A set of middle school student heights are normally distributed with a mean of \[150\] centimeters and a standard deviation of \[20\] centimeters. Uma is a middle school student with a height of \[165\] centimeters.

**What proportion of student heights are higher than Uma's height?**
*You may round your answer to four decimal places.*

To determine the proportion of middle school student heights that are higher than Uma's height of \[165\] centimeters, we'll follow these steps:

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

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

Where:
- \(X\) is Uma's height (165 cm)
- \(\mu\) is the mean height (150 cm)
- \(\sigma\) is the standard deviation (20 cm)

Plugging in the values:

\[
Z = \frac{165 - 150}{20} = \frac{15}{20} = 0.75
\]

### Step 2: Find the Cumulative Probability
Next, we find the cumulative probability associated with a Z-score of \(0.75\).

A Z-score of \(0.75\) corresponds to a cumulative probability of approximately \(0.7734\).

### Step 3: Calculate the Proportion of Heights Higher Than Uma's
The cumulative probability \(0.7734\) represents the proportion of students shorter than Uma. To find the proportion of students taller than Uma, subtract this value from 1:

\[
\text{Proportion of heights higher than Uma's} = 1 - 0.7734 = 0.2266
\]

### Final Answer
The proportion of student heights that are higher than Uma's height is approximately \(0.2266\), or 22.66%.

Do you want more details or have any questions?

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Here are five related questions:
1. What does the cumulative probability represent in this context?
2. How would the proportion change if Uma were shorter, say 145 cm?
3. What is the Z-score for a student who is 180 cm tall?
4. What proportion of student heights are below 140 cm?
5. How does a change in the standard deviation affect the distribution of heights?

**Tip:** To calculate the proportion of values above or below a certain point, always remember to subtract the cumulative probability from 1 when needed.

Calculate the proportion of student heights higher than Uma's height of 165 cm in a normal distribution with mean 150 cm and standard deviation 20 cm. Understand the concept of Z-score and cumulative probability.

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