Final exam scores in a Math class with large number of students have mean 155 and standard deviation 6.3. Provided the scores of this Final exam have a nearly symmetric/bell-curve distribution, what percent of students score below 136.1 OR above 142.4

We are given that the final exam scores follow a normal distribution with:

- Mean (\(\mu\)) = 155
- Standard deviation (\(\sigma\)) = 6.3

The problem asks for the percentage of students scoring **below 136.1** or **above 142.4**.

To solve this, we need to convert these raw scores into **z-scores** and use the properties of the standard normal distribution to find the corresponding probabilities.

### Step 1: Calculate the z-scores
The z-score is calculated using the formula:

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

#### For 136.1:
\[
z = \frac{136.1 - 155}{6.3} = \frac{-18.9}{6.3} \approx -3
\]

#### For 142.4:
\[
z = \frac{142.4 - 155}{6.3} = \frac{-12.6}{6.3} \approx -2
\]

### Step 2: Find the probabilities
We use standard normal distribution tables or a calculator to find the probabilities corresponding to these z-scores.

- The z-score of \(-3\) corresponds to approximately 0.00135, which means about 0.135% of students score **below 136.1**.
- The z-score of \(-2\) corresponds to approximately 0.0228, which means about 2.28% of students score **below 142.4**.

### Step 3: Calculate the percentage of students
Since we are looking for the percentage of students who score **below 136.1** or **above 142.4**, we add the two probabilities together:

\[
\text{Total percentage} = 0.135\% + 2.28\% = 2.415\%
\]

Thus, approximately **2.42%** of students score either below 136.1 or above 142.4.

---

Would you like further details on how to interpret these probabilities or additional clarification? 

Here are some related questions:
1. How do we interpret z-scores in the context of normal distributions?
2. What is the empirical rule, and how does it relate to this problem?
3. Can we use the empirical rule for this problem instead of z-scores?
4. What percentage of students score between 136.1 and 142.4?
5. How would the calculation change if the distribution was not normal?

**Tip:** The z-score represents how many standard deviations a value is from the mean. Negative z-scores indicate values below the mean.

Final exam scores in a Math class with large number of students have mean 155 and standard deviation 6.3. Provided the scores of this Final exam have a nearly symmetric/bell-curve distribution, what percent of students score below 136.1 OR above 142.4?

Solve a normal distribution problem where final exam scores with mean 155 and standard deviation 6.3 are analyzed. Learn how to calculate the percentage of students scoring below 136.1 or above 142.4 using z-scores. This solution covers key statistical concepts and the properties of normal distribution. Suitable for Grades 10-12.

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