Final answer:
To find the probability of a starting salary of at least $32,000, calculate the z-score, find the corresponding area under the normal distribution curve, and subtract from 1. The percentage of employees with salaries less than $11,200 can be found by calculating the z-score and finding the corresponding area. The minimum and maximum salaries of the middle 95% of employees can be calculated using the z-scores. To find the total number of employees, use the z-score formula and solve for the total number of employees.
Step-by-step explanation:
(a) To find the probability that a randomly selected employee will have a starting salary of at least $32,000, we need to calculate the z-score for $32,000 and then find the corresponding area under the standard normal distribution curve. The z-score is calculated as (x - mean) / standard deviation, so in this case, it is (32000 - 25000) / 5000 = 1.4. Using a standard normal table or technology, we find that the area to the left of 1.4 is approximately 0.9192. Since we are interested in the probability of having a salary of at least $32,000, we subtract this value from 1: 1 - 0.9192 = 0.0808. Therefore, the probability is approximately 0.0808 or 8.08%.
(b) To find the percentage of employees with salaries less than $11,200, we can again calculate the z-score for $11,200 and find the corresponding area to the left under the standard normal distribution curve. The z-score is (11200 - 25000) / 5000 = -2.76. Using a standard normal table or technology, we find that the area to the left of -2.76 is approximately 0.0026. Therefore, the percentage of employees with salaries less than $11,200 is approximately 0.26%.
(c) The middle 95% of employees' salaries can be found by calculating the z-scores for the two ends of the middle 95% and then finding the corresponding salaries. Since the middle 95% is symmetric around the mean, we need to find the z-score that leaves 2.5% in each tail. Using a standard normal table or technology, we find that the z-score for 2.5% in one tail is approximately -1.96. So, we can calculate the minimum and maximum salaries as follows: Minimum salary = mean + (z-score * standard deviation) = 25000 + (-1.96 * 5000) ≈ $15,200 and Maximum salary = mean + (z-score * standard deviation) = 25000 + (1.96 * 5000) ≈ $34,800.
(d) To find the total number of employees in the corporation, we can use the z-score formula to find the z-score for a salary of $35,600 and then calculate the number of employees corresponding to that z-score. The z-score is (35600 - 25000) / 5000 = 2.12. Using a standard normal table or technology, we find that the area to the left of 2.12 is approximately 0.9832. Since we are interested in the 51 employees with salaries of at least $35,600, we can multiply the area by the total number of employees: 0.9832 * Total number of employees = 51. Solving for the total number of employees, we get Total number of employees ≈ 51 / 0.9832 ≈ 51.92. Therefore, there are approximately 52 employees in the corporation.