Math  /  Data & Statistics

QuestionFor a population with μ=100\mu = 100 and σ=20\sigma = 20, what is the XX value corresponding to z=0.25z = 0.25? 90 110 105 95 Need Help? Read It

Studdy Solution

STEP 1

What is this asking? We're looking for a mystery number (XX) 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 XX and zz! XX is the actual value, and zz tells us how far XX 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 (XX) is from the mean (μ\mu) in terms of standard deviations (σ\sigma).

STEP 4

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

STEP 5

We know that μ=100\mu = 100, σ=20\sigma = 20, and z=0.25z = 0.25.
Let's plug these values into our **z-score formula**: 0.25=X10020 0.25 = \frac{X - 100}{20}

STEP 6

To find XX, 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. 0.2520=X1002020 0.25 \cdot 20 = \frac{X - 100}{20} \cdot 20 5=X100 5 = X - 100

STEP 7

Now, we **add 100** to both sides of the equation to get XX all by itself.
This is like "adding to zero" but in reverse! 5+100=X100+100 5 + 100 = X - 100 + 100 105=X 105 = X

STEP 8

So, the value of XX corresponding to z=0.25z = 0.25 is **105**!

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