Answer the following question, keeping in mind that your answer must be a number between 0 and 1, i.e., do not use percentages. Give your final answer with at least 3 dp.

Compute the probability that an observation y drawn from a normal distribution with mean μ=49.87 and standard deviation σ=7.16 is in the interval (41.99,57.75), i.e., Prob(41.99<y<57.75).

To compute the probability that an observation \( y \) drawn from a normal distribution with mean \( \mu = 49.87 \) and standard deviation \( \sigma = 7.16 \) falls within the interval \( (41.99, 57.75) \), we follow these steps:

### Step 1: Compute the z-scores
We need to calculate the z-scores for both bounds of the interval, \( y = 41.99 \) and \( y = 57.75 \).

The z-score formula is:
\[
z = \frac{y - \mu}{\sigma}
\]

#### For \( y = 41.99 \):
\[
z_1 = \frac{41.99 - 49.87}{7.16} = \frac{-7.88}{7.16} \approx -1.1006
\]

#### For \( y = 57.75 \):
\[
z_2 = \frac{57.75 - 49.87}{7.16} = \frac{7.88}{7.16} \approx 1.1006
\]

### Step 2: Find the probabilities
Now we find the probabilities corresponding to these z-scores using the standard normal distribution table or a calculator.

- The probability corresponding to \( z_1 = -1.1006 \) is approximately \( 0.1358 \).
- The probability corresponding to \( z_2 = 1.1006 \) is approximately \( 0.8642 \).

### Step 3: Calculate the probability of the interval
The probability that \( y \) lies within the interval \( (41.99, 57.75) \) is the difference between the two probabilities:
\[
P(41.99 < y < 57.75) = P(z_2) - P(z_1) = 0.8642 - 0.1358 = 0.7284
\]

### Final Answer
The probability that an observation \( y \) falls within the interval \( (41.99, 57.75) \) is **0.728** (to three decimal places).

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Would you like more details or have any questions? Here are some related questions:

1. What is the probability that \( y \) is less than 41.99?
2. How would the probability change if the interval were wider?
3. What happens to the probability if the mean \( \mu \) is increased?
4. How would a smaller standard deviation \( \sigma \) affect this probability?
5. How can you visualize this probability on a normal distribution curve?

**Tip:** Understanding the z-score and how it relates to areas under the standard normal curve is key to solving problems involving normal distributions.

Learn how to compute the probability that an observation y drawn from a normal distribution with mean μ=49.87 and standard deviation σ=7.16 falls within the interval (41.99, 57.75). This problem involves calculating z-scores, using the standard normal distribution, and understanding probability in normal distributions.

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