Math Snap

STEP 1

What is this asking?
We're looking for a mystery number ($X$) in a group of numbers where the average is 100 and the typical spread is 20.
We know this mystery number is special because it's 0.25 standard deviations above the average.
Watch out!
Don't mix up $X$ and $z$! $X$ is the actual value, and $z$ tells us how far $X$ is from the average in terms of standard deviations.

STEP 2

1. Understand the z-score formula
2. Plug in the values
3. Solve for X

STEP 3

The z-score formula is like a magical translator between regular numbers and standard deviations.
It tells us how far a particular value ($X$) is from the mean ($\mu$) in terms of standard deviations ($\sigma$).

STEP 4

Here's the formula:
$$z = \frac{X - \mu}{\sigma}$$ Where:
$z$ is the z-score (how many standard deviations away from the mean our value is).
$X$ is the value we're interested in.
$\mu$ is the population mean (the average of all the numbers).
$\sigma$ is the population standard deviation (how spread out the numbers are).

STEP 5

We know that $\mu = 100$, $\sigma = 20$, and $z = 0.25$.
Let's plug these values into our z-score formula:
$$0.25 = \frac{X - 100}{20}$$

STEP 6

To find $X$, we need to isolate it.
First, let's multiply both sides of the equation by 20.
This is like "dividing to one" but in reverse!
We're multiplying by 20 on both sides to keep the equation balanced.
$$0.25 \cdot 20 = \frac{X - 100}{20} \cdot 20$$ $$5 = X - 100$$

STEP 7

Now, we add 100 to both sides of the equation to get $X$ all by itself.
This is like "adding to zero" but in reverse!
$$5 + 100 = X - 100 + 100$$ $$105 = X$$

So, the value of $X$ corresponding to $z = 0.25$ is 105!