Midterm 2: 2.1-2.2

Question 1

What is a random experiment?

Answer 1

A process that leads to a single outcome which cannot be predicted with certainty.

Question 2

What is an element in probability?

Answer 2

A specific outcome of an experiment.

Question 3

What is the sample space denoted as S?

Answer 3

The collection of all possible outcomes for a given experiment.

Question 4

Define an event in probability.

Answer 4

A collection of elements that are of interest or share a common characteristic in a given experiment.

Question 5

What does P(A) represent?

Answer 5

The probability of event A occurring when an experiment is performed.

Question 6

What does the complement of event A, denoted as A’, represent?

Answer 6

The set of all elements in the sample space S that do not belong to A.

Question 7

Probability of X

Answer 7

Question 9

What is the formula for the complement of event A?

Answer 9

P(A) + P(A’) = 1.

Question 10

What is the union of events A and B denoted as?

Answer 10

A ∪ B.

Question 11

What does the union of A and B represent? + formula

Answer 11

The set of all elements that belong to A or B or both.

Question 12

What is the formula for the intersection of events A and B?

Answer 12

P(A ∩ B) = P(A) + P(B) - P(A ∪ B).

Question 13

What is mutual exclusivity in probability?

Answer 13

Events A and B are mutually exclusive if there are no elements that are in both A and B.

Question 14

What is conditional probability?

Answer 14

A probability that reflects additional knowledge that may affect the outcome of an experiment.

Question 15

How is conditional probability denoted?

Answer 15

P(A|B) represents the probability of A given B.

Question 16

What is the formula for calculating conditional probability?

Answer 16

P(A|B) = P(A ∩ B) / P(B).

Question 17

What is the law of total probability?

Answer 17

P(A) = P(A ∩ B) + P(A ∩B ^c).

Question 18

What is a contingency table?

Answer 18

A table that organizes the probability relationships between two events A and B.

Question 19

What does it mean for two events A and B to be independent?

Answer 19

The occurrence of A does not affect the probability of B, and vice-versa.

Question 20

What is one way to prove independence between events A and B?

Answer 20

If P(A|B) = P(A) and P(B|A) = P(B), then A and B are independent.

Question 21

Fill in the blank: The probability of event A is conditional upon another event occurring (event B), so now, P(A|B) is the ratio of _______.

Answer 21

P(A ∩ B) over P(B).

Question 22

What is DeMorgan’s law in probability?

Answer 22

P(A’ ∩ B’) = P(A ∪ B)’ = 1 - P(A ∪ B).

Question 23

True or False: If events A and B are mutually exclusive, then P(A ∩ B) > 0.

Answer 23

False.

Question 24

What keyword indicates conditional probability?

Answer 24

If, given, assuming, knowing.

Question 25

Define event A as 'the sum of the numbers is odd.' What is u(A)?

Answer 25

u(A) represents the number of elements in event A.

Question 26

What is the probability that a randomly selected person will test 'negative' for the disease?

Answer 26

Calculated based on the constructed contingency table.

Question 27

What is the probability that a randomly selected person will have the disease given that they test negative for the disease?

Answer 27

Calculated using conditional probability.

Question 28

What is a random variable?

Answer 28

A characteristic, attribute, or outcome that can vary from observation to observation.

Question 29

What is the purpose of a probability distribution table?

Answer 29

To display all the values a discrete random variable can assume along with the associated probabilities.

Question 30

What are the key summary values used to describe a random variable's behavior?

Answer 30

Expected value, variance, and standard deviation.

Question 31

What is the expected value of a random variable A?

Answer 31

The average value of A over an infinite number of repetitions of an experiment.

Question 32

How is variance defined for a discrete random variable A?

Answer 32

It is a measure of the variability of A over an infinite number of repetitions of an experiment.

Question 33

What is the standard deviation of a discrete random variable?

Answer 33

The positive square root of the variance.

Question 34

What defines a binomial random variable?

Answer 34

An experiment with a fixed number of identical, independent trials with the same probability of success.

Question 35

What are the parameters needed to compute probabilities for a binomial random variable?

Answer 35

* n — the number of trials in the experiment * p — the probability of success

Question 36

What is the probability mass function (pmf) for a binomial random variable?

Answer 36

The equation used to find the probability that a random variable assumes a particular value x.

Question 37

What is the R function to find P(X = x) for a binomial random variable?

Answer 37

dbinom(x, n, p)

Question 38

What is a continuous random variable?

Answer 38

A variable that can take on an infinite number of values.

Question 39

What is the probability density function (pdf)?

Answer 39

The function that defines the curve representing the distribution of a continuous random variable.

