Question

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.

Answer 1

It is a false statement

The correct statement is -The margin of error from the smaller sample will be
`√(2)` times the margin of error from the larger sample.

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 ∝
`(1)/(√(n) )`

For smaller margin, E₁ =
`(1)/(√(50) )`

For larger margin , E₂ =
`(1)/(√(100) )`

Now,

`(E_(1) )/(E_(2) ) = (√(100) )/(√(50) ) = \sqrt{(100)/(50) } = √(2)\\`

⇒E₁ =
`√(2)` E₂

⇒The margin of error from the smaller sample will be
`√(2)` times the margin of error from the larger sample.

So,

It is a false statement.