Math  /  Data & Statistics

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A normal population has mean μ=37\mu=37 and standard deviation σ=14\sigma=14. Find the value that has 25%25 \% of the population above it. Round the answer to at least one decimal place.
The value that has 25%25 \% of the population above it is \square .

Studdy Solution

STEP 1

What is this asking? Find the value where 25% of a normal distribution with mean 37 and standard deviation 14 lies above it. Watch out! Don't mix up "above" and "below." We want the top 25%, not the bottom!

STEP 2

1. Find the z-score.
2. Calculate the value.

STEP 3

Alright, so we're dealing with a **normal distribution** here!
We know the **mean** μ=37\mu = 37 and the **standard deviation** σ=14\sigma = 14.
We're looking for a mystery value, let's call it xx, where **25%** of the population is *greater* than this value.

STEP 4

That means **75%** of the population is *less* than xx.
We need to find the **z-score** that corresponds to this **0.75 probability**.
We can look this up in a **z-table** or use a calculator!

STEP 5

Looking up **0.75** in our **z-table**, we find a z-score of approximately **0.68**.
This means that the value we're looking for is **0.68 standard deviations** above the mean.

STEP 6

Now that we have our **z-score**, we can use the **z-score formula** to find our mystery value xx.
Remember, the z-score formula is: z=xμσ z = \frac{x - \mu}{\sigma} where zz is the **z-score**, xx is the **value** we're looking for, μ\mu is the **mean**, and σ\sigma is the **standard deviation**.

STEP 7

We know z=0.68z = 0.68, μ=37\mu = 37, and σ=14\sigma = 14.
Let's plug those values into our formula: 0.68=x3714 0.68 = \frac{x - 37}{14}

STEP 8

To solve for xx, we can **multiply** both sides of the equation by σ=14\sigma=14: 0.6814=x371414 0.68 \cdot 14 = \frac{x - 37}{14} \cdot 14 9.52=x37 9.52 = x - 37

STEP 9

Now, we just need to **add 37** to both sides to isolate xx: 9.52+37=x37+37 9.52 + 37 = x - 37 + 37 x=46.52 x = 46.52

STEP 10

The value that has 25% of the population above it is approximately 46.5.

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