A quality control engineer measured the lengths of precision bolts manufactured across fiv… — ACT

A quality control engineer measured the lengths of precision bolts manufactured across five production lines. The table below shows the number of bolts tested from each line and the mean length, in millimeters, of the bolts from that line.

Production Line	Number of Bolts Tested	Mean Length (mm)
Line 1	$15$	$24.8$
Line 2	$25$	$25.2$
Line 3	$20$	$24.6$
Line 4	$10$	$25.4$
Line 5	$x$	$25.6$

If the overall mean length of all the bolts tested across all five lines is $25.1$ millimeters, what is the value of $x$ , the number of bolts tested from Line 5?

A
$9$
B
$15$
$18$ Answer
D
$25$
E
$30$

Answer

To find the number of bolts

x

tested from Line 5, compute the sum of the bolt lengths for the first four lines by multiplying each count by its respective mean:

(15 \times 24.8) + (25 \times 25.2) + (20 \times 24.6) + (10 \times 25.4) = 372 + 630 + 492 + 254 = 1748

mm. The total number of bolts across all five lines is

15 + 25 + 20 + 10 + x = 70 + x

. Setting the weighted average equal to the target overall mean of

25.1

yields the equation

\frac{1748 + 25.6x}{70 + x} = 25.1

. Multiplying both sides by

(70 + x)

gives

1748 + 25.6x = 1757 + 25.1x

. Subtracting

25.1x

and

1748

from both sides gives

0.5x = 9

, which simplifies to

x = 18

Step-by-Step Solution

Calculate the total sum of lengths for Lines 1 through 4.

Line 1:

15 \times 24.8 = 372

; Line 2:

25 \times 25.2 = 630

; Line 3:

20 \times 24.6 = 492

; Line 4:

10 \times 25.4 = 254

. Total sum for Lines 1-4 =

372 + 630 + 492 + 254 = 1748

mm.

The total length of bolts in a line equals the line count multiplied by its mean length.

Determine the total count of bolts tested across all five lines.

Total count =

15 + 25 + 20 + 10 + x = 70 + x

Adding the number of bolts from each of the 5 production lines gives the overall sample size.

Set up the overall weighted mean equation and solve for

x

\frac{1748 + 25.6x}{70 + x} = 25.1 \implies 1748 + 25.6x = 25.1(70 + x) \implies 1748 + 25.6x = 1757 + 25.1x \implies 0.5x = 9 \implies x = 18

Overall mean is equal to the total sum of all bolt lengths divided by the overall number of bolts.

Key Concept

Weighted Mean and Combined Data Sets

Estimated Time:2m 0s

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