https://file.aiquickdraw.com/mathbot/file/17273307986143og88cli.jpg

The problem described in the image can be interpreted as follows:

We are given a normally distributed random variable (height of students in a particular school) with:
- Mean (\(\mu\)) = 170 cm
- Standard deviation (\(\sigma\)) = 10 cm

We are asked to find the probability that a randomly selected student's height is between 160 cm and 180 cm.

### Solution:

We need to calculate the probability that the height lies between 160 cm and 180 cm. For a normally distributed variable, this can be calculated using the **z-score formula**:

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

1. **For 160 cm:**
   \[
   z_1 = \frac{160 - 170}{10} = \frac{-10}{10} = -1
   \]

2. **For 180 cm:**
   \[
   z_2 = \frac{180 - 170}{10} = \frac{10}{10} = 1
   \]

Next, we use standard normal distribution tables or a calculator to find the area under the normal curve between these two z-scores.

- The probability corresponding to \(z_1 = -1\) is approximately 0.1587.
- The probability corresponding to \(z_2 = 1\) is approximately 0.8413.

The probability that a student's height is between 160 cm and 180 cm is the difference between these two values:

\[
P(160 \leq X \leq 180) = P(z_2) - P(z_1) = 0.8413 - 0.1587 = 0.6826
\]

Thus, the probability that a randomly selected student's height is between 160 cm and 180 cm is approximately **0.6826** or **68.26%**.

---

Would you like more details or have any other questions? Here are some related questions for further exploration:

1. What is the probability that a student's height is greater than 180 cm?
2. How would the probability change if the standard deviation were 15 cm instead of 10 cm?
3. What is the probability that a student's height is less than 160 cm?
4. Can you calculate the probability that a student's height is between 150 cm and 190 cm?
5. How does the shape of the normal distribution curve change if the standard deviation is reduced?

**Tip:** When using z-scores, it's useful to remember that about 68% of the data falls within 1 standard deviation of the mean in a normal distribution.

The height of students in a particular school is normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What is the probability that a randomly selected student is between 160 cm and 180 cm?

Calculate the probability that a randomly selected student's height is between 160 cm and 180 cm. The problem involves a normal distribution with a mean of 170 cm and a standard deviation of 10 cm, and it uses z-scores to find the required probability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation