Question

ClearBags is an online company that sells packaging materials for photographers. Suppose their average order size is $208.19 with a standard deviation of $53.44. A random sample of 35 customer orders has been selected. The standard error of the mean for this sample is ________.

Answer 1

Let X the random variable that represent the order size of a population, and for this case we know the distribution for X is given by:

`X \sim N(208.19,53.44)`

Where
`\mu=208.19` and
`\sigma=53.44`

We select a sample size of n =35.

We know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And for this case the standard error is given by:

`\sigma_(\bar x)= (53.44)/(√(35))= 9.033`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the order size of a population, and for this case we know the distribution for X is given by:

`X \sim N(208.19,53.44)`

Where
`\mu=208.19` and
`\sigma=53.44`

We select a sample size of n =35.

We know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And for this case the standard error is given by:

`\sigma_(\bar x)= (53.44)/(√(35))= 9.033`

Answer 2

The standard error of the mean for this sample is 9.033

Explanation:

The Central Limit Theorem estabilishes 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.

In this problem, we have that:

`\sigma = 53.44, n = 35`

So

`s = (53.44)/(√(35)) = 9.033`

The standard error of the mean for this sample is 9.033