A variable  
x
  is normally distributed with mean 21 and standard deviation 5.
Round your answers to the nearest hundredth as needed.

a) Determine the  
z
 -score for  
x
=
26
 .
z
=
 
b) Determine the  
z
 -score for  
x
=
17
 .
z
=
 
c) What value of  
x
  has a  
z
 -score of  
1.8
 ?
x
=
 
d) What value of  
x
  has a  
z
 -score of  
−
0.4
 ?
x
=
 
e) What value of  
x
  has a  
z
 -score of  
0
 ?
x
=
 


To solve this, we use the z-score formula:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- \( x \) is the value of the variable,
- \( \mu \) is the mean of the distribution, and
- \( \sigma \) is the standard deviation.

In this case, \( \mu = 21 \) and \( \sigma = 5 \).

---

### Solution:

#### Part (a): Determine the z-score for \( x = 26 \).

\[
z = \frac{26 - 21}{5} = \frac{5}{5} = 1.0
\]

**Answer:** \( z = 1.0 \)

---

#### Part (b): Determine the z-score for \( x = 17 \).

\[
z = \frac{17 - 21}{5} = \frac{-4}{5} = -0.8
\]

**Answer:** \( z = -0.8 \)

---

#### Part (c): What value of \( x \) has a \( z \)-score of \( 1.8 \)?

Rearrange the formula to solve for \( x \):
\[
x = z \cdot \sigma + \mu
\]
\[
x = 1.8 \cdot 5 + 21 = 9 + 21 = 30
\]

**Answer:** \( x = 30 \)

---

#### Part (d): What value of \( x \) has a \( z \)-score of \( -0.4 \)?

\[
x = -0.4 \cdot 5 + 21 = -2 + 21 = 19
\]

**Answer:** \( x = 19 \)

---

#### Part (e): What value of \( x \) has a \( z \)-score of \( 0 \)?

\[
x = 0 \cdot 5 + 21 = 21
\]

**Answer:** \( x = 21 \)

---

Would you like further details or have any questions?

---

### Related Questions:
1. What is the z-score formula and how is it derived?
2. How do we interpret positive and negative z-scores?
3. If the mean changes to 25 and standard deviation to 6, how would the z-scores change?
4. How do z-scores relate to percentiles in a normal distribution?
5. What are the uses of z-scores in real-world applications?

### Tip:
The z-score provides a way to measure how far any value is from the mean, making it easier to compare different data points on the same scale.

A variable x is normally distributed with mean 21 and standard deviation 5. a) Determine the z-score for x=26. b) Determine the z-score for x=17. c) What value of x has a z-score of 1.8? d) What value of x has a z-score of -0.4? e) What value of x has a z-score of 0?

Explore how to calculate z-scores for a normal distribution with mean 21 and standard deviation 5. This guide includes detailed solutions for values x=26 and x=17, and finds corresponding x values for z-scores of 1.8, -0.4, and 0. Perfect for high school students studying statistics.

Math Problem Statement

a) Determine the
z -score for
x

26 . z

b) Determine the
z -score for
x

17 . z

c) What value of
x has a
z -score of
1.8 ? x

d) What value of
x has a
z -score of
− 0.4 ? x

e) What value of
x has a
z -score of
0 ? x

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