The math department at a university gives all of the students in a certain math class the same final exam. The scores are normally distributed with a mean of 80 and a standard deviation of 10. What percentage of the students is expected to receive a score lower than 70?

To determine the percentage of students expected to receive a score lower than 70, we need to find the probability that a student's score falls below 70 in a normal distribution with a mean of 80 and a standard deviation of 10.

### Step 1: Standardize the Score
First, we convert the score of 70 into a standard score (also known as a z-score) using the formula:

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

where:
- \( X = 70 \) (the score we are interested in),
- \( \mu = 80 \) (the mean of the distribution),
- \( \sigma = 10 \) (the standard deviation of the distribution).

Plugging in the values:

\[
z = \frac{70 - 80}{10} = \frac{-10}{10} = -1
\]

So, the z-score for a score of 70 is -1.

### Step 2: Find the Probability
Next, we use the z-score to find the corresponding probability. The z-score of -1 corresponds to the area to the left of this z-score on the standard normal distribution curve.

Looking up a z-score of -1 in the standard normal distribution table or using a calculator, we find:

\[
P(Z < -1) \approx 0.1587
\]

### Step 3: Interpret the Result
The probability \( 0.1587 \) means that approximately **15.87%** of the students are expected to receive a score lower than 70.

### Summary:
- **Percentage of students scoring below 70**: Approximately **15.87%**.

Would you like more details on this process or any other questions related to normal distribution?

Here are 5 relative questions that expand on this topic:

1. What percentage of students is expected to score higher than 90?
2. How would the percentage change if the standard deviation was 5 instead of 10?
3. What score corresponds to the top 10% of the students in the class?
4. What is the probability that a student scores between 70 and 90?
5. How would you calculate the z-score for a score of 85?

**Tip:** When dealing with normally distributed data, z-scores help convert raw data into a standardized form, making it easier to calculate probabilities and compare different data points.

Explore how to calculate the probability of scoring below 70 in a normal distribution with a mean of 80 and a standard deviation of 10. Understand the z-score concept and its application in probability calculations. Ideal for university-level students studying statistics and probability.

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