To find each student’s exam score given their $z$-score, we can use the formula:

$X = \mu + z \cdot \sigma$

where:

• $X$ is the exam score we are trying to find,
• $\mu = 154$ is the mean score,
• $\sigma = 21$ is the standard deviation,
• $z$ is the z-score for each student.

Let's calculate $X$ for each student.

1. Robert

• $z = 1.28$ $X = 154 + (1.28 \cdot 21) \approx 154 + 26.88 \approx 181$

2. Juan

• $z = 1.62$ $X = 154 + (1.62 \cdot 21) \approx 154 + 34.02 \approx 188$

3. Haley

• $z = -1.90$ $X = 154 + (-1.90 \cdot 21) \approx 154 - 39.9 \approx 114$

4. Joel

• $z = 0.00$ $X = 154 + (0 \cdot 21) = 154$

5. Jan

• $z = -0.88$ $X = 154 + (-0.88 \cdot 21) \approx 154 - 18.48 \approx 136$

6. Linda

• $z = 1.58$ $X = 154 + (1.58 \cdot 21) \approx 154 + 33.18 \approx 187$

Summary of Scores:

• Robert: 181
• Juan: 188
• Haley: 114
• Joel: 154
• Jan: 136
• Linda: 187

Would you like more details on the calculations, or do you have any questions?

Related Questions:

1. How do z-scores relate to standard deviations?
2. What does a positive or negative z-score signify in terms of exam scores?
3. How would you calculate the score if the mean or standard deviation changed?
4. How is this method used in comparing scores across different tests?
5. Why is the normal distribution important in statistics?

Tip:

Remember that a z-score indicates how many standard deviations a value is from the mean. A positive z-score is above the mean, while a negative z-score is below the mean.