Math / Data & Statistics

QuestionSuppose $X \sim N(2, 6)$ . What value of $x$ has a $z$ -score of three?

Studdy Solution

STEP 1

What is this asking? If a random variable $X$ follows a normal distribution with a mean of 2 and a standard deviation of the square root of 6, what value of $x$ is three standard deviations above the mean? Watch out! Don't forget that the second parameter of $N(\mu, \sigma^2)$ is the variance, not the standard deviation!

STEP 2

1. Find the standard deviation.
2. Calculate the z-score.
3. Solve for x.

STEP 3

We're given that $X$ follows a normal distribution: $X \sim N(2, 6)$ .
This tells us the mean $\mu = 2$ and the variance $\sigma^2 = 6$ .

STEP 4

To work with z-scores, we need the standard deviation, $\sigma$ , which is the square root of the variance.
So, $\sigma = \sqrt{6}$ .
Awesome!

STEP 5

Remember that the z-score tells us how many standard deviations a value $x$ is away from the mean.
The formula is $z = \frac{x - \mu}{\sigma}$ .

STEP 6

We're given that the z-score is three, so $z = 3$ .
We also know $\mu = 2$ and $\sigma = \sqrt{6}$ .
Let's plug those values into our formula: $3 = \frac{x - 2}{\sqrt{6}}$ .
Look at that, an equation just waiting to be solved!

STEP 7

Our goal is to find the value of $x$ .
Let's isolate $x$ in the equation $3 = \frac{x - 2}{\sqrt{6}}$ .

STEP 8

First, we can multiply both sides of the equation by $\sqrt{6}$ to get rid of the denominator: $3 \cdot \sqrt{6} = \frac{x - 2}{\sqrt{6}} \cdot \sqrt{6}$ .
This simplifies to $3\sqrt{6} = x - 2$ .
We're almost there!

STEP 9

Now, we just need to add 2 to both sides of the equation: $3\sqrt{6} + 2 = x - 2 + 2$ .
This gives us $x = 3\sqrt{6} + 2$ .
Boom! We found $x$ !

STEP 10

The value of $x$ with a z-score of three is $3\sqrt{6} + 2$ .

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QuestionSuppose $X \sim N(2, 6)$ . What value of $x$ has a $z$ -score of three?

Studdy Solution

STEP 1

STEP 2

1. Find the standard deviation.
2. Calculate the z-score.
3. Solve for x.

STEP 3

We're given that $X$ follows a normal distribution: $X \sim N(2, 6)$ .
This tells us the mean $\mu = 2$ and the variance $\sigma^2 = 6$ .

STEP 4

To work with z-scores, we need the standard deviation, $\sigma$ , which is the square root of the variance.
So, $\sigma = \sqrt{6}$ .
Awesome!

STEP 5

Remember that the z-score tells us how many standard deviations a value $x$ is away from the mean.
The formula is $z = \frac{x - \mu}{\sigma}$ .

STEP 6

We're given that the z-score is three, so $z = 3$ .
We also know $\mu = 2$ and $\sigma = \sqrt{6}$ .
Let's plug those values into our formula: $3 = \frac{x - 2}{\sqrt{6}}$ .
Look at that, an equation just waiting to be solved!

STEP 7

Our goal is to find the value of $x$ .
Let's isolate $x$ in the equation $3 = \frac{x - 2}{\sqrt{6}}$ .

STEP 8

First, we can multiply both sides of the equation by $\sqrt{6}$ to get rid of the denominator: $3 \cdot \sqrt{6} = \frac{x - 2}{\sqrt{6}} \cdot \sqrt{6}$ .
This simplifies to $3\sqrt{6} = x - 2$ .
We're almost there!

STEP 9

Now, we just need to add 2 to both sides of the equation: $3\sqrt{6} + 2 = x - 2 + 2$ .
This gives us $x = 3\sqrt{6} + 2$ .
Boom! We found $x$ !

