A set of magical wand prices are normally distributed with a mean of \[50\] dollars and a standard deviation of \[4\] dollars. A blackthorn wand has a price of \[45.20\] dollars.

**What proportion of wand prices are lower than the price of the blackthorn wand?**
*You may round your answer to four decimal places.*

To determine the proportion of wand prices that are lower than the price of the blackthorn wand, we need to find the cumulative probability associated with the price of \[45.20\] dollars under the normal distribution.

### Step 1: Calculate the Z-score
The Z-score indicates how many standard deviations a data point (in this case, the price of the blackthorn wand) is from the mean of the distribution.

The formula for the Z-score is:

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

Where:
- \(X\) is the value of interest (45.20 dollars)
- \(\mu\) is the mean of the distribution (50 dollars)
- \(\sigma\) is the standard deviation of the distribution (4 dollars)

Plugging in the values:

\[
Z = \frac{45.20 - 50}{4} = \frac{-4.80}{4} = -1.2
\]

### Step 2: Find the Cumulative Probability
The cumulative probability associated with a Z-score of \(-1.2\) can be found using a standard normal distribution table or a calculator.

A Z-score of \(-1.2\) corresponds to a cumulative probability of approximately \(0.1151\).

### Final Answer
The proportion of wand prices that are lower than the price of the blackthorn wand is approximately \(0.1151\), or 11.51%.

Do you want more details or have any questions?

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Here are five related questions:
1. How do you interpret the Z-score in the context of this problem?
2. What would be the Z-score for a wand priced at 55 dollars?
3. How does the standard deviation affect the proportion of prices lower than a given value?
4. What proportion of wand prices are higher than the blackthorn wand?
5. How would the answer change if the standard deviation was larger?

**Tip:** Understanding how to calculate and interpret Z-scores can be very useful for comparing data points in different normal distributions.

Explore how to calculate the proportion of wand prices lower than a specified value using the normal distribution. Learn about Z-scores, cumulative probability, and their application in statistics. Suitable for advanced high school students and beyond.

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