Question

To celebrate 24 years in business, a clothing store's marketing executive is ordering scratch-off discount coupons to give to customers. She would like 40% of customers in the population to receive the highest possible discount, with an SEP of 0.01 for this population. How many coupons should she order?

Answer 1

She should order 2400 coupons.

Explanation:

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

She would like 40% of customers in the population to receive the highest possible discount, with an SEP of 0.01 for this population.

This means that
`p = 0.4, s = 0.01`

How many coupons should she order?

We have to find n. So

`s = \sqrt{(p(1-p))/(n)}`

`0.01 = \sqrt{(0.4*0.6)/(n)}`

`0.01√(n) = √(0.4*0.6)`

`√(n) = (√(0.4*0.6))/(0.01)`

`(√(n))^2 = ((√(0.4*0.6))/(0.01))^2`

`n = 2400`

She should order 2400 coupons.