Three textbooks have a mean weight of 17.5 pounds, with a standard deviation of 1.68 pounds. A cardboard shipping box has a mean weight of 0.64 pounds with a standard deviation of 0.12 pounds. What are the mean and standard deviation of the weight of a shipping box packed with three textbooks?
Mean = [?]
S.D. = $\square$
Round to the nearest hundredth.
Answer (2)
Calculate the mean weight by summing the mean weight of the textbooks and the box: 17.5 + 0.64 = 18.14 .
Calculate the variance of the total weight by summing the variances of the textbooks and the box: ( 1.68 ) 2 + ( 0.12 ) 2 = 2.8224 + 0.0144 = 2.8368 .
Calculate the standard deviation by taking the square root of the total variance: 2.8368 ≈ 1.68428 .
Round the mean and standard deviation to the nearest hundredth: Mean = 18.14 pounds, S.D. = 1.68 pounds. M e an = 18.14 , S . D . = 1.68 .
Explanation
Understand the problem and provided data Let X be the weight of the three textbooks and Y be the weight of the shipping box. We are given the following information:
Mean weight of the textbooks, E [ X ] = 17.5 pounds
Standard deviation of the textbooks, S D ( X ) = 1.68 pounds
Mean weight of the shipping box, E [ Y ] = 0.64 pounds
Standard deviation of the shipping box, S D ( Y ) = 0.12 pounds
We want to find the mean and standard deviation of the total weight, which is the weight of the textbooks plus the weight of the shipping box.
Calculate the mean The mean weight of the shipping box packed with three textbooks is the sum of the mean weights of the textbooks and the shipping box:
E [ X + Y ] = E [ X ] + E [ Y ] = 17.5 + 0.64 = 18.14
So, the mean weight of the packed box is 18.14 pounds.
Calculate the standard deviation The standard deviation of the weight of the shipping box packed with three textbooks is calculated assuming the weights of the textbooks and the shipping box are independent. Therefore, the variance of the sum is the sum of the variances:
Va r ( X + Y ) = Va r ( X ) + Va r ( Y )
We know that S D ( X ) = 1.68 and S D ( Y ) = 0.12 . The variance is the square of the standard deviation, so:
Va r ( X ) = ( 1.68 ) 2 = 2.8224 Va r ( Y ) = ( 0.12 ) 2 = 0.0144
Therefore,
Va r ( X + Y ) = 2.8224 + 0.0144 = 2.8368
The standard deviation is the square root of the variance:
S D ( X + Y ) = Va r ( X + Y ) = 2.8368 ≈ 1.68428
Rounding to the nearest hundredth, the standard deviation is 1.68 pounds.
State the final answer Therefore, the mean weight of a shipping box packed with three textbooks is 18.14 pounds, and the standard deviation is approximately 1.68 pounds.
Examples
This type of calculation is useful in logistics and shipping, where it's important to estimate the total weight and variability of shipments for planning and cost estimation. For example, a company might need to estimate the total weight of a shipment of books to determine the shipping costs. By knowing the mean and standard deviation of the weight of individual books and boxes, they can estimate the mean and standard deviation of the total shipment weight, which helps them to budget for shipping costs and plan logistics effectively. If a bookstore is shipping multiple boxes of books, understanding the mean and standard deviation helps them predict overall shipping costs and manage inventory efficiently.
The mean weight of a shipping box packed with three textbooks is 18.14 pounds, and the standard deviation is approximately 1.68 pounds.
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