Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let X be the random variable representing the time it takes her to complete one review. Assume X is normally distributed. Let X be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.

Find the probability that the mean of a month's reviews will take Yoonie from 3.5 to 4.25 hrs.

(b) Give the probability statement and the probability. (Enter exact numbers as integers, fractions, or decimals for the probability statement. Round the probability to four decimal places.)

To solve this problem, we can utilize the properties of the normal distribution and the Central Limit Theorem since the number of reviews \( n = 16 \) is reasonably large.

Given:
- The average time to complete one review is 4 hours.
- Population standard deviation \( \sigma = 1.2 \) hours.
- Sample size \( n = 16 \).
  
Let \( \bar{X} \) be the sample mean time for completing one review out of a set of 16 reviews.

### Step 1: Determine the Mean and Standard Deviation of \( \bar{X} \)

Since \( X \) is normally distributed with a mean \( \mu = 4 \) hours and standard deviation \( \sigma = 1.2 \) hours, the distribution of the sample mean \( \bar{X} \) will have:
- Mean \( \mu_{\bar{X}} = \mu = 4 \).
- Standard deviation \( \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{1.2}{\sqrt{16}} = \frac{1.2}{4} = 0.3 \) hours.

Thus, \( \bar{X} \sim N(4, 0.3) \).

### Step 2: Convert the Time Range to Z-scores

We are interested in the probability that \( \bar{X} \) falls between 3.5 and 4.25 hours. We convert these values to Z-scores using the standard normal distribution formula:

\[
Z = \frac{\bar{X} - \mu_{\bar{X}}}{\sigma_{\bar{X}}}
\]

1. **For \( \bar{X} = 3.5 \):**
   \[
   Z = \frac{3.5 - 4}{0.3} = \frac{-0.5}{0.3} \approx -1.67
   \]

2. **For \( \bar{X} = 4.25 \):**
   \[
   Z = \frac{4.25 - 4}{0.3} = \frac{0.25}{0.3} \approx 0.83
   \]

### Step 3: Find the Probability

Now, we find the probability that \( Z \) is between -1.67 and 0.83.

Using the standard normal distribution table (or a calculator):
- The probability \( P(Z < 0.83) \approx 0.7967 \).
- The probability \( P(Z < -1.67) \approx 0.0475 \).

Thus, the probability that \( \bar{X} \) falls between 3.5 and 4.25 is:
\[
P(3.5 < \bar{X} < 4.25) = P(-1.67 < Z < 0.83) = 0.7967 - 0.0475 = 0.7492
\]

### Answer

**Probability statement:** \( P(3.5 < \bar{X} < 4.25) = 0.7492 \).

**Probability:** 0.7492 (rounded to four decimal places).

Would you like a deeper explanation of any part, or do you have more questions?

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Here are some related questions for practice:

1. What is the probability that the mean review time will exceed 4.5 hours?
2. What is the probability that a single review time will take between 3 and 5 hours?
3. How would the probability change if the sample size increased to 25?
4. If Yoonie’s goal is to complete reviews within 3.8 hours on average, what is the probability she meets this goal?
5. What is the probability that Yoonie’s reviews will take less than 3.5 hours?

**Tip:** Always check if a sample size is large enough to apply the Central Limit Theorem, especially in real-world scenarios where assumptions may vary.

Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let X be the random variable representing the time it takes her to complete one review. Assume X is normally distributed. Let X be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews. Find the probability that the mean of a month's reviews will take Yoonie from 3.5 to 4.25 hrs. (b) Give the probability statement and the probability. (Enter exact numbers as integers, fractions, or decimals for the probability statement. Round the probability to four decimal places.)

Calculate the probability that the mean time for Yoonie's monthly employee reviews will fall between 3.5 and 4.25 hours. This problem involves understanding normal distribution, applying Z-scores, and using the Central Limit Theorem. The solution is ideal for students studying probability and statistics at the undergraduate level.

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