Test scores for a statistics class had a mean of
7
9
with a standard deviation of
4
.
5

.
Test scores for a calculus class had a mean of
6
9
with standard deviation
3
.
7
.
Suppose a stu

Question

Test scores for a statistics class had a mean of 
7
9
 ﻿with a standard deviation of 
4
.
5
 
.
 ﻿Test scores for a calculus class had a mean of 
6
9
 ﻿with standard deviation 
3
.
7
.
 ﻿Suppose a student gets 
8
4
 ﻿on the statistics test and 
8
0
 ﻿on the calculus test. On which test did the student perform better relative to the other students in each class?

VIPSolutions · Accepted Answer

Let's calculate the **z-scores** for the student's performances in both classes. The z-score tells us how many standard deviations a value is above or below the mean.

The formula for the z-score is:

$$
z = \frac{X - \mu}{\sigma}
$$

Where:
- $ X $ = student's score
- $ \mu $ = mean
- $ \sigma $ = standard deviation

Let's use your data:

---

### **Statistics test**
- Mean ($\mu$) = **79**
- Standard deviation ($\sigma$) = **4.5**
- Student's score ($X$) = **84**

$$
z_{stat} = \frac{84 - 79}{4.5} = \frac{5}{4.5} \approx 1.11
$$

---

### **Calculus test**
- Mean ($\mu$) = **69**
- Standard deviation ($\sigma$) = **3.7**
- Student's score ($X$) = **80**

$$
z_{calc} = \frac{80 - 69}{3.7} = \frac{11}{3.7} \approx 2.97
$$

---

## **Conclusion**

- **Statistics test z-score:** ≈ **1.11**
- **Calculus test z-score:** ≈ **2.97**

**The student performed better relative to their classmates on the calculus test**, because their score was further above the class mean (in terms of standard deviations) than on the statistics test.

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