A variable xx is normally distributed with mean 24 and standard deviation 5.
*Round your answers to the nearest hundredth as needed.*

a) Determine the zz-score for x=27x=27.

z=z=

b) Determine the zz-score for x=22x=22.

z=z=

c) What value of xx has a zz-score of 1.21.2?

x=x=

d) What value of xx has a zz-score of 00?

x=x=

e) What value of xx has a zz-score of 00?

x=x=

To solve these problems, we need to use the **z-score formula**, which is:

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

Where:
- \( z \) is the z-score,
- \( x \) is the value of the variable,
- \( \mu \) is the mean, and
- \( \sigma \) is the standard deviation.

In this case, the mean \( \mu = 24 \) and the standard deviation \( \sigma = 5 \).

---

### a) Determine the \( z \)-score for \( x = 27 \)

Using the formula:

\[
z = \frac{{27 - 24}}{{5}} = \frac{3}{5} = 0.60
\]

So, \( z = 0.60 \).

---

### b) Determine the \( z \)-score for \( x = 22 \)

\[
z = \frac{{22 - 24}}{{5}} = \frac{-2}{5} = -0.40
\]

So, \( z = -0.40 \).

---

### c) What value of \( x \) has a \( z \)-score of 1.2?

Here, we solve for \( x \) using the z-score formula:

\[
1.2 = \frac{{x - 24}}{{5}}
\]

Multiplying both sides by 5:

\[
x - 24 = 6
\]

So, \( x = 30 \).

---

### d) What value of \( x \) has a \( z \)-score of 0?

If the z-score is 0, the value of \( x \) is equal to the mean, since:

\[
z = \frac{{x - 24}}{{5}} = 0
\]

So, \( x = 24 \).

---

### e) What value of \( x \) has a \( z \)-score of 0? (This part repeats d)

Again, \( x = 24 \), as explained in part d.

---

Let me know if you need further details or clarifications!

---

### 5 Related Questions:
1. How is the z-score used to find probabilities in a normal distribution?
2. Can you calculate the probability that \( x \) falls between two values given their z-scores?
3. What does a negative z-score signify in terms of data position?
4. How do you interpret a z-score of exactly 1 or -1?
5. How would the z-scores change if the standard deviation was smaller or larger?

### Tip: 
A z-score of 0 means the data point is exactly at the mean of the distribution!

A variable x is normally distributed with mean 24 and standard deviation 5. a) Determine the z-score for x = 27. b) Determine the z-score for x = 22. c) What value of x has a z-score of 1.2? d) What value of x has a z-score of 0? e) What value of x has a z-score of 0?

Learn how to calculate z-scores for a normally distributed variable with mean 24 and standard deviation 5. This tutorial covers how to find z-scores for specific values and how to reverse calculate the variable given a z-score. Suitable for students in Grades 10-12.

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