Download SMAM 314 11AM Exam 1 Name_________ 1. A - RIT

SMAM 314 11AM
Exam 1
Name_________
1. A computer programmer is asked to code a standard function and the
time in minutes is recorded for 12 programmers. They are
18 14 19 11 23 21 10 13 19 24 15 20
A. Complete the stem and leaf display (7 points)
1
1
1
1
1
2
2
2
1 0
3
4 5
8 9 9
1 0
3
4
B. Make a five number summary (7 points)
18.5
20
13.5
24
10
C. Make a boxplot
(7 points)
10 11 12 13 14 15 16 17 18 19 20 21 22 23
24
2. An important characteristic of water is the concentration of suspended
solid material in mg./l . Six measurements on suspended solids from a
certain lake are as follows
57.0 68.7 67.3 76.3 54.3 54
Use your calculator to obtain the mean and the standard deviation. (6
points)
x = 62.93,s = 9.17
3. Let X be a random variable with the probability
 91 x2
f(x) = 
 0
0< x < 3
esewhere
Find
A. P( .5 < X <2.4)
(7 points)
P(.5 < X < 2.4) = ∫
2.4
.5
=
1
27
1
9
x 2dx =
1
27
2.4
x 3 .5
(2.4 − .5 ) = .507
3
3
B. Find the mean and the variance of X. (12 points)
EX =
∫
x dx =
1 3
0 9
EX 2 =
σ2 =
3
∫
3
1
36
x dx =
1 4
0 9
3
9
= 2.25
4
243 27
=
=
45
5
x4 0 =
1
45
3
x5 0
27 81 27
−
=
= .3375
5 16 80
density function
4. The time to failure in thousands of hours of a machine part is a
random variable with cumulative distribution function
F(x) = 1− e− x / 4 ,x > 0
A. What is the probability that the part lasts at least 4000 hours?
(7 points)
F(x) = 1− e− x / 4 ,x > 0
P(X > 4) = 1− (1− e− 1) = .368
B. What is the probability density function of X? (7 points)
f(x) =
1 −x/4
e ,x > 0
4
C. What is the median time to failure? (7 points)
F(x) = 1− e− x / 4 = .5
e − x/4 = .5
x
− = ln.5 = − ln2
4
x = 4ln2 = 2.772
5. The reaction time of a driver to a visual stimulus is normally
distributed with a mean of 0.4 second and a standard deviation of 0.05
second.
A. What is the probability that a reaction requires at least 0.5 seconds?
(7 points)
.5 −.4 

P(X > .5) = P Z >
= P(Z > 2) = .0228

.05 
B. What is the probability that a reaction requires between 0.25 and 0.45
seconds? (7 points)
.45 −.4 
 .25 − .4
P(.25 < X < .45) = P
<Z<
 .05
.05 
= P(−3 < Z < 1) = .8413 −.0014 = .8399
C. What is the reaction time that is exceeded 85% of the time? (7 points)
c −.4 
P(X > c) = P Z >
= .85

.05 
c − .4 
P Z ≤
= .15

.05 
c − .4
= −1.04
.05
c = .4 −1.04(.05) = .348
6. The distribution of resistance for resistors of a certain type is known to
be normal with 10% of all resistors having a resistance exceeding 10.256
ohms and 5% having a resistance smaller than 9.671 ohms . What are the
mean value and the standard deviation of the resistance distribution? (12
points)
P(X > 10.256) = .1
P(X < 9.671) = .05
10.256 −µ 

P Z>
= .1


σ
9.671− µ 

P Z<
= .05


σ
10.256 −µ
= 1.28
σ
9.671−µ
= −1.645
σ
µ +1.28σ = 10.256
µ −1.645σ = 9.671
2.925σ = .585
σ = .2
µ −1.645(.2) = 9.671
µ = 9.671+ 1.645(.2) = 10