Question

The time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes.

a.	What is the probability that a guitar neck can be carved between 95 and 165 minutes?
b.	What is the probability that the guitar neck can be carved between 120 and 200 minutes?
c.	Determine the expected completion time for carving the guitar neck.
d.	Compute the standard deviation.

Answer 1

Consider, X= The time it takes to hand carve a guitar neck

$X\sim U(a=110,b=190)$

$f(x)=\frac{1}{b-a}=\frac{1}{190-110}=\frac{1}{80}$ for $110\leq x\leq 190$

=0 otherwise.

a)  the probability that a guitar neck can be carved between 95 and 165 minutes

$\int_{110}^{165}\frac{1}{80}dx=\frac{1}{80}\int_{110}^{165}dx=\frac{165-110}{80}=0.6875$

b) the probability that the guitar neck can be carved between 120 and 200 minutes

$\int_{120}^{190}\frac{1}{80}dx=\frac{1}{80}\int_{120}^{190}dx=\frac{190-120}{80}=0.875$

c) the expected completion time for carving the guitar neck.

$\frac{b+a}{2}=\frac{190+110}{2}=150$

d)  the standard deviation.

$=\sqrt{\frac{(b-a)^2}{12}}$

$=\sqrt{\frac{(190-110)^2}{12}}$

=23.094