Unit 6: Anticipating Patterns

**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.

**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

**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

**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

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

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`

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 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

**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**: 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

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 distributionsample size = n

2nd DISTR

3:InvNorm

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

*k*= the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

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

2nd DIStR

3:invNorm

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

*k*is the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

**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.

**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

**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

**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

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

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`

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 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

**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**: 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

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 distributionsample size = n

2nd DISTR

3:InvNorm

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

*k*= the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

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

2nd DIStR

3:invNorm

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

*k*is the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n