A set of shirt prices are normally distributed with a mean of 
\[45\] dollars and a standard deviation of 
\[5\] dollars.
What proportion of shirt prices are between 
\[37\] dollars and 
\[59.35\] dollars?
You may round your answer to four decimal places.


To find the proportion of shirt prices between $ 37 $ dollars and $ 59.35 $ dollars, we'll use the properties of the normal distribution.

### Step 1: Standardize the Values
We'll convert the prices $ 37 $ dollars and $ 59.35 $ dollars to their corresponding z-scores using the formula:

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

Where:
- $ x $ is the value we want to standardize,
- $ \mu = 45 $ dollars is the mean,
- $ \sigma = 5 $ dollars is the standard deviation.

#### For $ x = 37 $ dollars:
\[
z_1 = \frac{37 - 45}{5} = \frac{-8}{5} = -1.6
\]

#### For $ x = 59.35 $ dollars:
\[
z_2 = \frac{59.35 - 45}{5} = \frac{14.35}{5} \approx 2.87
\]

### Step 2: Find the Corresponding Probabilities
Next, we'll find the probabilities corresponding to these z-scores using the standard normal distribution table or a calculator.

#### Probability for $ z_1 = -1.6 $:
\[
P(Z \leq -1.6) \approx 0.0548
\]

#### Probability for $ z_2 = 2.87 $:
\[
P(Z \leq 2.87) \approx 0.9979
\]

### Step 3: Calculate the Proportion
The proportion of shirt prices between $ 37 $ dollars and $ 59.35 $ dollars is the difference between these two probabilities:

\[
P(37 \leq X \leq 59.35) = P(Z \leq 2.87) - P(Z \leq -1.6) = 0.9979 - 0.0548 = 0.9431
\]

### Final Answer:
The proportion of shirt prices between $ 37 $ dollars and $ 59.35 $ dollars is approximately $ \boxed{0.9431} $.

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

Here are 5 related questions:
1. What proportion of shirt prices are less than 37 dollars?
2. What proportion of shirt prices are greater than 59.35 dollars?
3. What proportion of shirt prices are between 40 dollars and 50 dollars?
4. How would the proportion change if the standard deviation were 3 dollars instead of 5 dollars?
5. How can we find the price corresponding to the 75th percentile?

**Tip:** Always check if the z-score table or calculator gives cumulative probabilities (from the far left up to the z-score), as this is standard.

Learn how to calculate the proportion of shirt prices between $37 and $59.35 in a normal distribution with mean μ = $45 and standard deviation σ = $5. This problem involves standardizing values using the z-score formula and applying properties of the normal distribution.

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