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EGR 140 – Lab 9: Statistics II
Topics to be covered :
• Probability Distributions
• Correlation
Practice :
•
The average life of a certain battery is 36 months with a standard deviation of 6
months. Assuming a normal distribution, what is the probability that these
batteries should be expected to last from 27 to 41 months?
EDU>> normcdf140(41,36,6)-normcdf140(27,36,6)
ans =
0.730864417767499
•
Write three (3) functions that will assist you in solving probability problems dealing
with normal distributions:
- probLT (a, µ, σ ) % will output P ( X < a )
- probGT (a, µ, σ ) % will output P ( X > a )
- probInBetween (a, b, µ, σ ) % will output P ( a < X < b )
•
The average life of a certain battery is 36 months with a standard deviation of 6
months. Assuming a normal distribution, what is the probability that these
batteries should be expected to last from 27 to 41 months?
EDU>> probInBetween(27,41,36,6)
ans =
0.730864417767499
•
A radar unit is used to measure speeds of cars on a motorway. The speeds are
normally distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr.
What is the probability that a car picked at random is travelling at more than 100
km/hr?
EDU>> probGT(100,90,10)
ans =
0.158655253931457
•
For a certain type of computers, the length of time bewteen charges of the battery is
normally distributed with a mean of 50 hours and a standard deviation of 15 hours.
What is the probability that the length of time will be between 50 and 70 hours?
EDU>> probInBetween(50,70,50,15)
ans =
0.408788780274132
Lab Exercises :
•
Answer the previous problem using simulation.
EDU>> probInBetweenSIM(50,70,50,15,1000000)
ans =
0.407640000000000
•
(Poisson) The New York Times reports that large meteorites strike our atmosphere with the
intensity of atomic bombs an average of eight times a year. What is the probability of no
such meteorite striking in one year? Of 5 strikes in 1 year? Of eight strikes in one year?
EDU>> poisscdf140(0,8)
ans =
3.354626279025119e-04
EDU>> poisscdf140(5,8)- poisscdf140(4,8)
ans =
0.091603661592579
EDU>> poisscdf140(8,8)- poisscdf140(7,8)
ans =
0.139586531950597
•
It is given that the lifetime of a mechanical assembly in a vibration test is exponentially
distributed with a mean of λ = 400 hours. Let X denote the time to fail. Find the probability
that an assembly fails in less than 100 hrs .
EDU>> expcdf140(100,400)
ans =
0.221199216928595
•
Investgate the correlation between temperature and the heat capacity of gaseous propane
Temperature
(K)
50
100
150
200
273.16
298.15
300
400
500
600
700
800
900
1000
1100
1200
1300
(kJ/kg-mol-K)
34.06
41.3
48.79
56.07
68.74
73.6
73.93
94.01
112.59
128.7
142.67
154.77
163.35
174.6
182.67
189.74
195.85
EDU>> x=[50 100 150 200 273.16 298.15 300 400 500 600 700 800 900 1000
1100 1200 1300];
EDU>> y=[34.06 41.3 48.79 56.07 68.74 73.6 73.93 94.01 112.59 128.7
142.67 154.77 163.35 174.6 182.67 189.74 195.85];
EDU>> corr140(x,y)
ans =
0.988914270552431
Assignment :
You are to submit a clean hard copy of the solutions to the following problems.
1. (12 points)
In the manufacture of certain light bulbs, it is found that 2% are defective.
Assuming a normal distribution for defectives in lots of 1,000 of these bulbs,
what is the probability that 15 will be defective if σ = 3?
2. (12 points)
The greater the sulfur content of coal, the less desirable it is as a heating fuel. Given that the
variability among assays for sulfur in coal from a certain mine is σ = 6 lb per ton, and that
they follow a normal distribution, answer the following:
a) Mines that assay 80 lb of sulfur per ton are considered worthless for heating fuel. How
likely is it that a mine with a mean sulfur content of µ = 62 lb per ton will be placed
in the worthless category?
b) Some cities will not permit the sale coal; within city limits if its assay for sulfur is as
great as 34 lb per ton. How likely is it that coal with µ = 40 lb per ton will be
allowed to be sold within city limits?
3. (12 points)
The probability that a cathode ray tube manufactured by a machine is defective is 0.4. If a
sample of sixty tubes is inspected, find the probability that
a) exactly half the tubes will be defective.
b) over one half the tubes will be defective
4. (12 points)
(Binomial) An inspection procedure at a manufacturing plant involves picking three items at
random and then accepting the whole lot if at least two of the three items are in perfect
condition. If in reality 90% of the whole lot are perfect, what is the probability that the lot
will be accepted? If 75% of the whole are perfect? If 60% are perfect?
5. (12 points)
(Exponential) The number of miles that a particular car can run before its battery wears out
is exponentially distributed with an average of 10,000 miles. The owner of the car needs to
take a 5000-mile trip. What is the probability that he will be able to complete the trip without
having to replace the car battery?
6. (14 points)
Answer questions 4 and 5 using simulated data and compare the results.
7. (14 points)
An engineer at an aluminum castings plant assesses the relationship between the hydrogen
content and the porosity of aluminum alloy castings, and the hydrogen content and the
strength of the castings. The engineer collects a random sample of 14 castings and measures
the following properties of each casting: hydrogen content, porosity, and strength.
Hydrogen P o r o s i t y
Strength
0.1800
0.3300
0.8393
0.2100
0.4100
1.1225
0.2100
0.4500
1.1131
0.2100
0.5500
1.1000
0.2200
0.4400
0.7071
0.2200
0.2400
0.4975
0.2300
0.4700
0.5300
0.2300
0.7000
0.5206
0.2400
0.8000
0.1929
0.2400
0.2200
0.5400
0.2500
0.8800
0.3909
0.2600
0.7200
0.4200
0.2700
0.7500
0.1835
0.2800
0.7000
0.2400
8. (14 points)
Here are the classic 1910 observations of Rutherford, Geiger, and Bateman for the number
of alpha particles emitted by a film of polonium, as observed over intervals of one-eighth of a
minute (7.5 seconds). The lefthand column gives the number of particles observed in one
such time unit, and the righthand column gives the number of units in which such an
observation was made:
From this table, calculate the rate of occurrence of some event, r, and for that rate, using the
Poisson formula, calculate the Expected (E) number of instances of each value. Then
compare Rutherford's Actual (A) values, and tell us what you think about the nature of the
process behind Rutherford's data. of units in which such an observation was made:
# observed in one unit
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 or more
# of such units
57
203
383
525
532
408
273
139
45
27
10
4
0
1
1
0