Math 439 Diabetic patients

1. (18 pts.) Files of diabetic patients at a local medical practice were randomly selected and used as part of study to determine if the type of diabetes (Type 1 or Type 2) is related to a person’s gender.  The table below summarizes the data collected.

1. (2 pts.) How many patients were included in this study?  Please be sure to briefly explain how you got this answer.

1. (4 pts.) Based on this data, what percentage of diabetes patients have Type 2 diabetes?  Provide both an exact answer and a decimal approximation rounded to 3 decimal places for the probability.  Please be sure to answer the question in a complete sentence with a value rounded to 1 decimal place.

1. (4 pts.) Based on this data, what percentage of diabetes patients are men with Type 2 diabetes?  Provide both an exact answer and a decimal approximation rounded to 3 decimal places for the probability.  Please be sure to answer the question in a complete sentence with a value rounded to 1 decimal place.

1. (6 pts.) Based on this data, what percentage of diabetes patients are either women or they have Type 1 diabetes?  Provide both an exact answer and a decimal approximation rounded to 3 decimal places for the probability.  Please be sure to answer the question in a complete sentence with a value rounded to 1 decimal place.

1. (2 pts.) Let A represent the event that a diabetes patient is male.  Let B represent the event that a diabetes patient has Type 2 diabetes.  Are the events A and B mutually exclusive?  That is, are these events disjoint?  Justify your answer being sure to include the definition of mutually exclusive/disjoint and any relevant calculations.

2. (24 pts.) The number of TVs per household in Japan (X) has the distribution given in the table below.  Use this table to answer the following questions.

1. (5 pts.) Is the distribution given in the table above a valid probability distribution?  Justify your answer by addressing what the properties are of a valid probability distribution and how/why each property is either met or not referencing the specific values given for both the variable and the probabilities.

1. (1 pt.) What is the probability that a randomly selected Japanese household has exactly 3 TVs?  No justification or work necessary.

1. (3 pts.) What percentage of households in Japan have 2 or more TVs?  Provide your final answer rounded to 1 decimal place.  Justify your answer by referring specifically to the work necessary to find this value and why you used the method that you did to find this value.

1. (6 pts.) How many TVs would one expect a randomly selected Japanese household to have?  Provide both an exact answer and a rounded answer that makes more sense in the context of the question.  Justify your answer by showing your work and explaining why you chose your rounded answer in relation to the value of your exact answer.

1. (6 pts.) What is the standard deviation of X = the number of TVs per Japanese household?  You do not need to answer the question in a sentence, but you do need to show all of your work.  Please round to 5 decimal places within your work and your final answer to 3 decimal places.

1. (3 pts.) Would it be unusual for a randomly selected Japanese household to have no TVs?  Please be sure to justify your answer addressing what it means to be “unusual” and how your work specifically relates to both this definition and your final answer.

3. (19 pts.) A box contains 10 items, of which 3 are defective and 7 are not defective.  Suppose that whether or not any randomly selected part is defective does not affect whether or not any other part is defective.  Two items are randomly selected, one at a time, with replacement, and X represents the number of defectives in the sample of two.  Answer the following questions to determine if X is a binomial random variable.

1. (4 pts.) Is X, as defined above, a discrete random variable?  Justify your answer by addressing both the definition of a discrete random variable and the possible values of X for this situation.

1. (3 pts.) Are there a set number of trials or a specified sample size?  If so, what would be defined as a trial and what is the value of n?  If not, explain why the situation above does not have a fixed number of trials or sample size (thus violating this property of a binomial random variable).

1. (3 pts.) Are each of the trials (items in the sample) independent?  Justify your answer by specifically addressing the definition of independent trials and the particular detail(s) in the given situation that either support or contradict this definition.

1. (2 pt.) How many outcomes are there for each trial?  What are the possible outcomes for this example?

1. (2 pts.) What would be defined as a “success” in the situation given above?   Justify your answer by specifically addressing how a “success” is defined for a binomial random variable.

1. (2 pts.) Is the probability of “success” constant for all trials?  What specifically in the situation as given in the original problem statement either supports or contradicts this?

1. (3 pts.) Based on your answers in the above parts, is X, as defined above, a binomial random variable?  If so, indicate each of your answers above that support this conclusion (note that not all of the questions above are necessarily needed specifically for a variable to be defined as binomial).  If not, which properties of a binomial random variable (binomial experiment) are violated and state specifically how it/they is/are violated?

4. (34 pts.) The engines on an airliner operate independently.  There is a 97% chance that an individual engine operates for an entire trip.  A plane will be able to complete a trip successfully if at least one-half of its engines operate for the entire trip.

1. (6 pts.) Let X represent the number of engines in operation on a four-engine aircraft.  Suppose that X is a binomial random variable.  What are the values of n, p, and q?  Briefly justify your answers by indicating what each value represents in the context of this problem and what is defined as a “success.”

1. (13 pts.) Using appropriate probability notation and showing all of your work, find the probability that a four-engine aircraft will successfully complete an entire trip to 4 decimal places.  (Remember that you are rounding to 4 decimal places in the last step and should round to at least 6 decimal places within your calculations)

1. (6 pts.) Let Y represent the number of engines in operation on a two-engine aircraft. Suppose that Y is a binomial random variable.  What are the values of n, p, and q?  Briefly justify your answers by indicating what each value represents in the context of this problem and what is defined as a “success.”

1. (9 pts.) Using appropriate probability notation and showing all of your work, find the probability that a two-engine aircraft will successfully complete an entire trip to 4 decimal places.  (Remember that you are rounding to 4 decimal places in the last step and should round to at least 6 decimal places within your calculations).