Practice questions for Test 1
Practice questions for Test 1
Practice questions for Test 1
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1. Consider the sample data set<br />
<strong>Practice</strong> <strong>questions</strong> <strong>for</strong> <strong>Test</strong> 1<br />
51 42 38 10 27 65 51 34 39 17 35 43 13<br />
33 57 21 37 63 45 22 49 31 43 64 47 22<br />
(a) Construct a stem-and-leaf display of the data.<br />
(b) Complete the following frequency table.<br />
Class Boundaries<br />
10 to < 20<br />
20 to < 30<br />
30 to < 40<br />
40 to < 50<br />
50 to < 60<br />
60 to < 70<br />
Frequency<br />
TOTAL 26<br />
(c) Construct a frequency histogram <strong>for</strong> the data set using the boundaries of part (b).<br />
(d) Find the sample mean ¯x, median ˜x, variance s 2 and standard deviation s.<br />
2. Let E and F be two events such that P (E) = 0.3 and P (F ) = 0.5. Compute P (E ∪ F ) if<br />
(a) E and F are mutually exclusive.<br />
(b) E and F are independent.<br />
3. Let A and B be two events such that<br />
(a) Compute P (A ∩ B).<br />
(b) Compute P (B | A).<br />
(c) Compute P (A | A ∪ B).<br />
P (A) = 0.6, P (B) = 0.3, and P (A ∪ B) = 0.8.<br />
4. Let A, B, and C be three independent events with<br />
Compute P (A ∪ B ∪ C).<br />
P (A) = 1/2, P (B) = 1/3, and P (C) = 2/3.<br />
5. In how many distinct ways can a true-false test consisting of 9 <strong>questions</strong> be answered What<br />
is the probability of getting exactly 6 <strong>questions</strong> right by guessing<br />
6. In how many ways can 5 starting positions on a basketball team be filled from 8 players who<br />
can play any positions<br />
7. In a poker hand consisting of 5 cards, find the probability of holding<br />
(a) 3 aces;<br />
(b) 4 hearts and 1 club.
Math 264 <strong>Practice</strong> <strong>questions</strong> <strong>for</strong> <strong>Test</strong> 1 Page 2 of 2<br />
8. A small town has one fire engine and one ambulance available <strong>for</strong> emergencies. The probability<br />
that the fire engine is available when needed is 0.97 and the probability that the ambulance<br />
is available when called is 0.94. In the event of an injury resulting from a burning building,<br />
find the probability that both vehicles will be available. Clearly state what hypothesis you are<br />
making.<br />
9. A fire-detection device uses three temperature-sensitive cells acting independently of one another<br />
in such a manner that any cell can activate the alarm. Each cell has a probability p = 0.85<br />
of activating the alarm when the temperature reaches 100 ◦ C or more. Find the probability that<br />
the alarm will function when the temperature reaches 100 ◦ C.<br />
10. Assume that the probability of hitting a target is 1/5 and that 30 shots are fired independently.<br />
(a) What is the probability of the target being hit exactly 8 times<br />
(b) What is the probability of the target being hit at least twice<br />
(c) What is the expected number of times that the target will be hit<br />
11. The number of accidents occurring on a certain highway each day is a Poisson random variable<br />
with λ = 3. Find the probability that there will be 3 or more accidents today.<br />
12. Compare the Poisson approximation with the binomial probability <strong>for</strong> P (X = 6) if X is a<br />
binomial with n = 500 and p = 0.01.<br />
13. From a lot of 50 components, 6 are chosen at random <strong>for</strong> testing. If the lot contains 8 defective<br />
components, what is the probability that<br />
(a) all 6 components will be working.<br />
(b) at most 2 of the 6 will be defective.<br />
14. One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and<br />
and 5 black balls. One ball is drawn from the first bag and placed in the second. What is the<br />
probability that a ball now drawn from the second bag is black<br />
15. A certain steroid detection test is 99.5% effective, meaning that it will show a positive result <strong>for</strong><br />
99.5% of all steroid users. The test also gives a “false positive” result <strong>for</strong> 2% of the non-user,<br />
meaning that the probability of testing positive <strong>for</strong> a non-user is 2%.<br />
A confidential survey of Olympic swimmers tells us that about 10% of them are using steroids.<br />
Suppose one swimmer tests positive but claims he is not using steroids, what is the probability<br />
that he is telling the truth<br />
16. Consider a random variable X with the following probability distribution.<br />
(a) Find the expected value of X.<br />
x 1 2 3 4 5 6<br />
p(x)<br />
2<br />
15<br />
(b) Find the expected value of X 2 .<br />
3<br />
15<br />
2<br />
15<br />
(c) Find the variance and standard deviation of X.<br />
(d) Find the variance and expected value of the random variable Y = 5X + 3<br />
3<br />
15<br />
4<br />
15<br />
1<br />
15