Answer:
It is a false statement
The correct statement is -The margin of error from the smaller sample will be [tex]\sqrt{2}[/tex] times the margin of error from the larger sample.
Step-by-step explanation:
P.S - The exact question is -
Given - A large restaurant chain is curious what proportion of their
customers in a given day are new customers. They are thinking
of taking a sample of either n = 50 or nº= 100 customers and
building a one-sample z interval for a proportion using the data
from the sample.
To find - The margin of error from the smaller sample will be 2 times
the margin of error from the larger sample.
Proof -
We know that
E ∝ [tex]\frac{1}{\sqrt{n} }[/tex]
For smaller margin, E₁ = [tex]\frac{1}{\sqrt{50} }[/tex]
For larger margin , E₂ = [tex]\frac{1}{\sqrt{100} }[/tex]
Now,
[tex]\frac{E_{1} }{E_{2} } = \frac{\sqrt{100} }{\sqrt{50} } = \sqrt{\frac{100}{50} } = \sqrt{2}\\[/tex]
⇒E₁ = [tex]\sqrt{2}[/tex] E₂
⇒The margin of error from the smaller sample will be [tex]\sqrt{2}[/tex] times the margin of error from the larger sample.
So,
It is a false statement.
