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The college physical education department offered an advanced first aid course last semester. The scores on the comprehensive final exam were normally distributed, and the \(z\) scores for some of the students are shown below: $$ \begin{array}{lcl} \text { Robert, } 1.10 & \text { Juan, } 1.70 & \text { Susan, }-2.00 \\ \text { Joel, } 0.00 & \text { Jan, }-0.80 & \text { Linda, } 1.60 \end{array} $$ (a) Which of these students scored above the mean? (b) Which of these students scored on the mean? (c) Which of these students scored below the mean? (d) If the mean score was \(\mu=150\) with standard deviation \(\sigma=20\), what was the final exam score for each student?

Step by step solution

01

Identify Scores Above the Mean

Students with positive z-scores scored above the mean because a positive z-score indicates a distance above the mean. From the list, Robert (1.10), Juan (1.70), and Linda (1.60) have positive z-scores, indicating they scored above the mean.

02

Identify Score On the Mean

The student with a z-score of 0 scored exactly on the mean, as a z-score of 0 indicates no deviation from the mean. According to the data, Joel has a z-score of 0, meaning his score was exactly on the mean.

03

Identify Scores Below the Mean

Students with negative z-scores scored below the mean because a negative z-score indicates a distance below the mean. Susan (-2.00) and Jan (-0.80) have negative z-scores, indicating they scored below the mean.

04

Calculate Individual Exam Scores

Use the formula for converting z-scores to raw scores: \[ X = ext{Mean} + (z imes ext{Standard Deviation}) \]. Calculate for each student:- **Robert**: \[ X = 150 + (1.10 imes 20) = 150 + 22 = 172 \]- **Juan**: \[ X = 150 + (1.70 imes 20) = 150 + 34 = 184 \]- **Susan**: \[ X = 150 + (-2.00 imes 20) = 150 - 40 = 110 \]- **Joel**: \[ X = 150 + (0.00 imes 20) = 150 \]- **Jan**: \[ X = 150 + (-0.80 imes 20) = 150 - 16 = 134 \]- **Linda**: \[ X = 150 + (1.60 imes 20) = 150 + 32 = 182 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-Scores

Understanding **z-scores** is essential in statistics, especially when dealing with normally distributed data. A **z-score** represents the number of standard deviations a data point is from the mean. For example, in the exercise:

• A positive **z-score** (e.g., Robert's 1.10) means the score is above the mean.
• A **z-score** of 0 (e.g., Joel) indicates a score exactly at the mean.
• Negative **z-scores** (e.g., Susan's -2.00) imply the score is below the mean.

To calculate a student's actual score from a **z-score**, you'd use this formula: \
\[ X = \text{Mean} + (z \times \text{Standard Deviation}) \]
**Z-scores** are helpful for understanding where a score lies in a distribution and are crucial for comparing different data points across various distributions and contexts.

Normal Distribution

The concept of a **normal distribution** is fundamental in statistics, describing how data naturally distributes itself. **Normal distributions** are symmetric with a specific mean and standard deviation, forming a characteristic bell curve.

• The mean is the center, or peak, of the curve.
• Scores cluster around the mean and decrease in frequency as they move away.
• Approximately 68% of all scores fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

In the exercise mentioned, the exam scores follow this distribution, making **z-scores** particularly useful for identifying the student's performance in relation to the overall group. This makes it easier to understand whether a score is typical or unusual.

Standard Deviation

The **standard deviation** gives us an idea of how spread out the data is around the mean. A larger standard deviation indicates more spread.

• In our exercise, the standard deviation is 20.
• To find each student's score, multiply the **z-score** by this standard deviation and add the mean.
• This addition or subtraction from the mean adjusts the score by how far away it is, in terms of standard deviations.

For instance, if a student has a **z-score** of 1.10, like Robert,

\[ X = 150 + (1.10 \times 20) = 172 \]

**Standard deviation** provides critical insight into the variability of the scores around the mean. In summary, it tells you how much scores typically deviate from the average, fostering a better understanding of data variability.