The table below shows the scores on a math test.
David
75
Aaron
75
Ashley
60
Gina
75
Andrew
75
Kei
95
Monica
100
Anna
60
Lucas
75
Jennifer
65
 	 	 	 
 
The number of standard deviations between a score and the mean score is indicated by a z-score. What was Ashley’s z-score on the math test?
a.
1.1
c.
–1.1
b.
60
d.
–15.5

To find Ashley's z-score, we need to use the formula for a z-score:

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

where:
- \( X \) is Ashley's score (60),
- \( \mu \) is the mean score,
- \( \sigma \) is the standard deviation.

### Step 1: Calculate the Mean (\( \mu \))

First, we find the mean by summing all the scores and dividing by the total number of students.

Scores: 75, 75, 60, 75, 75, 95, 100, 60, 75, 65

\[
\text{Sum of scores} = 75 + 75 + 60 + 75 + 75 + 95 + 100 + 60 + 75 + 65 = 755
\]

Number of students = 10

\[
\text{Mean} (\mu) = \frac{755}{10} = 75.5
\]

### Step 2: Calculate the Standard Deviation (\( \sigma \))

To find the standard deviation, we use the formula:

\[
\sigma = \sqrt{\frac{1}{N} \sum (X_i - \mu)^2}
\]

Let's calculate the squared deviations from the mean:

- \( (75 - 75.5)^2 = 0.25 \)
- \( (75 - 75.5)^2 = 0.25 \)
- \( (60 - 75.5)^2 = 240.25 \)
- \( (75 - 75.5)^2 = 0.25 \)
- \( (75 - 75.5)^2 = 0.25 \)
- \( (95 - 75.5)^2 = 380.25 \)
- \( (100 - 75.5)^2 = 600.25 \)
- \( (60 - 75.5)^2 = 240.25 \)
- \( (75 - 75.5)^2 = 0.25 \)
- \( (65 - 75.5)^2 = 110.25 \)

Now, sum the squared deviations:

\[
\text{Sum of squared deviations} = 0.25 + 0.25 + 240.25 + 0.25 + 0.25 + 380.25 + 600.25 + 240.25 + 0.25 + 110.25 = 1572.5
\]

Now divide by the number of students:

\[
\frac{1572.5}{10} = 157.25
\]

Now take the square root to find the standard deviation:

\[
\sigma = \sqrt{157.25} \approx 12.54
\]

### Step 3: Calculate Ashley's z-score

Now that we have the mean and standard deviation, we can find Ashley's z-score using the formula:

\[
z = \frac{X - \mu}{\sigma} = \frac{60 - 75.5}{12.54} \approx \frac{-15.5}{12.54} \approx -1.24
\]

Ashley's z-score is approximately \(-1.24\).

### Closest Answer: None of the given choices exactly match this value, but the closest option is:

\[
\boxed{-1.1 \text{ (c)}}
\]

---

Would you like further details, or do you have any questions?

Here are five related questions:
1. How is a z-score interpreted in terms of standard deviations?
2. What does a positive z-score indicate?
3. How can standard deviation affect the spread of a dataset?
4. What does it mean when a z-score is negative?
5. Why is it important to understand z-scores in statistics?

**Tip:** Z-scores help compare values across different datasets by standardizing them based on their respective means and standard deviations.

The table below shows the scores on a math test.
David: 75
Aaron: 75
Ashley: 60
Gina: 75
Andrew: 75
Kei: 95
Monica: 100
Anna: 60
Lucas: 75
Jennifer: 65

The number of standard deviations between a score and the mean score is indicated by a z-score. What was Ashley’s z-score on the math test?
a. 1.1
b. 60
c. –1.1
d. –15.5

Determine Ashley's z-score on a math test by applying key statistical concepts, including the calculation of mean and standard deviation. This problem is suitable for high school students in Grades 9-12 and helps in understanding how z-scores indicate the number of standard deviations from the mean.

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