A materials scientist is analyzing how the force needed to bend a particular alloy…

MATHEMATICS · Multi-part · AP · AP Statistics · Exploring Two-Variable Data · Exploring Two-Variable Data

Problem

A materials scientist is analyzing how the force

F

needed to bend a particular alloy strip depends on its thickness

x

(in millimeters). For thin strips the data suggest a model of the form

F=m x

, where

m

(newtons per millimeter) is an unknown proportionality constant. Three experimental measurements are shown; because of differing equipment precision, each measurement carries a weight

w_i

indicating its relative reliability.

Measurement $i$	Thickness $x_i$ (mm)	Force $F_i$ (N)	Weight $w_i$
1	0.80	23.6	3
2	1.10	31.9	2
3	1.40	40.1	5

Your task is to determine the value of

m

that best aligns the linear model with the weighted data. (a) Write an explicit expression

E(m)

for the weighted sum of squared differences between the model prediction

m x_i

and the observed forces

F_i

. (b) Obtain

E'(m)

and set it equal to

0

to derive an equation satisfied by the optimal

m

. Show all algebraic steps. (c) Solve the equation from part (b) for the numerical value of

m

(units required). (d) Using your value of

m

, predict the force required to bend a strip of thickness

x=1.25\text{ mm}

. Include units and comment on whether this prediction is likely to be reliable based on the given weights.

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