Question 1191336
Here's how to solve this problem:

**a) 95% Confidence Interval for Mean Absences:**

Since the sample size is small (n=25) and the population standard deviation is unknown, we use the t-distribution.

1. **Find the critical t-value:** For a 95% confidence level and 24 degrees of freedom (n-1 = 25-1 = 24), the critical t-value is approximately 2.064 (you can find this using a t-table or calculator).

2. **Calculate the margin of error:**
   Margin of Error = t * (S / √n) = 2.064 * (4.0 / √25) = 2.064 * 0.8 = 1.6512

3. **Construct the confidence interval:**
   Confidence Interval = ȳ ± Margin of Error = 9.7 ± 1.6512

   The 95% confidence interval is approximately (8.05, 11.35) days.

**b) 95% Confidence Interval for Population Proportion:**

1. **Calculate the sample proportion (p̂):**
   p̂ = (Number of workers absent > 10 days) / (Total number of workers) = 12/25 = 0.48

2. **Find the critical z-value:** For a 95% confidence level, the critical z-value is 1.96.

3. **Calculate the margin of error:**
   Margin of Error = z * √(p̂(1 - p̂) / n) = 1.96 * √(0.48 * 0.52 / 25) ≈ 0.196

4. **Construct the confidence interval:**
   Confidence Interval = p̂ ± Margin of Error = 0.48 ± 0.196

   The 95% confidence interval is approximately (0.284, 0.676).

**c) Sample Size for Mean Absences:**

1. **Use the sample size formula:**
   n = (z * σ / E)²

   Where:
   * z is the critical z-value for 95% confidence (1.96)
   * σ is the estimated population standard deviation (4.5 days)
   * E is the desired margin of error (1.5 days)

2. **Calculate:**
   n = (1.96 * 4.5 / 1.5)² = (5.88)² = 34.57

   Since you can't have a fraction of a worker, round up to the nearest whole number.  Therefore, a sample size of 35 workers is needed.

**d) Sample Size for Population Proportion:**

1. **Use the sample size formula (when no prior estimate is available):**
   n = (z² * 0.25) / E²

   Where:
   * z is the critical z-value for 90% confidence (1.645)
   * E is the desired margin of error (0.075)

2. **Calculate:**
   n = (1.645² * 0.25) / 0.075² ≈ 120.278

   Round up to the nearest whole number. A sample size of 121 workers is needed.

**e) Sample Size for Single Survey:**

Since a single survey is being conducted, you need to choose the larger of the two sample sizes calculated in parts (c) and (d).  Therefore, a sample size of 121 workers is needed to satisfy both requirements.