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# Midterm Review Statistics for STAT28002010 S2023

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Statistics for Engineers Midterm Review
1. In a lot of 10 components, 2 are sampled at random for inspection. Assume that
in fact exactly 2 of the 10 components in the lot are defective. Let X be the
number of sampled components that are defective.
a) Find the probability mass function of X.
b) Find the mean of X. (Ans: 0.4)
c) Find the standard deviation of X. (Ans: 0.533)
d) Find the proportion of at most one component being defective. (Ans: 0.978)
2. Let A and B be events with P(A)=0.3 and P(AUB)=0.7.
a) For what value of P(B) will A and B be mutually exclusive?(Ans: 0.4)
b) For what value of P(B) will A and B be independent?(Ans: 4/7)
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3. Suppose A, B, and C are independent events, and that P(A)=1/3, P(B)=2/5, and
P(C)=3/4.
a) What is P( A  C ) ?
Ans: 5/6
b) What is P( A  B) ?
Ans: 1/5
c) What is P( A  ( B  C ) ) ?
Ans: 8/15
4. Find the values of a and b for the given pdf if the expected value is 3/5. (Ans:𝑎 =
3/5, 𝑏 = 6/5)
2
0≤𝑥≤1
𝑓(𝑥) = { 𝑎 + 𝑏𝑥
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
5. The strength observations for certain bars is given by the following stem and leaf
plot.
Stem
1
2
3
4
5
6
7
Leaf Unit = 10
Leaf
19
59
456689
4789
0
3457
23
Which of the following statements are true about this stem and leaf plot?
a)
b)
c)
d)
The smallest value in the data set is 11.
If redone with 2 leaf categories per stem the increment would be 5.
One of the data points is 50.
The median is 440 and mode is 360.
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6. Do the summary statistics for A provide enough information to construct a
boxplot?
7. A machine produces parts that are acceptable if they are larger than 10.5 mm
and smaller than 10.6 mm. From past tests, it is known that the parts produced
follow a normal distribution with a mean of 10.56 mm and variance of
0.0004 mm2 .
a) What proportion parts fall within the tolerance? (Ans:0.9759)
b) Out of a batch of 400 parts, how many would be acceptable? (Ans: 390)
c) A certain part is 0.5 standard deviations above the mean. What proportion of
parts produced are larger than that? (Ans: 0.3085)
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8. The lifetime of a bearing (in years) follows the Weibull distribution with
parameters  = 1.5 and  = 0.8 .
a) What is the probability that a bearing lasts more than 1 year? (Ans: 0.4889)
b) What is the probability that a bearing lasts less than 2 years? (Ans: 0.8679)
9. The lifetime (in days) of a certain electronic component is lognormally distributed
with  = 3.5 and  = 1.2 .
a) Find the mean lifetime. (Ans: 68.0)
b) Find the probability that a component lasts between 50 and 250 days.
(Ans:0.3204)
c) Find the median lifetime. (Ans: 33.1)
10. A commuter must pass through three traffic lights on her way to work. For each
light, the probability that it is green when she arrives is 0.8. The lights are
independent.
The commuter goes to work five days per week. Let X be the number of times
out of the five days in a given week that all three lights are green. Assume the
days are independent of one another. What is the distribution of X?
a)
b)
c)
d)
X ∼ Bin(5, 0.8)
X ∼ Bin(5, 0.512)
X ∼ Bin(3, 0.8)
X ∼ Bin(3, 0.512)
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11. Assume scores on an entrance test follows a normal distribution with mean of 64
points and standard deviation of 2.5. What is the 90th percentile?
a) 64.8
b) 66.5
c) 67.2
d) 90.0
12. The breaking strength of a rivet has a mean value of 10,000 psi and a standard
deviation of 500 psi. What is the probability that the sample mean breaking
strength for a random sample of 36 rivets is between 9900 and 10,200?
(Ans:0.8767)
Check out this web app to visualize a sampling distribution to understand CLT.
13. Aluminum cans received by a beverage company are tested for conformance to
a strength specification. From a very large lot of cans, 100 are sampled at
random and tested. The shipment is rejected if 12 or more of the cans fail the
test. Assume that 20% of the cans in the lot do not meet the specification. What
is the probability that the lot will be rejected? (Ans: 0.9834)
Check out this web app to visualize the binomial distribution for large n.
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14. It is known that cars arrive at a drive-through at a rate of three cars per minute
between 12 noon and 1:00 p.m. Assuming the number of cars that arrive in any
time interval has a Poisson distribution, what is the probability that exactly 9 cars
arrive between 12:10 and 12:15? (Ans: 0.0324)
15. Jones battery company makes 12-volt batteries. After many years of product
testing, Jones knows the average life of a battery is normally distributed, with a
mean age of 45 months, and a standard deviation of 7 months.
a) Jones guarantees to replace for free any battery that fails in less than 37
months. What percentage of the total production does Jones expect to
replace? (Ans: 12.71%)
b) If Jones wishes to replace no more than 3.75% of their production, how long
should the battery be guaranteed? (Ans: 32 months)
c) Find the probability that a randomly selected battery will have a lifetime over
37 months given that it has already lasted 30 months. (Ans: 88.7%)
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16. The number of telephone calls that pass through a switchboard has a Poisson
distribution with mean equal to 2 per minute.
a) Find the expected number of phone calls that pass through the switchboard in
one minute.
b) Find the probability that no telephone calls pass through the switchboard in
two consecutive minutes. (Ans: 0.0183)
17. An exam has two questions: I and II. The probability that a student will solve
only question I correctly is 0.4, and only question II is 0.45. The probability that a
student will solve either question I or II correctly is 0.9. Find the probability that
the student will correctly solve:
a) Both questions (Ans: 0.05)
b) Exactly one question (Ans: 0.85)
c) None of the questions (Ans: 0.1)
d) Question II given that he solved question I correctly (Ans: 0.11)
3
18. Find the missing lower bound so 𝑓(𝑡) is a pdf on [a, 4]. (Ans: 𝑎 = √63)
4
∫ 3𝑡 2 𝑑𝑡
𝑎
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