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Unit 6: Anticipating Patterns

Terms and Concepts

  • Probability: the chance of the outcome of an event

  • Sample space: a set of all possible outcomes

  • Tree diagram: representation is useful in determining the sample space for an experiment, especially if there are relatively few possible outcomes.

Basic Probability Rules and Terms

  • Rule 1: For any event A, the probability of A is always greater than or equal to 0 and less than or equal to 1

  • Rule 2: The sum of the probabilities for all possible outcomes in a sample space is always 1

  • Impossible event: If an event can never occur, its probability is 0

  • Sure event: Of an event must occur every time, its probability is 1

  • “Odds in favor of an event”: ratio of the probability of the occurrence of an event to the probability of the nonoccurrence of that event.

    • Odds in favor of an event = P(Event A occurs) / P(Event A does not occur) or P(Event A occurs) : P(Event A does not occur)

  • Complement: the set of all possible outcomes in a sample space that do not lead to the event

  • Disjoint or mutually exclusive events: events that have no outcome in common. In other words, they cannot occur together.

  • Union: events A and B is the set of all possible outcomes that lead to at least one of the two events A and B

  • Intersection: events A and B is the set of all possible outcomes that lead to both events A and B

  • Conditional Events: A given B is a set of outcomes for event A that occurs if B has occurred

Random Variables and Their Probability Distribution

  • Variable: quantity whose value varies from subject to subject

  • Probability experiment: an experiment whose possible outcomes may be known but whose exact outcome is a random event and cannot be predicted with certainty in advance

  • Random variables: The outcome of a probability experiment takes a numerical value

  • Discrete random variable: quantitative variable that takes a countable number of values

  • Continuous random variable: a quantitative variable that can take all the possible values in a given range

Discrete Random Variable

  • Expected value: Computed by multiplying each value of the random variable by its probability and then adding over the sample space

  • Variance: sum of the product of squared deviation of the values of the variable from the mean and the corresponding probabilities

Combinations

  • Combination: the number of ways r items can be selected out of n items if the order of selection is not important.

Binomial Distribution

  • 3 Characteristics of a binomial experiment

    • There are a fixed number of trials

    • There are only 2 possible outcomes

    • The n trials are independent and are repeated using identical conditions

  • Binomial probability distribution:

    • Mean: μ = np

    • Variance: σ2 = npq

    • Standard deviation: σ = √npq

Geometric Distribution

  • 3 Characteristics of a geometric experiment

    • There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success.

    • In theory, the number of trials could go on forever. There must be at least one trial.

    • The probability, p, of a success and the probability, q, of a failure is the same for each trial. p + q = 1 and q = 1 − p.

  • X = the number of independent trials until the first success

  • Mean: μ = 1/p

  • Standard Deviation: σ = √1/𝑝(1/𝑝−1)

The Probability Distribution of Continuous Random Variables

  • The continuous probability distribution (cdf): graph or a formula giving all possible values taken by a random variable and the corresponding probabilities

Let X be a continuous random variable taking values in the range (a, b)

  • The area under the density curve is equal to the probability

  • P(L < X < U) = the area under the curve between L and U, where a ≤ L ≤ U ≤ b

  • The total probability under the curve = 1

  • The probability that X takes a specific value is equal to 0, i.e., P(X = x0) = 0

Sampling Distribution

  • Parameter: a numerical measurement describing some characteristic of a population.

  • Statistic: a numerical measurement describing some characteristic of a sample.

  • Sampling distribution: the probability distribution of all possible values of a statistic, different samples of the same size from the same population will result in different statistical values

  • Standard error: standard deviation of the distribution of the statistics.

Central Limit Theorem

  • Central limit theorem: If the sample size is large enough then we can assume it has an approximately normal distribution.

    • The sample size has to be greater than 30 to assume an approximately normal distribution

  • The shape of the distribution of “X bar” becomes more symmetrical and bell-shaped

  • The center of the distribution of “X bar” remains at μ

  • The spread of the distribution “X bar” decreases, and the distribution becomes more peaked

Calculator Steps

Probabilities for means on the calculator

  • 2nd DISTR

  • 2:normalcdf

  • normalcdf (lower value of the area, upper value of the area, mean, standard deviation / √sample size)

  • where

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Percentiles for means on the calculator

  • 2nd DISTR

  • 3:InvNorm

  • k = invNorm (area to the left of 𝑘, mean, standard deviation / √sample size)

  • Where→

    • k = the kth percentile

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Probabilities for sums on the calculator

  • 2nd DISTR

  • 2: normalcdf (lower value of the area, upper value of the area, (n)(mean), (√n)(standard deviation))

  • where:

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Percentiles for sums on the calculator

  • 2nd DIStR

  • 3:invNorm

  • k = invNorm (area to the left of k, (n)(mean), (√n)(standard deviation)

  • where:

    • k is the kth percentile

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Unit 6: Anticipating Patterns

Terms and Concepts

  • Probability: the chance of the outcome of an event

  • Sample space: a set of all possible outcomes

  • Tree diagram: representation is useful in determining the sample space for an experiment, especially if there are relatively few possible outcomes.

