high-frequency tones can be generated?
      
      

   2. How many different sequences of five high-frequency tones can be generated?
      

   3. How many different sequences of five tones can be generated if the first two must be of low, the next two mid, and the final one high frequency?
      

   4. How many different sequences of nine tones can be generated if there must be three from each frequency range, but the tones can be generated in any order?

2. Compute the following values given a subject pool of seven female adults, four female children, twelve male adults, and three male children.
   1. Six subjects are to be selected from the pool for a series of hearing tests. How many different groups of four males and two females can be formed?
      

   2. Eight subjects are to be selected for participation in a research study. How many different groups can be formed if each category (f-a, f-c, m-a, and m-c) must be equally represented?
      

   3. Six subjects are selected at random. What is the probability that all six subjects are female?
      

   4. Four subjects are selected at random. What is the probability that three are female and one is a male child?

3. 1. Draw a tree diagram representing the following scenario. For all the hearing-impaired people in one community, only 60% of the people who notice problems with their hearing seek help from a clinic.  Of those who do come into a clinic, 80% are outfitted with hearing aids, and of those outfitted with hearing aids, 75% actually use them.

   2. Given the information in 3(a), Calculate the following conditional probability. You meet a person at random and learn that she has problems with her hearing but is not wearing a hearing aid.  What is the probability that she sought help at the clinic: P(sought help | has a hearing problem and is not wearing an aid)?

   3. Based on the information presented in 3(a) and 3(b) calculate P(outfitted with aid | has a hearing problem and is not wearing an aid).

4. Suppose that during a hearing test we expect a subject to correctly identify a stimulus tone with probability 0.7 each time it is generated. Calculate the following probabilities.
   1. Assume the tone is generated 20 times; what is the probability that the subject will identify 10 of the tones?
      

   2. Assume the tone is generated 20 times; what is the probability that the subject will identify 18 of the tones?
      

   3. Assume the tone is generated 20 times; what is the probability that the subject will identify fewer than eight of the tones?

5. 1. Draw a tree diagram showing the expected outcomes (label the numbers of TP, FN, FP, TN) of using the fictitious Stryker test (sensitivity = 0.95, specificity = 0.85) on a population of 1200 patients with a disease prevalence of 0.12. Also create a decision matrix to accompany the tree diagram.

   2. Calculate the positive predictive value and negative predictive value for the fictitious Stryker test (sensitivity = 0.95, specificity = 0.85) for three different populations with prevalence values of 0.02, 0.12, and 0.34. Compare, contrast, and interpret the differences.

6. Analysis of Two Testing Protocols
   You are the director of a health clinic that is required by the state to screen for a speech disorder having a prevalence of 5% in the population served by the clinic. Two of your clinicians have proposed different testing protocols. You are to analyze the characteristics of the two protocols and make a decision as to which one your clinic will implement. To conduct the analysis:
   

   1. Draw tree diagrams.
      

   2. Fill in decision matrices for each protocol.
      

   3. Calculate the expected total cost of each protocol if used on the 400 people targeted by the state for testing.
      

   4. Use what you find out about cost, true and false positives, true and false negatives, and your own reasoning about the costs and benefits to choose which protocol to implement; defend your choice in a concise paragraph. 
      

      The disease and tests are fictitious; however, assume that the disease can lead to serious irreparable speech impairment but is not life threatening, and assume that the treatment is time consuming but not expensive, nor is it dangerous.

   Protocol I
   Administer Stryker tests to everyone in the target population. Treat all of the patients who test positive. Characteristics of the Stryker test: cost = $120 per administration, sensitivity = 0.95, specificity = 0.85.

   Protocol II
   Administer Iffans tests to everyone in the target population. Next, administer Stryker tests to everyone who tests positive on an Iffans test. Treat everyone who tests positive on both tests. Characteristics of the Iffans test: Cost = $80 per administration, sensitivity = 0.8, specificity = 0.95. Characteristics of the Stryker test: cost = $120 per administration, sensitivity = 0.95, specificity = 0.85.