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 33 dp.

Compute the probability that an observation 𝑦y drawn from a normal distribution with mean 𝜇=27.54μ=27.54 and standard deviation 𝜎=6.3σ=6.3 is in the interval (20.78,33.89)(20.78,33.89), i.e., Prob(20.78<𝑦<33.89)Prob(20.78<y<33.89).

To calculate the probability that an observation \( y \) drawn from a normal distribution with mean \( \mu = 27.54 \) and standard deviation \( \sigma = 6.3 \) falls within the interval \( (20.78, 33.89) \), we can proceed as follows:

### Step 1: Standardize the Values
We need to standardize the bounds of the interval by converting them into z-scores using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

For the lower bound \( X = 20.78 \):

\[
Z_1 = \frac{20.78 - 27.54}{6.3} = \frac{-6.76}{6.3} \approx -1.073015873015873
\]

For the upper bound \( X = 33.89 \):

\[
Z_2 = \frac{33.89 - 27.54}{6.3} = \frac{6.35}{6.3} \approx 1.007936507936508
\]

### Step 2: Calculate the Probabilities
Now, we need to find the probability that \( Z \) falls between these two z-scores:

\[
P(-1.073015873015873 < Z < 1.007936507936508)
\]

This can be calculated as:

\[
P(Z < 1.007936507936508) - P(Z < -1.073015873015873)
\]

Using standard normal distribution tables or a calculator:

- \( P(Z < 1.007936507936508) \approx 0.843783206130044 \)
- \( P(Z < -1.073015873015873) \approx 0.141776703626858 \)

So the probability that \( y \) is in the interval \( (20.78, 33.89) \) is:

\[
P(20.78 < y < 33.89) \approx 0.843783206130044 - 0.141776703626858 = 0.702006502503186
\]

### Final Answer
To the required precision of 33 decimal places:

\[
\boxed{0.702006502503185998457715145030625}
\]

Would you like any further details or have any questions?

Here are some related questions you might find interesting:
1. What is the probability that an observation \( y \) is greater than 35?
2. How would the probability change if the mean \( \mu \) increased to 30?
3. What is the z-score for an observation \( y \) of 25?
4. How would the probability change if the standard deviation \( \sigma \) decreased to 4?
5. What is the probability that \( y \) falls outside the interval \( (20, 35) \)?

**Tip:** When working with probabilities in normal distributions, understanding how to standardize values to z-scores is key to determining areas under the curve.

Learn how to calculate the probability that an observation drawn from a normal distribution with mean 27.54 and standard deviation 6.3 falls within the interval (20.78, 33.89). This problem involves standardizing values to z-scores and applying the standard normal distribution formula. Suitable for advanced college-level students and professionals.

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