Question 1170719
Let's break down this problem step by step.

**Given:**

* Mean (μ) = $16,000
* Standard Deviation (σ) = $800
* Normal distribution

**(a) Probability that salary is higher than $18,000**

1.  **Calculate the z-score:**
    * z = (x - μ) / σ
    * z = (18000 - 16000) / 800
    * z = 2000 / 800
    * z = 2.5

2.  **Find the probability:**
    * We want to find P(X > 18000), which is the same as P(Z > 2.5).
    * Using a z-table or calculator, we find P(Z < 2.5) ≈ 0.9938.
    * P(Z > 2.5) = 1 - P(Z < 2.5) = 1 - 0.9938 = 0.0062

3.  **Answer:**
    * The probability that the salary is higher than $18,000 is approximately 0.0062 or 0.62%.

**(b) Find t such that P(X < t) = 0.80**

1.  **Find the z-score:**
    * We need to find the z-score that corresponds to a cumulative probability of 0.80.
    * Using a z-table or calculator, we find z ≈ 0.84.

2.  **Use the z-score formula:**
    * z = (t - μ) / σ
    * 0.84 = (t - 16000) / 800

3.  **Solve for t:**
    * t - 16000 = 0.84 * 800
    * t - 16000 = 672
    * t = 16000 + 672
    * t = 16672

4.  **Answer:**
    * The value of t is $16,672.

**(c) Mean salary of 20 employees and claim of 20% > $16,200**

1.  **Distribution of Sample Means:**
    * The distribution of sample means (x̄) is also normal with:
        * Mean (μ_x̄) = μ = $16,000
        * Standard Deviation (σ_x̄) = σ / √n = 800 / √20 ≈ 178.89

2.  **Calculate the z-score:**
    * z = (x̄ - μ_x̄) / σ_x̄
    * z = (16200 - 16000) / 178.89
    * z = 200 / 178.89
    * z ≈ 1.12

3.  **Find the probability:**
    * We want to find P(x̄ > 16200), which is the same as P(Z > 1.12).
    * Using a z-table or calculator, we find P(Z < 1.12) ≈ 0.8686.
    * P(Z > 1.12) = 1 - P(Z < 1.12) = 1 - 0.8686 = 0.1314.

4.  **Convert to Percentage:**
    * 0.1314 * 100% = 13.14%

5.  **Compare to Manager's Claim:**
    * The manager claims that over 20% of the mean salaries are higher than $16,200.
    * We found that approximately 13.14% are higher than $16,200.

6.  **Answer:**
    * No, we do not agree with the manager's claim. The calculated percentage (13.14%) is less than 20%.