Question

A business woman wants to open a coffee stand across the street from a competing coffee company. She notices that the competing company has an average of 170 customers each day, with a standard deviation of 45 customers. Suppose she takes a random sample of 31 days. Identify the following to help her decide whether to open her coffee stand, rounding to the nearest whole number when necessary:

μ =_____customers per day
σ =_____customers per day
n =____
μ-x =____
σ-x =_____customers per day

Answer 1

μ = 170 customers per day

σ = 45 customers per day

n = 31

`\mu_x = 170`

`\sigma_x = 8`

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.

She notices that the competing company has an average of 170 customers each day, with a standard deviation of 45 customers.

This means that
`\mu = 170, \sigma = 45`

Suppose she takes a random sample of 31 days.

This means that
`n = 31`

For the sample:

By the Central Limit Theorem, the mean is
`\mu_x = 170` and the standard deviation is
`\sigma_x = (45)/(√(31)) = 8`