Question

Q4. You conduct a poll comparing two candidates. In a poll of 400 people, 190 prefer candidate A, and 210 prefer B.

Q4a. What is the point estimate for the probability that a voter will choose candidate A?

Q4b. What is the stdev for that point estimate?

Q4c. What is the 95% Confidence interval?

Q4d. What is the probability that candidate B loses in the actual election?

Q4e. If you want to shrink one stdev error to 1%, how many people do you need to survey?

Answer 1

4a)

the point estimate for the probability that a voter will choose candidate A = 190/400
= 0.4750

4b)

the stdev for that point estimate = sqrt(p*(1-p)/n)
= sqrt(0.4750 *(1-0.4750)/400)
= 0.0250

4c)
z value at 95% confidence interval is ,

alpha = 1 -0.95 = 0.05
alpha/2 = 0.05/2 = 0.025

Zalpha/2 = Z0.025 = 1.96

Margin of error = E = z *sqrt(p*(1-p)n)
= 1.96 * 0.0250
= 0.0490

The 95% confidence interval is,

point estiamte - E < p < point estiamte + E

0.4750 - 0.0490 < p < 0.4750 + 0.0490
0.4260 < p < 0.5240

4d)
p = 210/400 = 0.5250

P(x < 0.5250)
= P(z < (0.5250 - 0.4750)/0.0250)
= P(z < 2)
= 0.9772

4e)

z value at 95% = 1.96
ME = 0.01, p = 0.4750

By using ME formula,

ME = z *sqrt(p*(1-p)/n)
0.01 = 1.96 *sqrt(0.4750 *(1-0.4750)/n)
n = (1.96/0.01) ^2 * 0.4750 *(1-0.4750)
n = 9580