Download Key for Practice Problems

MID-TERM STUDY GUIDE Key
1. A teacher gives a four-question true-false quiz. How many outcomes are in the sample
space? What is the probability of each one, assuming that they are independent? What is
the probability that a student who is just guessing will get all four correct? If getting three
correct out of four is passing, what is the probability of passing?
a) 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16
b) 1 / 16
c) 1 / 16
d) 4 C 3 ⋅ 0.5 3 ⋅ 0.5 ( 4−3) = 0.25
2. The probability of being a freshman in this class is 0.20. The probability of being a
senior is 0.05. What is the probability that a randomly selected student will be something
other than a freshman or senior? If we were selecting just one student at random, what
would be your expectation of who was selected?
a) 1 − 0.2 − 0.05 = 0.75
b) other than a freshman or senior
3. At FAU, 10% of students live in the residence halls, 47% receive financial aid, and 7%
do both. What is the probability that a student randomly chosen lives in the residence
halls but does not receive financial aid?
10% − 7% = 3%
4. The probability that a student at a certain college is male is 0.45. The probability that a
student at that college has a job off campus is 0.33. The probability that a student at the
college is male and has a job off campus is 0.15. If a student is chosen at random from
the college, what is the probability that the student is male or has an off campus job?
0.45 + 0.33 − 0.15 = 0.63
5. The probability that it will rain on game day is 60%. The probability that the Owls will
beat their next opponent in football is 75%. Assuming that these events are independent,
what is the probability that it will rain and that the Owls beat their next opponent? What
is the probability that the Owls lose on a clear day?
a) 0.6 ⋅ 0.75 = 0.45
b) 0.4 ⋅ 0.25 = 0.1
6 – 10: College students were given three choices of pizza toppings and asked to choose
one favorite. The following table shows the results:
toppings freshman
sophomore
junior
senior Total
cheese
13
15
18
27
73
meat
19
27
15
13
74
veggie
15
13
19
27
74
Total
47
55
52
67
221
6. What is the probability that a randomly-chosen student prefers veggie pizza?
74 / 221
7. What is the probability that a randomly-chosen junior prefers meat?
15 / 221
8. What is the probability that a randomly-chosen student is a freshman or prefers cheese
pizza?
47 / 221 + 73 / 221 − 13 / 221 = 107 / 221
9. What is the conditional probability that a randomly-chosen student is a freshman given
that he or she prefers cheese pizza?
13 / 73
10. Does pizza preference depend on class level?
YES.
11 – 12: The random variable X is the number of houses sold by a realtor in a single
month at Caldwell-Banker Realtors. Its probability distribution is given in the table.
x
P( x)
x*P(x)
0
0.24
0
1
0.01
0.01
2
0.12
0.24
3
0.16
0.48
4
0.01
0.04
5
0.14
0.7
6
0.11
0.66
7
0.21
1.47
11. What is µ?
μ = E ( X ) = ∑ [ x ⋅ P( x)] = 0 + 0.01 + 0.24 + 0.48 + 0.04 + 0.7 + 0.66 + 1.47 = 3.6
12. What is the expected value of X?
Same as µ.
13. A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be
reported in a given day are 0.50, 0.40, 0.09, and 0.01 respectively. What is the
expected number of burglaries on an average day?
μ = E ( X ) = ∑ [ x ⋅ P( x)] = 0 ⋅ 0.5 + 1 ⋅ 0.4 + 2 ⋅ 0.09 + 3 ⋅ 0.01 = 0.61
14. The probability that a person has immunity to a particular disease is 0.6. Find the
mean (expected value) for the random variable X, the number who have immunity in
samples of size 26.
μ = n ⋅ p = 26 ⋅ 0.6 = 15.6
15. Police estimate that 25% of drivers drive without their seat belts. If they stop 6 drivers
at random, find the probability that all of them are wearing their seat belts.
n = 6, p = 0.75, x = 6
6
C 6 ⋅ 0.75 6 ⋅ 0.25 ( 6 −6 )
16. Suppose that 11% of people are left handed. If 6 people are selected at random, what
is the probability that exactly 2 of them are left handed?
n = 6, p = 0.11, x = 2
6
C 2 ⋅ 0.112 ⋅ 0.89 ( 6− 2 )
17. On a multiple choice test with 16 questions, each question has four possible answers,
one of which is correct. For students who guess at all answers, find the standard
deviation for the random variable X, the number of correct answers.
n = 16, p = 1 / 4 = 0.25
σ = n ⋅ p ⋅ (1 − p) = 16 ⋅ 0.25 ⋅ 0.75 ≈ 1.73
18 – 23: Finding probabilities for normal (bell-shaped) distributions:
18. Find P(Z < 0.21) = 0.5832
19. Find P(Z > -1.23) = 1 - 0.1093 = 0.8907
20. Find P(Z > 1.23) = 1 – 0.8907 = 0.1093
21. Find P(-0.43 < Z < 0.50) = 0.6915 – 0.3336 = 0.3579
22. What is the Z-score above which approximately 82% of a normal distribution falls?
Z = - 0.92
23. What Z-score is at the 48th percentile of a normal distribution?
Z = - 0.05
24 – 28: The weekly salaries of teachers in one state are normally distributed with a mean
of $490 and a standard deviation of $45.
24. What is the probability that a randomly selected teacher earns more than $525 a
week?
x−μ
525 − 490
= 0.78
σ
45
P( z > 0.78) = 1 − 0.7823 = 0.2177
z=
=
25. What proportion of teachers earn less than $600 a week?
x−μ
600 − 490
= 2.44
σ
45
P( z < 2.44) = 0.9927
z=
=
26. If a teacher’s salary is at the 25th percentile, what is that salary?
z = −0.67
x = μ + z ⋅ σ = 490 + (−0.67) ⋅ 45 = 459.85
27. Find the salary that separates the highest 75% of salaries from the others.
z = 0.67
x = μ + z ⋅ σ = 490 + 0.67 ⋅ 45 = 520.15
28. Using the Empirical Rule, find the symmetric interval that contains about 95% of the
salaries.
( μ − 2 ⋅ σ , μ + 2 ⋅ σ ) = (490 − 2 ⋅ 45,490 + 2 ⋅ 45) = ( 400,580)
HINTS FOR SOLVING PROBABILITY PROBLEMS WITH DISTRIBUTIONS
Ask yourself:
1) What is the problem asking for? Is it a probability (p)?
Is it a mean (µ) ?
Is it an individual score (x)?
Is it a standard deviation (σ)?
2) Is the outcome in question binary? (use formulas for binomial)
Discrete but not binary? (use formulas for discrete random variable X)
Continuous? (use formulas for standard normal distribution)
OTHER TOPICS THAT YOU SHOULD STUDY
Ask yourself:
Can I read a frequency table?
Do I know the types of variables? (categorical, discrete, continuous)
Can I read a dot plot and interpret it?
Can I read a box plot and interpret it?
Do I know the measures of central tendency? (mean, median, mode) Which one is a
better representative, under different circumstances?
Do I know the measures of spread? (IQR, standard deviation, variance)
Do I know how to compute a z-score and what it means?
Can I look at a scatterplot and decide whether it shows a strong, weak, or no association?
Can I interpret the correlation coefficient, r? (its properties)
Can I compute a margin of error for a sample survey?
Do I know how to distinguish the bias of sampling?
Do I know what the Law of Large Numbers says? (section 5.1, page 197)
OTHER SOURCE THAT YOU NEED TO GO OVER:
All the quizzes (1 – 6)
All the lecture notes, especially those examples