AP Statistics / Mr. Hansen 11/29/2000 (rev. 2002, 12/1/2012, 11/27/2017)

Name: ______________________ Name of collaborator: _________________

Mini-Project (in 2002-03 and 2017-18, do #3 through #6 only)

1.	List the two statistics (both name and symbol) whose sampling distributions we will learn the most about in this class. _____ = ____________________ _____ = ____________________
2.	Remember the claim that was made in the article we read yesterday. The author claimed that an error rate of 0.05% would produce a 3000-vote error in 6 million votes. Is this true? Design a simulation to test this assertion. Work with your group members to estimate what the actual discrepancy would likely be in a case such as this.
3.	Let S = the event that a randomly selected school day in Washington, D.C., is sunny, let U = the event that a randomly selected school-day resident of Washington, D.C., is an Upper School student at STA, and let G be the probability that the randomly selected school-day resident of Washington, D.C., is wearing galoshes on the randomly selected school day. Estimate each of the following probabilities (we will compare answers for a, b, and c before proceeding): (a) P(S) (b) P(U) (c) P(G) (d) P(U | S) (e) P(S | U) (f) Are S and U independent? ____ How do you know? _________________ (g) P(G | U) (h) P(U | G) (i) P(G | ~S) (j) P(G | S U) (k) Are U and G independent? ____ How do you know? _________________ (l) Are G and S independent? ____ How do you know? _________________ (m) P(S G) (n) P(S G)
4.	Let H = the event of drawing a heart, A = the event of drawing an ace, K = the event of drawing a king, and R = the event of drawing a red card. In each case a single draw is performed from a well-shuffled standard deck of cards. Compute the following, showing your work or justification for each one: (a) P(H | A) (b) P(K | ~A) (c) P(K | A) (d) Are K and A mutually exclusive? ____ How do you know? _________________ (e) Are K and A independent? ____ How do you know? _________________ (f) P(~A | ~H) (g) P(~A) (h) Are A and H mutually exclusive? ____ How do you know? _________________ (i) Are A and H independent? ____ How do you know? _________________ (j) P(R | A) (k) P(R | H) (l) P(K | ~R ~H) (m) Are H and R mutually exclusive? ____ How do you know? _________________ (n) Are H and R independent? ____ How do you know? _________________
5.	A pair of factoids appeared in an issue of Time, Dec. 4, 2000, p.29: Of all adult Americans, 20% have one or more diagnosable psychological conditions (anxiety, substance abuse, bipolar disorder, etc.), and of all cigarette-smoking adult Americans, 44% have diagnosable psychological conditions. [I have altered the wording slightly in order to make the problem more amenable to our statistical analysis.] (a) Define your events: S = ____________________________________________ , P = ______________________________________________________________ (b) Use probability notation to state the two factoids above. (c) How much more likely (i.e., how many times more likely) is a randomly selected American adult with psychological conditions to be a cigarette smoker than a randomly selected American adult from the overall population is? Is there enough information to answer this question? Justify your answer. (Don’t give up without a solid try! Hints are available by email.)
6.	Explain in your own words (no copying from collaborators) why a “scattershot” screening approach to medical diagnosis among individuals with no symptoms or risk factors can be inappropriate even if the tests that the doctor is using are extremely accurate. For example, tests for HIV-AIDS antibodies are typically 99% (or more) accurate in detecting viral infection in infected individuals, as well as being 99% (or more) accurate in producing a negative result in non-infected individuals, yet the PPV (positive predictive value, i.e., the conditional probability of infection given a positive reading) of an HIV-AIDS screening test may be below 50%. Why is the PPV so low, even when the tests are so accurate?