Question

To qualify for the December 2019 debate, a Democratic candidate for President must attract 4% support in four qualifying polls taken between October 16 and December 12.

For the purposes of part (A), you may assume:

• Each participant within a poll is independent
• A candidate’s true support is 3%; that is, 3% (0.03) of Democratic voters would vote for this candidate for President in a Democratic primary

A) If we take a poll of 1000 randomly selected Democratic voters, what is the probability that the candidate has at least 4% support in that poll?

For the purposes of part (B), you may assume:

• There are a total of 26 qualifying polls between October 16 and December 12
• The probability of earning at least 4% support in a poll is given by your answer to part (A); you may make up an answer to part (A) if you cannot solve it
• Each poll is independent; that is, the result of one poll does not affect any other polls

B) What is the probability that that this candidate qualifies for the debate by earning at least 4% support in 4 (or more) qualifying polls?

Answer 1

A)

True proportion, p = 0.03

Standard error of proportion, SE = = 0.00539

p(1 - p/n = 0.03 * (1 - 0,03)/1000

Probability that the candidate has at least 4% support in that poll = P(p $\ge$ 0.04)

= P[Z $\ge$ (0.04 - 0.03)/0.00539]

= P[Z $\ge$ 1.86]

= 0.0314

B)

Let X be the number of polls qualified by the debate. Then X ~ Binomial(n = 26, p = 0.0314)

Using Normal approximation to the binomial distribution X will follow Normal distribution with mean = np = 26 * 0.0314 = 0.8164 and standard deviation , SD = = 0.8892

Vnp(1 - p) = 26 * 0.0314 * (1 – 0.0314)

Probability that that this candidate qualifies for the debate by earning at least 4% support in 4 (or more) qualifying polls

= P(X $\ge$ 4)

= P(X > 3.5) (Using Continuity correction)

= P[Z > (3.5 - 0.8164) / 0.8892]

= P[Z > 3.02]

= 0.0013