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.