Question 1170718
Let's solve this problem step-by-step.

**Given:**

* Sample size (n) = 150
* Sample mean (x̄) = 6 kg
* Population standard deviation (σ) = 0.8 kg

**(a) Construct a 95% Confidence Interval**

1.  **Find the critical z-value (z_c):**
    * For a 95% confidence interval, z_c = 1.96 (from a z-table or calculator).

2.  **Calculate the margin of error (E):**
    * E = z_c * (σ / √n)
    * E = 1.96 * (0.8 / √150)
    * E ≈ 1.96 * (0.8 / 12.247)
    * E ≈ 1.96 * 0.0653
    * E ≈ 0.128 kg

3.  **Construct the confidence interval:**
    * Confidence Interval = x̄ ± E
    * Confidence Interval = 6 ± 0.128
    * Confidence Interval = (6 - 0.128, 6 + 0.128)
    * Confidence Interval = (5.872, 6.128) kg

4.  **Answer:**
    * The 95% confidence interval for the population mean weight of apples is (5.872, 6.128) kg.

**(b) Find the sample size for a 98% Confidence Interval**

1.  **Given Confidence Interval:**
    * (5.8835, 6.1165) kg

2.  **Calculate the margin of error (E):**
    * E = (Upper Bound - Lower Bound) / 2
    * E = (6.1165 - 5.8835) / 2
    * E = 0.233 / 2
    * E = 0.1165 kg

3.  **Find the critical z-value (z_c):**
    * For a 98% confidence interval, z_c ≈ 2.33 (from a z-table or calculator).

4.  **Use the margin of error formula to solve for n:**
    * E = z_c * (σ / √n)
    * 0.1165 = 2.33 * (0.8 / √n)
    * √n = (2.33 * 0.8) / 0.1165
    * √n = 1.864 / 0.1165
    * √n ≈ 16.00
    * n = (16.00)²
    * n = 256

5.  **Answer:**
    * The sample size should be 256 apples.