A student models the heights of people in their class and finds that it follows a normal distribution with mean 151 cm.

They find that 24 percent of students have heights between 151 cm and 156 cm.

According to the model, what percentage of students are expected to be smaller than 146 cm? give to 2dp



To solve this problem, we will use the properties of the **normal distribution** and the **standard normal distribution (Z-distribution)**.

### Step 1: Determine the Standard Deviation (σ)

We know that the heights follow a normal distribution with:
- Mean (\(\mu\)) = 151 cm
- 24% of the students have heights between 151 cm and 156 cm.

To find the percentage of students smaller than 146 cm, we first need to calculate the standard deviation (\(\sigma\)) of the distribution.

The 24% of students between 151 cm and 156 cm corresponds to the probability in a standard normal distribution between \( Z = 0 \) and some positive Z-value, where the Z-score for 156 cm is:
\[
Z = \frac{156 - 151}{\sigma}
\]

From standard normal distribution tables, the probability that corresponds to a Z-score of 0.24 (24%) between the mean and this Z-value is approximately \( Z \approx 0.68 \). Hence, we have:
\[
\frac{156 - 151}{\sigma} = 0.68
\]
\[
\sigma = \frac{5}{0.68} \approx 7.35 \, \text{cm}
\]

### Step 2: Find the Z-score for 146 cm

Now that we have the standard deviation, we calculate the Z-score for 146 cm:
\[
Z = \frac{146 - 151}{7.35} = \frac{-5}{7.35} \approx -0.68
\]

### Step 3: Find the Probability for Z = -0.68

Using the standard normal distribution table, the cumulative probability for \( Z = -0.68 \) is approximately 0.2483.

### Final Answer

Thus, about **24.83%** of students are expected to have heights smaller than 146 cm.

---

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How do we interpret Z-scores in a normal distribution?
2. What are the properties of a standard normal distribution?
3. How do we use the cumulative distribution function (CDF) in probability?
4. What is the empirical rule in normal distributions?
5. How can we estimate probabilities for values not in Z-tables?

**Tip:** The standard deviation tells us how spread out the data is from the mean, which affects the shape of the normal distribution.

A student models the heights of people in their class and finds that it follows a normal distribution with mean 151 cm. They find that 24 percent of students have heights between 151 cm and 156 cm. According to the model, what percentage of students are expected to be smaller than 146 cm? (give to 2dp)

Learn how to solve a normal distribution problem where a student's class height data is modeled with a mean of 151 cm. The problem asks for the percentage of students smaller than 146 cm, using Z-scores and the standard normal distribution. Suitable for students in Grades 10-12.

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