Assume the random variable X is normally distributed with mean 

mu equals 50μ=50 

and standard deviation 

sigma equals 7σ=7. 

Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.

Upper P left parenthesis 35 less than Upper X less than 60 right parenthesis

To compute the probability \( P(35 < X < 60) \) for a normally distributed random variable \( X \) with mean \( \mu = 50 \) and standard deviation \( \sigma = 7 \), we'll follow these steps:

### Step 1: Standardize the values
We need to convert the values 35 and 60 into their corresponding \( z \)-scores using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]

- For \( X = 35 \):
\[
z_1 = \frac{35 - 50}{7} = \frac{-15}{7} \approx -2.14
\]

- For \( X = 60 \):
\[
z_2 = \frac{60 - 50}{7} = \frac{10}{7} \approx 1.43
\]

### Step 2: Find the area under the standard normal curve
Now, we need to find the area under the standard normal curve between \( z_1 = -2.14 \) and \( z_2 = 1.43 \). This area corresponds to the probability \( P(35 < X < 60) \).

1. Using the standard normal distribution table:
   - The area to the left of \( z_1 = -2.14 \) is approximately 0.0162.
   - The area to the left of \( z_2 = 1.43 \) is approximately 0.9236.

2. The probability \( P(35 < X < 60) \) is the difference between these two areas:
\[
P(35 < X < 60) = 0.9236 - 0.0162 = 0.9074
\]

So, the probability \( P(35 < X < 60) \) is approximately **0.9074**.

### Step 3: Draw a normal curve and shade the area
Here’s a description of the curve:
- The normal curve is centered at \( \mu = 50 \) with inflection points at \( \mu - \sigma = 43 \) and \( \mu + \sigma = 57 \).
- The area between \( X = 35 \) (corresponding to \( z_1 = -2.14 \)) and \( X = 60 \) (corresponding to \( z_2 = 1.43 \)) is shaded.

Would you like me to generate an image of the curve with the shaded area?

Here are five related questions:
1. What is the probability \( P(X > 60) \) for the same distribution?
2. How would you calculate \( P(40 < X < 55) \)?
3. What is the value of \( X \) such that \( P(X < x) = 0.90 \)?
4. How do you find the \( z \)-score for a given probability in the upper tail?
5. How does the standard deviation affect the shape of the normal distribution?

**Tip:** When working with normal distributions, always sketch the curve to visualize the problem better.

Learn how to compute the probability \( P(35 < X < 60) \) for a normal distribution with mean \( \mu = 50 \) and standard deviation \( \sigma = 7 \). Detailed steps and calculations provided. Suitable for undergraduate-level statistics and mathematics.

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