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Difficulty: MediumData Distributions and Measures

A shipping company records the weights, in pounds, of 1010 packages in a delivery batch. The table below shows the distribution of the weights.

Weight (pounds)Frequency
4433
6633
8833
262611

If the heaviest package is removed from the batch, by how much will the mean weight of the remaining packages decrease, in pounds?

Answer: 2 pounds

Answer

The mean weight of the remaining packages will decrease by 22 pounds.
The original mean weight of the 1010 packages is calculated by finding the total weight and dividing it by 1010. The total weight is (3×4)+(3×6)+(3×8)+(1×26)=12+18+24+26=80(3 \times 4) + (3 \times 6) + (3 \times 8) + (1 \times 26) = 12 + 18 + 24 + 26 = 80 pounds, so the initial mean is 8010=8\frac{80}{10} = 8 pounds. After removing the heaviest package of 2626 pounds, the remaining 99 packages have a total weight of 80−26=5480 - 26 = 54 pounds. The new mean weight is 549=6\frac{54}{9} = 6 pounds. The decrease in the mean weight is 8−6=28 - 6 = 2 pounds.

Step-by-Step Solution

1
Calculate the total weight of the 1010 packages.
8080 pounds
By summing the products of each weight and its corresponding frequency.
2
Calculate the initial mean weight.
88 pounds
By dividing the total weight of 8080 pounds by the total count of 1010 packages.
3
Calculate the sum of the weights of the remaining packages after removing the heaviest package.
5454 pounds
By subtracting the weight of the heaviest package (2626 pounds) from the original total weight (8080 pounds).
4
Calculate the new mean weight of the remaining 99 packages.
66 pounds
By dividing the remaining weight (5454 pounds) by the new count of packages (99).
5
Find the decrease in the mean weight.
22 pounds
By subtracting the new mean weight (66 pounds) from the original mean weight (88 pounds).

Key Concept

Effect of outlier removal on the mean of a data distribution
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Topics

Data Distributions and Measures

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