**Question #2:** What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

## Answer

**Answer**: p-value = 0.9664

## Explanation

To find our p-value, we're first going to calculate the z-score for a sample mean IQ score of 105, then we'll find the corresponding p-value in the z-table!

### Calculating the z-score

We're going to utilize the below formula to calculate z-score with a sampling distribution:

What do each of these variables mean?

**x-bar** is the **sample mean** that you want to obtain the z-score of.**μ **is the **mean** value of the given population.**σ** is the **standard deviation** of the given population’s values.**n** is the **sample size**.

The only major difference between this formula and the one in Question #1 Explanation | Finding p-value of z-score with population distribution is that instead of just **σ**, we now have **σ** divided by **n**!

This probably looks familiar... this is the formula for Standard Error (SE) from Z-scores with sampling distributions!

Your standard error (SE) will occur in the **denominator** of your z-score (and t-score) equations.

When we were working with z-score in Question #1 Explanation | Finding p-value of z-score with population distribution, our formula for z-score was as follows:

Lemme ask you: what was our sample size in that article?

1! Because we were dealing with one, single IQ score.

And based on the z-score formula above when working with standard error (σ/sqrt(n))...

...our sample size was 1...

...any number divided by 1 equals itself...

...so we've really been working with standard error all along, even in What is a z-score, in relation to data points?!

#### Plugging in the mean (**μ**) and standard deviation (**σ**)

In the Z-scores article, we stated the following about the population of all IQ scores:

"Well, it is known that all IQ scores follow on a normal distribution like this...

...with a mean of 100...

...and a standard deviation of 15."

Therefore, we'll once again plug in 100 for **μ**...

...and 15 for **σ**.

#### Plugging in the sample mean (x-bar)

Our prompt states that we're looking for a sample mean of less than 105...

What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

We'll handle the whole "less than" part of this later; all we need to do here for now is plug in 105 for **x-bar**.

#### Plugging in the sample size (n)

Our prompt states that our sample size is 30...

What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

So, we'll plug in 30 for **n**!

#### Solving for z-score

When we solve this out, we get a z-score of 1.83!

### How to associate a p-value to your z-score

This process is *literally* the exact same as what we did in Question #1 Explanation | Finding p-value of z-score with population distribution.

Using the z-table...

...find "1.8" in the left-hand column (representing 1.83)...

...and then "0.03" in the top row (representing 1.83)...

...to locate our p-value of 0.9664!

#### Understanding your p-value visually

Once again... this process is *literally* the exact same as what we did in Question #1 Explanation | Finding p-value of z-score with population distribution.

Notice how at the top of the z-table, the area to the *left* of the "z" on the x-axis is filled in?

This corresponds to the "less than" part of our prompt. We're looking for the area under our sampling distribution to the *left*, or *less than* 105.

What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

Therefore, our p-value of 0.9664 means that on our sampling distribution, the area under the curve to the left of a sample mean IQ score of 105...

...is equal to 0.9664, or 96.64%.

In other words, to answer our original question...

**Question #2:** What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

...the probability of finding a random sample of 30 people with a mean IQ score less than 105 is 96.64%! Therefore, our answer is a p-value of 0.9664!

**Answer**: p-value = 0.9664