Math 2526 Applied Statistics Exam 1 Review Sheet Spring 2023
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Math 2526-03
Applied Statistics
Exam 1 Review Sheet
Spring 2023
Type I. Simple probability
Problem 1. Our class is comprised of 10 female and 15 male students. Find the following probabilities: a). pick one randomly, this student is male.
15
25
b). pick two randomly, both are male.
, or,
c). pick five randomly, three are female and two are male.
╱ 3(1〇)、╱ 2(15)、
╱5(25)、
d). pick five randomly, at least one male.
1 ) ╱(╱)、(、)
Problem 2. Consider a deck of 52 cards, find the following probabilities:
a). Pick one card, you get a diamond.
13
52
b). Pick two cards, you get a diamond and a heart.
╱ 11(3)、╱ 11(3)、
╱2(52)、
c). Pick four cards, you get two distinct pairs.
╱ 2(13)、╱ 2(4)、╱ 2(4)、
╱4(52)、
d). Pick five cards, you get a triple and a pair.
╱ 11(3)、╱ 、╱、╱ 2(4)、
╱5(52)、
Type II. Conditional Probability
Problem 1. Our class is comprised of 10 female and 15 male students. Among 10 female students, three are honor students. Among 15 male students, four are honor students. Pick one student randomly, denote M = the student is male, and H = the student is honor student. Find:
a). P (M) b). P (H) c). P (H~M) d). P (M~H)
Problem 2. A normal poker consists of 52 cards (13 hearts, 13 diamonds, 13 clubs and 13 spades). Someone erroneously put one extra card into the poker (but you don’t know which one). Now from these 53 cards pick one card, what is the probability for getting a heart?
P = 1 ( 14 + 3 ( 13
Problem 3. Suppose in our class, male students count for 60%, while female count for 40%. The chance to get grade A among male student is 20%, the chance to get grade A for all student is 25%. What is the chance to get grade A for female students?
|
A |
A |
Total |
M |
|
|
60 |
F |
|
|
40 |
Total |
|
|
100 |
Type III. Distribution and Expectation
Problem 1. Consider this simple game. Just pick one card from a poker with 52 cards. If you get an Ace, you will win. Otherwise, you will lost $1. If the game is fair, how much you should get when you win?
w |
g |
|
P (w) |
4 52 |
48 52 |
Problem 2. Our class is comprised of 10 female and 15 male students. From our class, pick 3 students randomly. Let x be the number of female students among the three picked one. Give out the distribution of x .
x
P (x)
3 3 3 3
Problem 3. Imagine a game between you and someone. Assume the guy has a box, inside the box, there are ten $1 bills, four $20 bills, and one $100 bill. You are charged $12 to pick one bill from the box. Is the game in favor of you?
Type IV. Binomial Distribution
Problem 1. The ABC Company manufactures toy robots. About 1 toy robot per 100 does not work. You purchase 50 ABC toy robots. What is the probability that exactly 5 do not work?
P = ╱5(50)、0.015 0.9945
Problem 2. Our exam consists of 30 questions, each has five choices. If you answer the questions by random guessing, what is the chance that you will get exact 12 question correctly?
P = ╱1(3)2(0)、╱、12 ╱ 、18
Type V. Normal Distribution
Problem 1. Assume Z follows standard normal distribution. Find the following probabilities:
a). P (Z < 1.25) b). P (Z > )1.32) c). P (Z > 2.12)
Problem 2. Assume Z follows standard normal distribution.
a). If P(Z > g) = 0.05, find the value of g.
b). If P(Z < g) = 0.85, find the value of g.
Problem 3. Suppose that the SAT Math score (in a certain region) follows a normal distribution with mean 560 and standard deviation 90. Your score is 620. Find your ranking (the percentage of all students whose scores are below your).
P = P (x s 620) = P╱x s 620 )90560 、= ( ( (
Problem 4. Suppose that the SAT Math score (in a certain region) follows a normal distribution with mean 560 and standard deviation 90. You are going to take the test. Find your lowest score if you want to be among the top 10%.
P (x s a) = 0.90.
P (Z s a )90(5)60 ) = 0.9000.
a ) 560
Type VI. Central Limit Theorem
Problem 1. Suppose a test consists of 100 questions, each has 4 choices (A, B, C, D). A very nice professor sets the passing score to be 35. Assume you just randomly pick your answers. Find the probability that you can pass the test without any study.
u = n夕 = 100 ) (1/4) = 25. a2 = n夕p = 100 ) (1/4) ) (3/4)\ a = 4.33.
P = P (x 2 35) = P(Z 2 34(5))33(2)5 ) = 0.01.
Problem 2. Consider this unfair game. Toss a coin, if face up, you win $1; if face down, you lose $1.5. Suppose you play such a game 100 times, find the chance that you can win some money.
|
|
P (w) |
0.5 0.5 |
T = w1 + w2 + ( ( ( + w1〇〇
T ~ N (nu\ ^na2 )
u = 1 ( 0.5 + ()1.5) ( 0.5 = )0.25\ \ a2 = E (x ) u)2 = (1 + 0.25)2 ( 0.5 + ()1.5 + 0.25)2 ( 0.5 = 1.5625.
So,
T ~ N ()25\ 12.5)\ P (T 2 0) = 0.02.
2023-05-07