Question 40

What is the probability that a continuous random variable equals a specific value?

Answer 40

It is equal to zero.

Question 41

What does the area under the curve of a pdf represent?

Answer 41

It represents the probabilities for intervals of values.

Question 42

What is the normal distribution often assumed for?

Answer 42

Many variables in the natural and social sciences when their exact distributions are not known.

Question 43

True or False: Discrete random variables can take on any value within a range.

Answer 43

False

Question 44

Fill in the blank: A probability distribution table is appropriate for _______.

Answer 44

discrete random variables.

Question 45

What is the notation for a binomial random variable?

Answer 45

X ~ Binomial(n, p)

Question 46

What does the notation P(A) represent?

Answer 46

The probability of event A occurring.

Question 47

What is the formula for calculating the expected value of a binomial random variable?

Answer 47

E(X) = n * p

Question 48

What is the formula for calculating the variance of a binomial random variable?

Answer 48

VAR(X) = n * p * (1 - p)

Question 49

What does standardizing a normal variable involve?

Answer 49

Calculating its z-score.

Question 50

What does a probability distribution graph help to observe?

Answer 50

The shape of the distribution and the behavior of a random variable.

Question 51

What is the normal (or Gaussian) distribution often assumed for?

Answer 51

A wide variety of variables in the natural and social sciences when exact distributions are not known. ## Footnote Examples include scores on the Stanford-Binet IQ test, adult heights, and gestation time of elephants.

Question 52

What two parameters are needed to compute probabilities for a normal random variable?

Answer 52

* Mean (y) * Standard deviation (σ)

Question 53

How is the notation for a normal distribution expressed?

Answer 53

X ~ N(μ, σ) where μ is the mean and σ is the standard deviation.

Question 54

What is the shape of normal distributions?

Answer 54

Bell-shaped and symmetric about the mean.

Question 55

What does the Probability Density Function (pdf) define?

Answer 55

The shape of a normal random variable.

Question 56

What does the Cumulative Density Function (cdf) allow us to do?

Answer 56

Define and calculate probabilities for a normal random variable.

Question 57

What R function is used to find P(X ≤ x)?

Answer 57

pnorm(x, mean, sd)

Question 58

What R function is used to find a specific value of X given a known probability?

Answer 58

qnorm(p, mean, sd)

Question 59

What are the expected value and variance for a normal random variable Z?

Answer 59

* E(X) = μ * VAR(X) = σ²

Question 60

What is the normal distribution used to model in the context of the Lilac Bloomsday Run?

Answer 60

The finishing time of participants.

Question 61

How do you calculate the proportion of participants finishing in under 90 minutes?

Answer 61

Using the R function pnorm(x, mean, sd)

Question 62

What is the maximum finishing time for the bottom 20% of participants?

Answer 62

Calculated using the R function qnorm(p, mean, sd)

Question 63

What is the standard normal distribution characterized by?

Answer 63

Mean of 0 and standard deviation of 1.

Question 64

What are the three main purposes of the standard normal distribution?

Answer 64

* Ease of calculating probabilities * Use of standardized scores (z-scores) * Use in statistical inference

Question 65

What does a z-score represent?

Answer 65

How many standard deviations a value is from the mean.

Question 66

What does the empirical rule state about a normal distribution?

Answer 66

* Approximately 68% of observations within +/- 1 standard deviation * Approximately 95% within +/- 2 standard deviations * Approximately 99.7% within +/- 3 standard deviations

Question 67

What is another name for the empirical rule?

Answer 67

The 68-95-99.7 rule.

Question 68

What were standard normal tables developed for?

Answer 68

To compute probabilities without the need for calculus.

Question 69

What is the formula for calculating the z-score?

Answer 69

z = (X - μ) / σ

Question 70

How can a z-score be calculated in R?

Answer 70

Using the scale function: scale(x, center, scale)

Question 71

What is the mean and standard deviation of the Stanford-Binet IQ test?

Answer 71

* Mean = 100 * Standard deviation = 15

Question 72

How do you standardize a score of 106 on the IQ test?

Answer 72

Using the formula z = (X - μ) / σ or the R function scale(x, center, scale)

Question 73

What does a z-score of 3 indicate?

Answer 73

The corresponding x value is 3 standard deviations above the mean.

Question 74

What does a z-score of -0.5 indicate?

Answer 74

The corresponding x value is half a standard deviation below the mean.

Question 75

What does a z-score of 0 represent?

Answer 75

A value equal to the mean.

Question 76

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Question 77

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Question 78

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Question 79

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Question 80

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Question 81

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