STEP 10

The value of $x$ with a z-score of three is $3\sqrt{6} + 2$ .

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QuestionSuppose X∼N(2,6)X \sim N(2, 6)X∼N(2,6). What value of xxx has a zzz-score of three?

Studdy Solution

STEP 1

STEP 2

1. Find the standard deviation.2. Calculate the z-score.3. Solve for x.

STEP 3

We're given that XXX follows a normal distribution: X∼N(2,6)X \sim N(2, 6)X∼N(2,6).This tells us the **mean** μ=2\mu = 2μ=2 and the **variance** σ2=6\sigma^2 = 6σ2=6.

STEP 4

To work with z-scores, we need the **standard deviation**, σ\sigmaσ, which is the square root of the variance.So, σ=6\sigma = \sqrt{6}σ=6​.Awesome!

STEP 5

Remember that the z-score tells us how many standard deviations a value xxx is away from the mean.The formula is z=x−μσz = \frac{x - \mu}{\sigma}z=σx−μ​.

STEP 6

We're given that the z-score is **three**, so z=3z = 3z=3.We also know μ=2\mu = 2μ=2 and σ=6\sigma = \sqrt{6}σ=6​.Let's plug those values into our formula: 3=x−263 = \frac{x - 2}{\sqrt{6}}3=6​x−2​.Look at that, an equation just waiting to be solved!

STEP 7

Our goal is to find the value of xxx.Let's **isolate** xxx in the equation 3=x−263 = \frac{x - 2}{\sqrt{6}}3=6​x−2​.

STEP 8

First, we can **multiply** both sides of the equation by 6\sqrt{6}6​ to get rid of the denominator: 3⋅6=x−26⋅63 \cdot \sqrt{6} = \frac{x - 2}{\sqrt{6}} \cdot \sqrt{6}3⋅6​=6​x−2​⋅6​.This simplifies to 36=x−23\sqrt{6} = x - 236​=x−2.We're almost there!

STEP 9

Now, we just need to **add** 2 to both sides of the equation: 36+2=x−2+23\sqrt{6} + 2 = x - 2 + 236​+2=x−2+2.This gives us x=36+2x = 3\sqrt{6} + 2x=36​+2.Boom! We found xxx!

STEP 10

The value of xxx with a z-score of three is 36+23\sqrt{6} + 236​+2.

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QuestionSuppose $X \sim N(2, 6)$ . What value of $x$ has a $z$ -score of three?

1. Find the standard deviation.
2. Calculate the z-score.
3. Solve for x.

We're given that $X$ follows a normal distribution: $X \sim N(2, 6)$ .
This tells us the mean $\mu = 2$ and the variance $\sigma^2 = 6$ .

To work with z-scores, we need the standard deviation, $\sigma$ , which is the square root of the variance.
So, $\sigma = \sqrt{6}$ .
Awesome!

Remember that the z-score tells us how many standard deviations a value $x$ is away from the mean.
The formula is $z = \frac{x - \mu}{\sigma}$ .

We're given that the z-score is three, so $z = 3$ .
We also know $\mu = 2$ and $\sigma = \sqrt{6}$ .
Let's plug those values into our formula: $3 = \frac{x - 2}{\sqrt{6}}$ .
Look at that, an equation just waiting to be solved!

Our goal is to find the value of $x$ .
Let's isolate $x$ in the equation $3 = \frac{x - 2}{\sqrt{6}}$ .

First, we can multiply both sides of the equation by $\sqrt{6}$ to get rid of the denominator: $3 \cdot \sqrt{6} = \frac{x - 2}{\sqrt{6}} \cdot \sqrt{6}$ .
This simplifies to $3\sqrt{6} = x - 2$ .
We're almost there!

Now, we just need to add 2 to both sides of the equation: $3\sqrt{6} + 2 = x - 2 + 2$ .
This gives us $x = 3\sqrt{6} + 2$ .
Boom! We found $x$ !

The value of $x$ with a z-score of three is $3\sqrt{6} + 2$ .