A. To find the probability a financial manager earns between $30 and $35 per hour, we need to find the standard score (z-score) for each of these values and use the standard normal distribution table to find the corresponding probabilities. The z-score is calculated as follows:
z = (x - mean) / standard deviation
For $30 per hour: z = (30 - 32.62) / 2.32 = -1.05
For $35 per hour: z = (35 - 32.62) / 2.32 = 1.05
Using a standard normal distribution table, we can find that the probability of a financial manager earning between $30 and $35 per hour is the area under the curve between the z-scores of -1.05 and 1.05. This is approximately 0.8413.
B. To find the hourly rate that puts a financial manager in the top 10% with respect to pay, we need to find the z-score that corresponds to a cumulative probability of 0.90. We can use the standard normal distribution table to find the z-score and then convert it back to the hourly pay rate using the formula:
x = mean + z * standard deviation
Using the standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 0.90 is approximately 1.28. So,
x = 32.62 + 1.28 * 2.32 = $37.49
C. To find the probability a financial manager earns less than $28 per hour, we need to find the standard score for $28 and then use the standard normal distribution table to find the cumulative probability. The z-score is calculated as follows:
z = (x - mean) / standard deviation
z = (28 - 32.62) / 2.32 = -2.02
Using a standard normal distribution table, we can find that the cumulative probability for a z-score of -2.02 is approximately 0.0228. So,
the probability a financial manager earns less than $28 per hour is approximately 0.0228