Ph 490

Feb. 26, 2003

Problem Set 5

Problem 1. Consider an presidential election between two candidates, Bore and Gush. The number of people who vote for each candidate is:

N_B = votes for Bore

N_C = votes for Gush

Let the margin of votes for Bore be defined as

M º N_B - N_G.

(a) Assuming the numbers of votes for the two candidates to be independent counts, so that

s_NB = N_B^1/2

and

s_NG= N_G^{/2 ,}

Derive the expression for the error s_M on the vote margin M,

.

(b) In 2000 the presidential election in Florida was decided by 387 votes, with 6,213,000 votes cast. Is this result consistent with M = 0 (which would mean that the voting populace is exactly divided between the two candidates)?

Problem 2. Suppose that a poll is conducted to determine the relative level of support for presidential candidates Bore and Gush. People are invited to call into a radio station and register a vote for either Bore or Gush. After a reasonable interval of time the phones are closed. The number of votes registered is

N_B number of votes for Bore,

N_G = number of votes for Gush.

Suppose that both N_B and N_G are large enough that the "square root of N" rule can be used for their standard errors.

Let

R = N_B/(N_B + N_G)

be the fraction of the people favoring Bore.

(a) Derive the expression for the error s_R on the vote margin R,

(b) It is often stated that the error on such figures in a poll is "three percentage points." If R is to be accurate to 3% (take this to mean that s_R = 0.03), calculate the number of total votes required, N = N_B + N_G, in the approximation that N_B » N_G.

Problem 3. Consider the medical study of the efficacy of a vaccine for AIDS, described in the attached newspaper article. Let the hypothesis to be tested be that the vaccine is ineffective - that is, that the number of AIDS infections is the same for vaccinated subjects and for those who received a placebo. For the vaccine to be approved as effective, the results of the test should be statistically inconsistent with the "ineffective" hypothesis.

Thus, if for a certain population sample there are P and V subjects receiving, respectively, the placebo and the vaccine, and if PI and VI are the corresponding numbers infected, the hypothesis is that the placebo-to-vaccine ratio of infected subjects,

.

be consistent with the ratio

of all subjects. Since the rate of infection is rather small, the ratio R₀ can be taken to be exact.

(a) Show that the standard error on R is given by

.

(b) Calculate the appropriate ratios for the samples tested in the study described, and discuss the statistical significance of the results.