A company has been studying the performance of their sales department. The measure of performance used is: average monthly sales amount in dollars. For the past 12 months, the average monthly sales amount has been $4,532 with a standard deviation of $386. Calculate the percentage of months that the average monthly sales amount exceeds $5,800.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score. Multiply by 100.
To calculate the percentage of months that the average monthly sales amount exceeds $5,800, we can use the z-score formula. The z-score is a measure of how many standard deviations a data point is away from the mean.
First, let's calculate the z-score for $5,800 using the formula:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for ($5,800)
- μ is the mean ($4,532)
- σ is the standard deviation ($386)
z = (5,800 - 4,532) / 386
z = 1,268 / 386
z ≈ 3.29
Now that we have the z-score, we can find the percentage of months that the average monthly sales amount exceeds $5,800 using a standard normal distribution table (also known as a z-table). The z-table gives us the area under the curve for a given z-score.
Since we want to find the percentage of months that exceed $5,800, we need to find the area to the right of the z-score on the z-table. This represents the percentage of months that have a higher average monthly sales amount.
Looking up the z-score of 3.29 in the z-table, we find the area to the right is approximately 0.9992. This means that approximately 99.92% of the months have an average monthly sales amount below $5,800.
To calculate the percentage that exceeds $5,800, we subtract this value from 1:
Percentage = 1 - 0.9992
Percentage ≈ 0.0008
Therefore, the percentage of months that the average monthly sales amount exceeds $5,800 is approximately 0.08%.