Basic Probability Rules and Terms

  • Rule 1: For any event A, the probability of A is always greater than or equal to 0 and less than or equal to 1

  • Rule 2: The sum of the probabilities for all possible outcomes in a sample space is always 1

  • Impossible event: If an event can never occur, its probability is 0

  • Sure event: Of an event must occur every time, its probability is 1

  • “Odds in favor of an event”: ratio of the probability of the occurrence of an event to the probability of the nonoccurrence of that event.

    • Odds in favor of an event = P(Event A occurs) / P(Event A does not occur) or P(Event A occurs) : P(Event A does not occur)

  • Complement: the set of all possible outcomes in a sample space that do not lead to the event

  • Disjoint or mutually exclusive events: events that have no outcome in common. In other words, they cannot occur together.

  • Union: events A and B is the set of all possible outcomes that lead to at least one of the two events A and B

  • Intersection: events A and B is the set of all possible outcomes that lead to both events A and B

  • Conditional Events: A given B is a set of outcomes for event A that occurs if B has occurred

Random Variables and Their Probability Distribution

  • Variable: quantity whose value varies from subject to subject

  • Probability experiment: an experiment whose possible outcomes may be known but whose exact outcome is a random event and cannot be predicted with certainty in advance

  • Random variables: The outcome of a probability experiment takes a numerical value

  • Discrete random variable: quantitative variable that takes a countable number of values

  • Continuous random variable: a quantitative variable that can take all the possible values in a given range

Discrete Random Variable

  • Expected value: Computed by multiplying each value of the random variable by its probability and then adding over the sample space

  • Variance: sum of the product of squared deviation of the values of the variable from the mean and the corresponding probabilities

Combinations

  • Combination: the number of ways r items can be selected out of n items if the order of selection is not important.

Binomial Distribution

  • 3 Characteristics of a binomial experiment

    • There are a fixed number of trials

    • There are only 2 possible outcomes

    • The n trials are independent and are repeated using identical conditions

  • Binomial probability distribution:

    • Mean: μ = np

    • Variance: σ2 = npq

    • Standard deviation: σ = √npq

Geometric Distribution

  • 3 Characteristics of a geometric experiment

    • There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success.

    • In theory, the number of trials could go on forever. There must be at least one trial.

    • The probability, p, of a success and the probability, q, of a failure is the same for each trial. p + q = 1 and q = 1 − p.

  • X = the number of independent trials until the first success

  • Mean: μ = 1/p

  • Standard Deviation: σ = √1/𝑝(1/𝑝−1)

The Probability Distribution of Continuous Random Variables

  • The continuous probability distribution (cdf): graph or a formula giving all possible values taken by a random variable and the corresponding probabilities

Let X be a continuous random variable taking values in the range (a, b)

  • The area under the density curve is equal to the probability

  • P(L < X < U) = the area under the curve between L and U, where a ≤ L ≤ U ≤ b

  • The total probability under the curve = 1

  • The probability that X takes a specific value is equal to 0, i.e., P(X = x0) = 0

Sampling Distribution

  • Parameter: a numerical measurement describing some characteristic of a population.

  • Statistic: a numerical measurement describing some characteristic of a sample.

  • Sampling distribution: the probability distribution of all possible values of a statistic, different samples of the same size from the same population will result in different statistical values

  • Standard error: standard deviation of the distribution of the statistics.

Central Limit Theorem

  • Central limit theorem: If the sample size is large enough then we can assume it has an approximately normal distribution.

    • The sample size has to be greater than 30 to assume an approximately normal distribution

  • The shape of the distribution of “X bar” becomes more symmetrical and bell-shaped

  • The center of the distribution of “X bar” remains at μ

  • The spread of the distribution “X bar” decreases, and the distribution becomes more peaked

Calculator Steps

Probabilities for means on the calculator

  • 2nd DISTR

  • 2:normalcdf

  • normalcdf (lower value of the area, upper value of the area, mean, standard deviation / √sample size)

  • where

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Percentiles for means on the calculator

  • 2nd DISTR

  • 3:InvNorm

  • k = invNorm (area to the left of 𝑘, mean, standard deviation / √sample size)

  • Where→

    • k = the kth percentile

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Probabilities for sums on the calculator

  • 2nd DISTR

  • 2: normalcdf (lower value of the area, upper value of the area, (n)(mean), (√n)(standard deviation))

  • where:

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n

Percentiles for sums on the calculator

  • 2nd DIStR

  • 3:invNorm

  • k = invNorm (area to the left of k, (n)(mean), (√n)(standard deviation)

  • where:

    • k is the kth percentile

    • mean is the mean of the original distribution

    • standard deviation is the standard deviation of the original distribution

    • sample size = n