Professor's Salaries the average salary for a Queens College full professor is $85,900. If the average salaries are normally distributed with a standard deviation of $11,000, find these probabilities
To find the probabilities, we need to use the concept of the standard normal distribution and the z-score. The z-score is a way to measure how many standard deviations an observation or value is away from the mean.
In this case, we want to find the probability for different salary ranges based on the given mean and standard deviation.
To find the probability that a professor's salary is below a certain value, we can calculate the z-score using the formula:
z = (x - μ) / σ
where x is the given salary, μ is the mean salary, and σ is the standard deviation.
We can then look up the corresponding probability in the standard normal distribution table.
1. Probability of a salary less than $75,000:
First, calculate the z-score:
z = ($75,000 - $85,900) / $11,000
Once you find the z-score, you can use the standard normal distribution table or a statistical calculator to find the probability.
2. Probability of a salary between $80,000 and $90,000:
Calculate the z-score for both values:
z1 = ($80,000 - $85,900) / $11,000
z2 = ($90,000 - $85,900) / $11,000
Then, subtract the probability of the lower value from the probability of the higher value to get the probability of being in that range.
3. Probability of a salary greater than $100,000:
Calculate the z-score:
z = ($100,000 - $85,900) / $11,000
Then, subtract the probability of being below the given value from 1 to get the probability of being greater.
By following these steps, you can find the probabilities you are looking for based on the given mean and standard deviation.