Essay:An analysis of another misleading poll/calculations
The following is a thorough list of mathematical calculations/explanations related to Essay:An analysis of another misleading poll.
Contents
Finding sample makeup by gender
According to the polling data for the 2020 Senate special election in Arizona, 34% of respondents intend to vote for Republican Martha McSally and 49% prefer Democrat Mark E. Kelly. For the two-way tables that involve the preferences by gender, 34% of men intended to vote for McSally, and 52% for Kelly; for women, 34% back McSally and 46% are for Kelly. Since it's unknown exactly what percent of the sample makeup is by gender, M can be used as a variable for male respondents and F for female respondents. Since the percent of men (out of the entire sample) that support McSally and the percent of women (again out of the entire sample) that support McSally must add up to 34%, the equation "34M + 34F = 34" can be set. With the same notion applying to Mark Kelly, the equation "52M + 46F = 49" can be set as well. And to solve:
34M + 34F = 34 → → → 34M = –34F + 34 | 52M + 46F = 49 → → → 52M = –46F + 49 ↓ ↓ ↓ ↓
− 34F = 34F → → → ÷ 34 = ÷ 34 | – 46F = 46F → → → ÷ 52 = ÷ 52 ↓ ↓ ↓ ↓
——————————————————————— → → → ————————————————— | ——————————————————————— → → → ————————————————————————— ↓ ↓ ↓ ↓
34M = –34F + 34 → → → M = –F + 1 | 52M + = –46F + 49 → → → M = –(46/52)F + 49/52 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓ ↓
↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓
↓ ↓ ↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ←
↓ ↓ ↓ ↓
–F + 1 = –(46/52)F + 49/52 → → → (6/52)F + 49/52 = 1 → → → (6/52)F = 3/52
+ F = (52/52)F → → → – 49/52 = 49/52 → → → × (52/6) = × (52/6)
——————————————————————————— → → → ———————————————————————— → → → ——————————————————
1 = (6/52)F + 49/52 → → → (6/52)F = 3/52 → → → F = 3/6
Since exactly half of the sample makeup are women, the other half must be men. Plugging these figures back into the original equations result in identities, so the answers yielded are correct.
Finding sample makeup by age
Given the way the data had been set up for age groups, it is impossible to find the exact, correct answer for the makeup. Six age groups are given (which means six variables to solve for), and since there are only four options to choose from for candidates ("McSally", "Kelly", "Other", and "Don't Know"), that means that only four equations (each involving the six variables) can be set up. Since there more variables than equations, it is impossible to cancel out all enough variables to find out what any one of them are equal to in terms of hard numbers (rather, each variable can only be expressed in terms of other variables); thus there are infinitely many solutions possible.[1]
Finding sample makeup by region
According to the statistics for preferences by region, 31% of those in Maricopa County back Martha McSally in addition to 40% in Pima County and 37% from the rest of the counties. For Mark Kelly, 51% in Maricopa County back him, along with 53% in Pima County and 40% in the other counties. Also, 2% in Maricopa answered "Other", along with another 2% from Pima and 6% from the rest of the counties. With these given numbers, a three-variable system can be set up. Since the 31% in Maricopa County that back McSally along with the 40% in Pima County and 37% in the other counties that follow as such must add up to 34% of all respondents, the equation "31M + 40P + 37T = 34" can be set (note that "T" is used for "Other" rather than "O", as the latter can be confused with the number zero). Following the same idea for Kelly and "Other", the respective equations "51M + 53P + 40T = 49" and "2M + 2P + 6T = 3" can set up as well. Now to solve:
31M + 40P + 37T = 34 | 51M + 53P + 40T = 49 → → 10(2M + 2P + 6T) = 10(3) | 20(2M + 2P + 6T) = 20(3) ↓ ↓ ↓ ↓
51M + 53P + 40T = 49 | – 31M + 40P + 37T = 34 → → – 20M + 13P + 3T = 15 | – 31M + 40P + 37T = 34 ↓ ↓ ↓ ↓
2M + 2P + 6T = 3 | —————————————————————— → → ——————————————————————————— | ———————————————————————————— ↓ ↓ ↓ ↓
| 20M + 13P + 3T = 15 → → 7P + 57T = 15 | 9M + 83T = 26 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓ ↓
↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓
↓ ↓ ↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ←
↓ ↓ ↓ ↓
7P + 57T = 15 → → 9M + 83T = 7P + 57T + 11 → → 9M + 26T = 7P + 11 → → 26T = 7P – 9M + 11
+ 11 = 11 → → – 57T = 57T → → – 9M = 9M → → ÷ 26 = ÷ 26
———————————————————— → → —————————————————————————— → → ————————————————————————— → → —————————————————————————————————
7P + 57T + 11 = 26 → → 9M + 26T = 7P + 11 → → 26T = 7P – 9M + 11 → → T = (7/26)P – (9/26)M + 11/26
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓ ↓
↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓
↓ ↓ ↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ←
↓ ↓ ↓ ↓
31M + 40P + 37((7/26)P – (9/26)M + 11/26) = 34 | 2M + 2P + 6((7/26)P – (9/26)M + 11/26) = 3 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
31M + 40P + (259/26)P – (333/26)M + 407/26 = 34 | 2M + 2P + (42/26)P – (54/26)M + (66/26) = 3 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
(1299/26)P + (473/26)M + 407/26 = 34 | (94/26)P – (2/26)M + (66/26) = 3 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
(1299/26)P + (473/26)M = (477/26) | (94/26)P – (2/26)M = 12/26 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
1299P + 473M = 477 | 94P – 2M = 12 ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
473M = 477 – 1299P | –2M = 12 – 94P ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓ | ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
–(1299/473)P + (477/473) = M | M = –6 + 47P ↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓ ↓
↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ↓ ↓
↓ ↓ ↓ ↓ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ←
↓ ↓ ↓ ↓
–(1299/473)P + 477/473 = –6 + 47P
↓ ↓ ↓ ↓
477/473 = –6 + (23530/473)P
↓ ↓ ↓ ↓
(3315/473) = (23530/473)P
↓ ↓ ↓ ↓
P = 3315/23530
↓ ↓ ↓ ↓
P = 51/362
Notes
- ↑ Further explanation: Normally in a 2-D graph, two lines would intersect at one point. In a 3-D graph, two planes would intersect at a line and three planes would intersect at one point, etc. Suppose a point can be considered to the 0-D analogue of a square, a line an infinitely extending 1-D analogue of a square, a plane a square extending infinitely, etc. Now suppose that two infinitely extending 5-D analogues of squares are graphed on a 6-D plane. The intersection would be an infinitely extending 4-D analogue of a square. Now graph another equation, which now yields three infinitely extending 5-D analogues of squares that intersect at an infinitely extending 3-D space. Graph yet another equation, and the four graphed infinite extensions of 5-D analogues of squares will intersect at a plane. Because that plane has infinitely many points lying on it, with each point representing a valid solution, there are an infinite amount of valid ordered pairs, thus an infinite amount of solutions.
