Based on what you have studied in this unit, reflect and answer the following questions:

What are the main characteristics of the normal distribution?
Why is it advantageous to use the Z scores?

For a standard normal distribution, about 68% of values falls within one standard deviation of the mean; about 95% falls within two standard deviations; and about 99.7% falls within three standard deviations. Suppose the distribution of math scores in the SAT follow a normal distribution with a mean µ=700 and a standard deviation σ=150.

Calculate the Z-score for an SAT Math score of 600. Interpret it in your own words.
Find the interval [a, b] that captures 68% of the SAT Math score in this example.
Find the interval [a, b] that captures 95% of the SAT Math score in this example.
Find the interval [a, b] that captures 99.7% of the SAT Math score in this example.

### Main Characteristics of the Normal Distribution:
1. **Symmetry**: The normal distribution is symmetric about its mean. This means the left and right sides of the graph are mirror images.
2. **Bell-Shaped Curve**: The distribution forms a bell-shaped curve where most values cluster around the mean and taper off towards the extremes.
3. **Mean, Median, and Mode Coincide**: For a perfectly normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
4. **Empirical Rule**: Approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
5. **Defined by Mean and Standard Deviation**: The distribution is fully characterized by two parameters: the mean (µ) and standard deviation (σ).

### Advantages of Using Z-Scores:
1. **Standardization**: Z-scores allow comparison between different data points and distributions by converting scores into a common scale, where the mean is 0 and the standard deviation is 1.
2. **Interpreting Relative Standing**: Z-scores tell us how many standard deviations a data point is from the mean, allowing for a clear interpretation of its position in the distribution.
3. **Identify Outliers**: Z-scores help identify outliers in the data since values with high absolute Z-scores are far from the mean.
4. **Simplifies Probability Calculations**: Z-scores allow the use of standard normal distribution tables (or software) to quickly find probabilities and percentiles.
5. **Universal Application**: Z-scores can be applied to any normal distribution, making them versatile for different types of data.

---

### Z-Score Calculation for SAT Math Score of 600:
Given:
- Mean \( \mu = 700 \)
- Standard deviation \( \sigma = 150 \)
- SAT Math score \( X = 600 \)

The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]

Substitute the values:
\[
Z = \frac{600 - 700}{150} = \frac{-100}{150} = -0.67
\]

**Interpretation**: The Z-score of -0.67 means that a score of 600 is 0.67 standard deviations below the mean. In other words, it is slightly below the average SAT Math score.

---

### Interval that Captures 68% of the SAT Math Scores:
For 68% of the distribution, we are considering 1 standard deviation from the mean (empirical rule):
\[
[a, b] = [\mu - \sigma, \mu + \sigma] = [700 - 150, 700 + 150] = [550, 850]
\]
Thus, 68% of SAT Math scores lie between **550 and 850**.

---

### Interval that Captures 95% of the SAT Math Scores:
For 95% of the distribution, we consider 2 standard deviations from the mean:
\[
[a, b] = [\mu - 2\sigma, \mu + 2\sigma] = [700 - 2(150), 700 + 2(150)] = [400, 1000]
\]
Thus, 95% of SAT Math scores lie between **400 and 1000**.

---

### Interval that Captures 99.7% of the SAT Math Scores:
For 99.7% of the distribution, we consider 3 standard deviations from the mean:
\[
[a, b] = [\mu - 3\sigma, \mu + 3\sigma] = [700 - 3(150), 700 + 3(150)] = [250, 1150]
\]
Thus, 99.7% of SAT Math scores lie between **250 and 1150**.

---

### Do you want further details or explanations?  
Here are 5 questions to deepen your understanding:
1. What does it mean if a Z-score is positive or negative?
2. How can Z-scores be used to compare different datasets?
3. What is the probability of scoring more than 850 in this SAT example?
4. How does the empirical rule relate to the shape of the normal curve?
5. How would you calculate probabilities for non-standard normal distributions?

#### Tip:
Always check if your data approximately follows a normal distribution before applying Z-scores for meaningful interpretation.

Learn the key characteristics of the normal distribution and how to calculate Z-scores. This example uses SAT Math scores with a mean of 700 and standard deviation of 150. Discover intervals capturing 68%, 95%, and 99.7% of SAT scores using the empirical rule.

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