Based on historical data, sales for a particular cosmetic li…

Based on historical data, sales for a particular cosmetic line follow a continuous uniform distribution with a lower limit of $2,500 and an upper limit of $5,000. A.  What   are the mean and standard deviation of this uniform distribution? B.  What is the probability that   sales exceed $4,000?

Answer

A. In order to determine the mean and standard deviation of the sales for the given cosmetic line, we need to use the properties of a continuous uniform distribution. The continuous uniform distribution is a probability distribution where all outcomes between two values are equally likely.

Given that the lower limit (a) is $2,500 and the upper limit (b) is $5,000, we can calculate the mean (μ) and standard deviation (σ) using the formulas:

Mean (μ) = (a + b) / 2

Standard Deviation (σ) = (b – a) / √12

Applying these formulas to the given data, we have:

Mean (μ) = ($2,500 + $5,000) / 2 = $3,750

Standard Deviation (σ) = ($5,000 – $2,500) / √12 ≈ $721.11

Therefore, the mean and standard deviation of this uniform distribution are $3,750 and approximately $721.11, respectively.

B. To determine the probability that sales exceed $4,000, we need to consider the relative position of this value within the given uniform distribution. Since the distribution is continuous and uniform, the probability is equal to the ratio of the segment of the distribution that exceeds $4,000 to the total range of the distribution.

The total range of the distribution is the difference between the upper and lower limits, which in this case is $5,000 – $2,500 = $2,500.

To calculate the probability, we divide the segment of the distribution that exceeds $4,000 by the total range. The segment that exceeds $4,000 is equal to the upper limit ($5,000) minus $4,000, which is $5,000 – $4,000 = $1,000.

Probability = Segment that exceeds $4,000 / Total range

Probability = $1,000 / $2,500 = 0.4 or 40%

Therefore, the probability that sales exceed $4,000 is 40%.

In summary, the mean (μ) and standard deviation (σ) of the sales for the given cosmetic line are $3,750 and approximately $721.11, respectively. Additionally, the probability that sales exceed $4,000 is 40%.