Compute the probability that an observation 𝑦 drawn from a normal distribution with mean 𝜇=31.68 and standard deviation 𝜎=8.26 is in the interval (19.18,41.03), i.e.,  Prob(19.18<𝑦<41.03)

To compute the probability that an observation \( y \) drawn from a normal distribution with mean \( \mu = 31.68 \) and standard deviation \( \sigma = 8.26 \) lies in the interval \( (19.18, 41.03) \), we'll use the properties of the normal distribution.

The probability \( P(19.18 < y < 41.03) \) can be calculated by finding the cumulative distribution function (CDF) values at the boundaries of the interval and then subtracting them.

### Steps:

1. **Standardize the bounds**: Convert the values 19.18 and 41.03 to their corresponding z-scores.
   \[
   z_1 = \frac{19.18 - \mu}{\sigma} = \frac{19.18 - 31.68}{8.26}
   \]
   \[
   z_2 = \frac{41.03 - \mu}{\sigma} = \frac{41.03 - 31.68}{8.26}
   \]

2. **Find the CDF for the z-scores**: Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to these z-scores.

3. **Compute the desired probability**:
   \[
   P(19.18 < y < 41.03) = P(z_1 < Z < z_2) = \Phi(z_2) - \Phi(z_1)
   \]
   where \( \Phi(z) \) is the CDF of the standard normal distribution.

Let's calculate this step by step.

### Step 1: Standardize the bounds
\[
z_1 = \frac{19.18 - 31.68}{8.26} \approx \frac{-12.5}{8.26} \approx -1.513
\]
\[
z_2 = \frac{41.03 - 31.68}{8.26} \approx \frac{9.35}{8.26} \approx 1.132
\]

### Step 2: Find the CDF values
Using a standard normal distribution table or calculator:

\[
\Phi(-1.513) \approx 0.0651
\]
\[
\Phi(1.132) \approx 0.8716
\]

### Step 3: Compute the probability
\[
P(19.18 < y < 41.03) = \Phi(1.132) - \Phi(-1.513) \approx 0.8716 - 0.0651 = 0.8065
\]

### Final Answer
The probability that an observation \( y \) lies in the interval \( (19.18, 41.03) \) is approximately **0.8065**.

Would you like more details on any part of this process or have any questions?

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Here are five related questions you might explore:
1. How do you calculate the probability of a z-score falling below a certain value?
2. What is the relationship between the standard normal distribution and any normal distribution?
3. How do you interpret the cumulative distribution function (CDF)?
4. How would the probability change if the interval were narrower or wider?
5. What are the applications of normal distribution in real-world scenarios?

**Tip:** Understanding z-scores is crucial in solving problems involving normal distributions as they allow us to use the standard normal distribution table for probability calculations.

Learn how to compute the probability that an observation drawn from a normal distribution with mean 31.68 and standard deviation 8.26 falls within the interval (19.18, 41.03). Understand the use of z-scores and the cumulative distribution function (CDF) in solving this probability problem. Suitable for advanced high school students and anyone interested in statistics.

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