To compute the probability that an observation $y$ drawn from a normal distribution with mean $\mu = 44.44$ and standard deviation $\sigma = 7.65$ lies within the interval $(35.82, 49.94)$, we follow these steps:

Step 1: Standardize the values to the Z-distribution

We convert the boundaries of the interval to their corresponding Z-scores using the formula: $Z = \frac{X - \mu}{\sigma}$ where $X$ is the boundary value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

For the lower bound $X = 35.82$:

$Z_1 = \frac{35.82 - 44.44}{7.65} = \frac{-8.62}{7.65} \approx -1.1275$

For the upper bound $X = 49.94$:

$Z_2 = \frac{49.94 - 44.44}{7.65} = \frac{5.50}{7.65} \approx 0.7183$

Step 2: Find the cumulative probabilities

We now find the cumulative probability corresponding to these Z-scores using the standard normal distribution table or a calculator.

• $P(Z \leq -1.1275) \approx 0.1296$
• $P(Z \leq 0.7183) \approx 0.7634$

Step 3: Calculate the probability that $y$ lies within the interval

The probability that $y$ is between 35.82 and 49.94 is given by: $P(35.82 < y < 49.94) = P(Z \leq 0.7183) - P(Z \leq -1.1275)$ $P(35.82 < y < 49.94) = 0.7634 - 0.1296 = 0.6338$

Thus, the probability that an observation $y$ lies within the interval $(35.82, 49.94)$ is 0.634 (to three decimal places).

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you interpret the Z-score in a normal distribution?
2. What is the significance of the cumulative distribution function in probability?
3. Can you find the probability for a different interval, such as $(40, 50)$, using the same normal distribution?
4. How would the probability change if the standard deviation were smaller?
5. What is the probability that $y$ is greater than 50 in this distribution?

Tip: When solving probability problems involving the normal distribution, always start by standardizing the variable to the Z-distribution to simplify your calculations.