Answer:
Step-by-step explanation:
Confidence interval is written as
Sample proportion ± margin of error
Margin of error = z × √pq/n
Where
z represents the z score corresponding to the confidence level
p = sample proportion. It also means probability of success
q = probability of failure
q = 1 - p
p = x/n
Where
n represents the number of samples
x represents the number of success
From the information given,
n = 1400
x = 872
p = 872/1400 = 0.62
q = 1 - 0.62 = 0.38
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.005 = 0.995
The z score corresponding to the area on the z table is 2.58. Thus, the z score for a confidence level of 99% is 2.58
Given margin of error = ±0.01,
Therefore,
0.01 = 2.58√(0.62)(0.38)/n
0.01/2.58 = √0.2356/n
0.0039 = √0.2356/n
Square both sides
0.00001521 = 0.2356/n
n = 0.2356/0.00001521
n = 15490
