Dee's store monthly sales for April, May, June are three independent random variables with means and variances as follow:
April:
Mean: $9500
Variance: $1250
May:
Mean: $7400
Variance: $1425
June:
Mean: $8600
Variance: $1610
a) find the mean and standard deviation of the total sales for the 2nd quarter (3 month period).
-for mean, do you just add the means together and for stdev, do you just add the variance together and square root it?
b) If the store manager is paid by a base monthly salary of $2000 plus 10% of the monthly sales as commission, what is the mean and standard deviation of his monthly salary for May?
-how do you do this part?
a) To find the mean and standard deviation of the total sales for the 2nd quarter (3-month period), you are correct in adding the means together and adding the variances together before taking the square root.
Let's calculate the mean first:
Mean of April + Mean of May + Mean of June = $9500 + $7400 + $8600 = $25,500
Now, let's calculate the variance:
Variance of April + Variance of May + Variance of June = $1250 + $1425 + $1610 = $4285
To get the standard deviation, we take the square root of the variance:
Standard Deviation = √4285 ≈ $65.5
Therefore, the mean of the total sales for the 2nd quarter is $25,500, and the standard deviation is approximately $65.5.
b) To find the mean and standard deviation of the store manager's monthly salary for May, we need to consider both the base monthly salary and the commission based on the monthly sales.
The base monthly salary is $2000.
Now, let's calculate the commission based on the monthly sales for May:
Commission = 10% × Sales for May = 10% × $7400 = $740
To find the mean, we add the base monthly salary and the commission:
Mean Salary for May = Base Monthly Salary + Commission = $2000 + $740 = $2740
As for the standard deviation, since the base monthly salary does not have any variability, it does not contribute to the standard deviation. Therefore, we only need to consider the standard deviation of the commission.
To find the standard deviation of the commission, we first need to calculate the variance of the commission. Since the commission is calculated based on a percentage of the sales, we can use the variance formula for a constant multiplied by a random variable:
Variance of the Commission = (Commission Percentage)^2 × Variance of Monthly Sales
= (0.1)^2 × $1425
= $14.25
To get the standard deviation, we take the square root of the variance:
Standard Deviation of Commission = √14.25 ≈ $3.77
Therefore, the mean of the store manager's monthly salary for May is $2740, and the standard deviation is approximately $3.77.