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Difficulty: MediumMeasures of Central Tendency

A structural mechanic is testing the tensile strength of a batch of aluminum rivets. The mean tensile strength of the first 44 rivets tested is 410410 MPa. If the quality standard requires the mean strength of 55 rivets to be exactly 415415 MPa, what tensile strength must the 55 th rivet have?

Answer: 435 MPa

Answer

The 5th rivet must have a tensile strength of 435 MPa.
The correct answer is derived by finding the total required sum for 5 rivets (2075) and subtracting the existing sum of the first 4 rivets (1640).

Step-by-Step Solution

1
Calculate the total tensile strength of the first 4 rivets.
1640 MPa
The sum of a data set is equal to its mean multiplied by the number of values.
2
Calculate the required total tensile strength for all 5 rivets.
2075 MPa
To achieve a target mean of 415 MPa with 5 items, the overall sum must equal the new mean multiplied by the new count.
3
Find the difference between the target total and the current total.
435 MPa
Subtracting the sum of the first 4 items from the required sum of all 5 items isolates the value needed for the 5th item.

Key Concept

Calculating a missing data value given an initial mean and a target mean.

Alternative Method

Use the deviation method: The first 4 rivets have a mean of 410 MPa, which is 5 MPa below the target mean of 415 MPa. This creates a total deficit of 4 × 5 = 20 MPa. The 5th rivet must provide enough strength to cover its own requirement (415 MPa) plus the 20 MPa deficit from the others. Therefore, 415 + 20 = 435 MPa.
Estimated Time:1m 0s
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Topics

Measures of Central Tendency

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