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

Answer:

She should order 2400 coupons.

Step-by-step 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 $s = \frac{\sigma}{\sqrt{n}}$.

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 $s = \sqrt{\frac{p(1-p)}{n}}$

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

$s = \sqrt{\frac{p(1-p)}{n}}$

$0.01 = \sqrt{\frac{0.4*0.6}{n}}$

$0.01\sqrt{n} = \sqrt{0.4*0.6}$

$\sqrt{n} = \frac{\sqrt{0.4*0.6}}{0.01}$

$(\sqrt{n})^2 = (\frac{\sqrt{0.4*0.6}}{0.01})^2$

$n = 2400$

She should order 2400 coupons.