A salesperson has found that the probability of making various numbers of sales per day is presented in the following table. Calculate the expected number of sales per day and the standard deviation of the number of sales.
Number of sales(Probability)
1(0.05) 2(0.16) 3(0.20) 4(0.25) 5(0.18) 6(0.10) 7(0.04) 8(0.02)
Find the mean first = sum of scores/number of scores
Sum of scores = (.05) + 2(.16)... + 8(.02) = ?
Number of scores = 1 + 2 + 3... + 8
Subtract each of the scores (.05, .16, .16, .2, .2, .2....) from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
μ = ∑[x * P (x)]
μ = [(0*0.06) + (1*0.25) + (2*0.37) + (3*0.25) + (4*0.06)]
μ = 0 + 0.25 + 0.74 + 0.75 + 0.24
μ = 1.98
σ2 = ∑[x2 * P(x)] – μ2
σ2 = [(02*0.06) + (12*0.25) + (22*0.37) + (32*0.25) + (42*0.06)] – 1.98
σ2 = [(0*0.06)+(1*0.25)+ (4*0.37) + (9*0.25) + (16*0.06)] – 3.92
σ2 = [0 + 0.25 + 1.48 + 2.25 + 0.96] – 3.92
σ2 = 4.94 - 3.92
σ2 = 1.02
σ = √σ2
σ = √1.02
σ = 1.009
To calculate the expected number of sales per day, you need to multiply each number of sales by its corresponding probability and then sum the results.
Expected number of sales per day = (1 x 0.05) + (2 x 0.16) + (3 x 0.20) + (4 x 0.25) + (5 x 0.18) + (6 x 0.10) + (7 x 0.04) + (8 x 0.02)
Calculating the above expression:
Expected number of sales per day = 0.05 + 0.32 + 0.60 + 1.00 + 0.90 + 0.60 + 0.28 + 0.16
Expected number of sales per day = 3.91
Therefore, the expected number of sales per day is approximately 3.91.
To calculate the standard deviation of the number of sales, you need to find the variance first. The variance is calculated as the sum of the squared differences between each number of sales and the expected number of sales, weighted by their respective probabilities.
Variance = (1-3.91)² x 0.05 + (2-3.91)² x 0.16 + (3-3.91)² x 0.20 + (4-3.91)² x 0.25 + (5-3.91)² x 0.18 + (6-3.91)² x 0.10 + (7-3.91)² x 0.04 + (8-3.91)² x 0.02
Calculating the above expression:
Variance = (-2.91)² x 0.05 + (-1.91)² x 0.16 + (-0.91)² x 0.20 + (0.09)² x 0.25 + (1.09)² x 0.18 + (2.09)² x 0.10 + (3.09)² x 0.04 + (4.09)² x 0.02
Variance = 0.846025 x 0.05 + 0.698881 x 0.16 + 0.8281 x 0.20 + 0.0081 x 0.25 + 1.1881 x 0.18 + 4.3681 x 0.10 + 9.5881 x 0.04 + 16.7681 x 0.02
Variance = 0.04230125 + 0.11182096 + 0.16562 + 0.002025 + 0.213858 + 0.43681 + 0.3835244 + 0.335362
Variance = 1.69052161
Next, to calculate the standard deviation, you take the square root of the variance.
Standard deviation = √(1.69052161)
Standard deviation ≈ 1.30
Therefore, the standard deviation of the number of sales is approximately 1.30.
To calculate the expected number of sales per day, you need to multiply each number of sales by its corresponding probability, and then sum up the results.
Expected number of sales = (1 * 0.05) + (2 * 0.16) + (3 * 0.20) + (4 * 0.25) + (5 * 0.18) + (6 * 0.10) + (7 * 0.04) + (8 * 0.02)
Expected number of sales = 0.05 + 0.32 + 0.60 + 1.00 + 0.90 + 0.60 + 0.28 + 0.16
Expected number of sales = 3.91
Therefore, the expected number of sales per day is 3.91.
To calculate the standard deviation of the number of sales, you need to use the formula:
Standard deviation = sqrt(sum[ (x_i - mean)^2 * P(x_i) ])
Where:
x_i = the number of sales
mean = the expected number of sales
P(x_i) = the probability of x_i
First, calculate the squared difference for each x_i:
(1 - 3.91)^2 * 0.05
(2 - 3.91)^2 * 0.16
(3 - 3.91)^2 * 0.20
(4 - 3.91)^2 * 0.25
(5 - 3.91)^2 * 0.18
(6 - 3.91)^2 * 0.10
(7 - 3.91)^2 * 0.04
(8 - 3.91)^2 * 0.02
Next, sum up all the squared differences multiplied by their corresponding probabilities.
Standard deviation = sqrt((1 - 3.91)^2 * 0.05 + (2 - 3.91)^2 * 0.16 + (3 - 3.91)^2 * 0.20 + (4 - 3.91)^2 * 0.25 + (5 - 3.91)^2 * 0.18 + (6 - 3.91)^2 * 0.10 + (7 - 3.91)^2 * 0.04 + (8 - 3.91)^2 * 0.02)
After performing the calculations, the standard deviation of the number of sales can be determined.