The mean weight of a pair of shoes is 575.00 g with a standard deviation of 32.00 g. The mean weight of an empty shoe box for a pair of shoes is 68.00 g with a standard deviation of 9.10 g. What are the mean and standard deviation of the weight of a shoe box with a pair of shoes inside?
Mean = $\square$ S.D. = $\square$
Round to the nearest hundredth.
Answer (2)
Calculate the mean of the total weight by summing the means of the shoes and the box: E [ X + Y ] = 575.00 + 68.00 = 643.00 .
Calculate the variance of the total weight by summing the variances of the shoes and the box: Va r [ X + Y ] = ( 32.00 ) 2 + ( 9.10 ) 2 = 1106.81 .
Calculate the standard deviation of the total weight by taking the square root of the variance: S D [ X + Y ] = 1106.81 ≈ 33.27 .
The mean weight is 643.00 g and the standard deviation is 33.27 g.
Explanation
Understand the problem and provided data Let's break down this problem step by step. We're given the mean and standard deviation of the weight of a pair of shoes and the mean and standard deviation of the weight of an empty shoe box. We want to find the mean and standard deviation of the total weight when the shoes are inside the box.
Define variables and given data Let X be the weight of the pair of shoes and Y be the weight of the empty shoe box. We are given the following:
E [ X ] = 575.00 g (mean weight of shoes)
S D [ X ] = 32.00 g (standard deviation of shoes)
E [ Y ] = 68.00 g (mean weight of the box)
S D [ Y ] = 9.10 g (standard deviation of the box)
We want to find the mean and standard deviation of X + Y , which is the weight of the shoe box with the shoes inside.
Calculate the mean of the total weight The mean of the total weight is simply the sum of the individual means:
E [ X + Y ] = E [ X ] + E [ Y ]
E [ X + Y ] = 575.00 + 68.00 = 643.00
So, the mean weight of the shoe box with the shoes inside is 643.00 g.
Calculate the variance of the total weight Since the weights of the shoes and the shoe box are independent, the variance of the total weight is the sum of the individual variances:
Va r [ X + Y ] = Va r [ X ] + Va r [ Y ]
We know that Va r [ X ] = ( S D [ X ] ) 2 and Va r [ Y ] = ( S D [ Y ] ) 2 , so:
Va r [ X + Y ] = ( 32.00 ) 2 + ( 9.10 ) 2
Va r [ X + Y ] = 1024 + 82.81 = 1106.81
Calculate the standard deviation of the total weight The standard deviation of the total weight is the square root of the variance:
S D [ X + Y ] = Va r [ X + Y ]
S D [ X + Y ] = 1106.81 ≈ 33.26875410952445
Rounding to the nearest hundredth, we get 33.27 g.
State the final answer Therefore, the mean weight of a shoe box with a pair of shoes inside is 643.00 g, and the standard deviation is 33.27 g.
Mean = 643.00 g S.D. = 33.27 g
Examples
This type of calculation is useful in logistics and shipping, where you need to estimate the total weight and variability of packaged items. For example, if a company ships many shoe boxes, knowing the mean and standard deviation of the total weight helps them plan for shipping costs and ensure that the packages do not exceed weight limits. This also helps in inventory management and warehouse organization.
The mean weight of a shoe box with a pair of shoes inside is 643.00 g. The standard deviation of the total weight is approximately 33.27 g. These calculations are derived from summing the means and variances of the individual items.
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