A set of chemistry exam scores are normally distributed with a mean of 
\[70\] points and a standard deviation of 
\[5\] points.
What proportion of exam scores are between 
\[68\] and 
\[73\] points?
You may round your answer to four decimal places.

To find the proportion of exam scores between 68 and 73 points, we need to use the properties of the normal distribution and the **z-scores**.

The formula for the z-score is:

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

where:
- \( x \) is the data point,
- \( \mu = 70 \) is the mean,
- \( \sigma = 5 \) is the standard deviation.

We'll calculate the z-scores for 68 and 73 points.

### Step 1: Calculate the z-scores
1. For \( x = 68 \):
\[
z_{68} = \frac{68 - 70}{5} = \frac{-2}{5} = -0.4
\]

2. For \( x = 73 \):
\[
z_{73} = \frac{73 - 70}{5} = \frac{3}{5} = 0.6
\]

### Step 2: Use the standard normal distribution table (or calculator)
We now need to find the cumulative probabilities corresponding to these z-scores.

- For \( z_{68} = -0.4 \), the cumulative probability is approximately 0.3446.
- For \( z_{73} = 0.6 \), the cumulative probability is approximately 0.7257.

### Step 3: Find the proportion between 68 and 73
To find the proportion of scores between 68 and 73, we subtract the cumulative probability at \( z = -0.4 \) from the cumulative probability at \( z = 0.6 \):

\[
P(68 \leq X \leq 73) = P(z = 0.6) - P(z = -0.4)
\]
\[
P(68 \leq X \leq 73) = 0.7257 - 0.3446 = 0.3811
\]

Thus, the proportion of exam scores between 68 and 73 points is approximately **0.3811** or **38.11%**.

Would you like further clarification or additional details?

Here are 5 related questions to consider:
1. How do we calculate the probability for a score above a certain value using z-scores?
2. What is the z-score of a score that is exactly at the mean?
3. How does changing the standard deviation affect the shape of the normal distribution?
4. How do we interpret a negative z-score versus a positive z-score?
5. Can we calculate the proportion of scores between two values using a non-standard normal distribution?

**Tip:** Always ensure you are working with a standard normal distribution when using z-scores, as this simplifies finding cumulative probabilities.

A set of chemistry exam scores are normally distributed with a mean of 70 points and a standard deviation of 5 points. What proportion of exam scores are between 68 and 73 points? You may round your answer to four decimal places.

Find the proportion of chemistry exam scores between 68 and 73 points in a normally distributed set of data with a mean of 70 and a standard deviation of 5. This problem uses z-scores and the properties of the normal distribution to calculate the probability, ideal for Grades 10-12.

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