Q: The mean annual salary for a sample of electrical engineers is $83,500, with a standard deviation
The mean annual salary for a sample of electrical engineers is $83,500, with a standard deviation of $1500. The data set has a bell-shaped distribution.
(a) Use the Empirical Rule to estimate the percent of electrical engineers whose annual salaries are between $80,500 and $86,500.
(b) In a sample of 40 additional electrical engineers, about how many electrical engineers would you expect to have annual salaries between $80,500 and $86,500?
(c) The salaries of three randomly selected electrical engineers are $90,500, $79,750, and $82,600. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.
(a) To estimate the percent of electrical engineers whose annual salaries are between $80,500 and $86,500 using the Empirical Rule, we need to calculate the z-scores for these values first.
The z-score formula is given by:
z = (x - μ) / σ
Where:
z is the z-score
x is the value
μ is the mean of the distribution
σ is the standard deviation of the distribution
Let's calculate the z-scores:
For $80,500:
z1 = ($80,500 - $83,500) / $1,500 = -2
For $86,500:
z2 = ($86,500 - $83,500) / $1,500 = 2
Now, we can use the Empirical Rule:
Approximately 68% of the data falls within 1 standard deviation (z-score) of the mean.
Approximately 95% of the data falls within 2 standard deviations (z-scores) of the mean.
Approximately 99.7% of the data falls within 3 standard deviations (z-scores) of the mean.
Since the salaries $80,500 and $86,500 fall within 1 standard deviation of the mean, we can estimate that approximately 68% of electrical engineers' annual salaries are between $80,500 and $86,500.
(b) To estimate the number of electrical engineers who would have annual salaries between $80,500 and $86,500 in a sample of 40 additional engineers, we can assume that the distribution remains the same.
Since the percentage of salaries that fall within the range $80,500 - $86,500 is approximately 68%, we can estimate that 68% of 40 (or 0.68 * 40) additional electrical engineers would have salaries within this range.
0.68 * 40 = 27.2
Therefore, we would expect around 27 electrical engineers to have annual salaries between $80,500 and $86,500 in a sample of 40 additional electrical engineers.
(c) To find the z-scores for the given salaries, we can use the z-score formula mentioned above:
For $90,500:
z1 = ($90,500 - $83,500) / $1,500 = 4.67
For $79,750:
z2 = ($79,750 - $83,500) / $1,500 = -2.5
For $82,600:
z3 = ($82,600 - $83,500) / $1,500 = -0.6
To determine whether these salaries are unusual, we can consider the general rule that if the z-score is greater than 2 or less than -2, the corresponding value is considered unusual or an outlier.
Based on this rule:
$90,500 has a z-score of 4.67, which is greater than 2. Therefore, it can be considered an unusual salary.
$79,750 has a z-score of -2.5, which is less than -2. Therefore, it can also be considered an unusual salary.
$82,600 has a z-score of -0.6, which is within the range of -2 to 2. Therefore, it is not considered an unusual salary.
To answer these questions, we can use the concept of z-scores, which measure the number of standard deviations a data point is from the mean. We can then use the Empirical Rule, also known as the 68-95-99.7 rule, to estimate the percentage of data within a certain range.
(a) To estimate the percent of electrical engineers whose annual salaries are between $80,500 and $86,500, we first need to calculate the z-scores for these two salaries.
Z-score formula: Z = (X - μ) / σ
Where:
- X is the value we want to convert to a z-score (salary in this case)
- μ is the mean of the dataset (mean annual salary)
- σ is the standard deviation of the dataset
For $80,500:
Z1 = ($80,500 - $83,500) / $1500
For $86,500:
Z2 = ($86,500 - $83,500) / $1500
Once we have the z-scores, we can use the Empirical Rule to estimate the percentage of data within a certain range. According to the Empirical Rule:
- Approximately 68% of the data falls within one standard deviation of the mean (between Z = -1 and Z = 1).
- Approximately 95% of the data falls within two standard deviations of the mean (between Z = -2 and Z = 2).
- Approximately 99.7% of the data falls within three standard deviations of the mean (between Z = -3 and Z = 3).
Since the salaries in the question follow a bell-shaped distribution, we can estimate that the percentage of electrical engineers whose annual salaries are between $80,500 and $86,500 is approximately 68%.
(b) To estimate how many electrical engineers would have annual salaries between $80,500 and $86,500 in a sample of 40 additional engineers, we can multiply the estimated percentage from part (a) by the sample size:
Estimated number = Percentage * Sample size
Estimated number = 0.68 * 40 = 27.2
Therefore, we would expect approximately 27 additional electrical engineers to have annual salaries between $80,500 and $86,500 in the sample of 40.
(c) For the salaries of $90,500, $79,750, and $82,600, we can find the z-scores using the same formula as in part (a).
Z1 = ($90,500 - $83,500) / $1500
Z2 = ($79,750 - $83,500) / $1500
Z3 = ($82,600 - $83,500) / $1500
After calculating the z-scores, we can compare them to determine if any of the salaries are unusual. Typically, a z-score greater than 3 or less than -3 is considered unusual. If any of the calculated z-scores fall outside of this range, then the corresponding salaries would be considered unusual.