Stat Exam 2

Bayes's Rule

-Suppose that A1,A2,...,AkA1,A2,...,Ak are disjoint events whose probabilities are not 0 and add to exactly 1. That is, any outcome is in exactly one of these events. Then, if BB is any other event whose probability is not 0 or 1, P(Ai|B)=P(B|Ai)P(Ai)P(B|A1)P(A1)+P(B|A2)P(A2)+⋯+P(B|Ak)P(Ak)

Variance of a Discrete Random Variable

-Suppose that X is a discrete random variable whose distribution is and that μx is the mean of X is: σ2X=(x1−μX)2p1+(x2−μX)2p2+⋯=∑(xi−μX)2pi -The standard deviation σX of X is the square root of the variance

Random phenomenon

-If individual outcomes are uncertain but there is, nonetheless, a regular distribution of outcomes in a large number of repetitions

Sample Size for Specified Margin of Error

-The confidence interval for a population mean will have a specified margin of error mm when the sample size is n = (z∗σ/m)**2

Mean and Standard Deviation of a Sample Proportion

-Let p̂ be the sample proportion of successes in an SRS of size n drawn from a large population having population proportion p of successes. The mean and standard deviation of p̂ are

Mean and Standard Deviation of a Sample Mean

-Let x bar be the mean of an SRS of size n from a population having mean μμ and standard deviationσσ. The mean and standard deviation of x bar are -σx=σ/n**.5

Rules for Means

-Rule 1. If X is a random variable and a and b are fixed numbers, then μa+bX=a+bμX -Rule 2. If X and Y are random variables, then μX+Y=μX+μY -Rule 3. If X and Y are random variables, then μX−Y=μX−μY

Rules for Variances and Standard deviations of Linear Transformations, Sums, and differences

-Rule 1. If X is a random variable and a and b are fixed numbers, then σ2a+bX=b2σ2X -Rule 2. If X and Y are independent random variables, then σ2X+Y=σ2X+σ2Y σ2X−Y=σ2X+σ2Y -Rule 3. If X and Y have correlation ρρ, then σ2X+Y=σ2X+σ2Y+2ρσXσYσ2X−Y=σ2X+σ2Y−2ρσXσY

Facts about Sample Means

-Sample means are less variable than individual observations. -Sample means are centered around the population mean. -Sample means are more Normal than individual observations

Sampling Distribution of pˆ

Draw an SRS of size n from a large population having population proportion p of successes. Let pˆp̂ be the sample proportion of successes. When n is large, the sampling distribution of pˆp̂ is approximately Normal:

Equally Likely outcomes

-If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is P(A) = count of outcomes in A/count of outcomes in S

Binomial Standard Deviation

(np(1-p))**.5

Addition Rule for disjoint Events

-If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(A or B or C)=P(A)+P(B)+P(C) -This rule extends to any number of disjoint events.

Sampling Distribution of a Sample Mean

-If a population has the N(μ,σ. distribution, then the sample mean x¯ of n independent observations has the N(μ,σ/n**.5) distribution

Continuous Random Variable

-A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event

Discrete random variable

-A discrete random variable XX has possible values that can be given in an ordered list. The probability distribution of XX lists the values and their probabilities -Every probability pi is a number between 0 and 1 -The sum of the probabilities is 1; p1+p2+⋯=1 -Find the probability of any event by adding the probabilities pi of the particular values xixi that make up the event

Confidence Interval

-A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter

Statistic

-A number that describes a sample. The value is known when we have taken a sample, but it can change from sample to sample. We often use this to estimate an unknown parameter

Parameter

-A parameter is a number that describes the population. A parameter is a fixed number, but in practice we do not know its value.

Distribution of Count of Successes in an SRS

-A population contains proportion pp of successes. If the population is much larger than the sample, the count XX of successes in an SRS of size nn has approximately the binomial distribution B(n,p). -The accuracy of this approximation improves as the size of the population increases relative to the size of the sample. As a rule of thumb, we use the binomial distribution for counts when the population is at least 20 times as large as the sample.

Random Variable

-A random variable is a variable whose value is a numerical outcome of a random phenomenon

Event

-An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space

Rules about probability

-Any probability is a number between 0 and 1 -All possible outcomes of the sample space together must have probability 1 -If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities -The probability that an event does not occur is 1 minus the probability that the event does occur

Probabilities in a Finite Sample Space

-Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum 1. -The probability of any event is the sum of the probabilities of the outcomes making up the event

Bias

-Bias concerns the center of the sampling distribution. A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated

Confidence Interval for a Population Mean

-Choose an SRS of size n from a population having unknown mean μ and known standard deviation σσ. The margin of error for a level C confidence interval for μ is m=z∗ * σ/n**.5 -Here, z* is the value on the standard Normal curve with area CC between the critical points -z* and z*. The level CC confidence interval for μ is x±m

Normal Approximation for Counts and Proportions

-Draw an SRS of size n from a large population having population proportion p of successes. Let X be the count of successes in the sample and pˆ=X/n be the sample proportion of successes. When n is large, the distributions of these statistics are approximately Normal X is approximately N(np,(np(1-p))**.5) pˆ is approximately N(p,(np(1-p))/n)**.5)

Central Limit Theorem

-Draw an SRS of size n from any population with mean μμ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x¯ is approximately Normal

Law of Large numbers

-Draw independent observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean x bar of the observed values becomes progressively closer to the population mean μ

Complement rule

-For any event A, P(Ac) = 1 — P(A)

Binomial Probability

-If X has the binomial distribution B(n,p)B(n, p), with n observations and probability p of success on each observation, the possible values of X are 0,1,2, . . . ,n0,1,2, . . . ,n. If k is any one of these values, the binomial probability is

Poisson Distribution

-The distribution of the count X of successes in the Poisson setting is the Poisson distribution with mean μ. The parameter μ is the mean number of successes per unit of measure. The possible values of X are the whole numbers 0, 1, 2, 3, .... If k is any whole number, then

Binomial Distribution

-The distribution of the count X of successes in the binomial setting with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that the distribution of X is B(n, p)

Intersection

-The event that all the events of a collection of events occur

Union

-The event that at least one of any collection of events occurs

Margin of error

-The margin of error is a numerical measure of the spread of a sampling distribution. It can be used to set bounds on the size of the likely error in using the statistic as an estimator of a population parameter

The Poisson Setting

-The number of successes that occur in two nonoverlapping units of measure are independent. -The probability that a success will occur in a unit of measure is the same for all units of equal size and is proportional to the size of the unit. -The probability that more than one event occurs in a unit of measure is negligible for very small-sized units. In other words, the events occur one at a time

Binomial coefficient

-The number of ways of arranging k successes among n observations -n!/(k!(n-k)!)

Population Distribution

-The population distribution of a variable is the distribution of its values for all cases of the population. The population distribution is also the probability distribution of the variable when we choose one case at random from the population

Probability

-The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions

Multiplication Rule

-The probability that both of two events A and B happen together can be found by P(A and B)=P(A)P(B|A) -Here P(B|A)P(B | A) is the conditional probability that B occurs, given the information that A occurs

Sample Space

-The sample space SS of a random phenomenon is the set of all distinct possible outcomes

Sampling Distribution

-The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population

Variability of a Statistic

-The variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and the sample size n. Statistics from larger probability samples have smaller spreads

The Binomial Setting

-There are a fixed number nn of observations. -The n observations are all independent. That is, knowing the result of one observation tells you nothing about the outcomes of the other observations. -Each observation falls into one of just two categories, which, for convenience, we call "success" and "failure." -The probability of a success, call it p, is the same for each observation

Mean of a Discrete Random Variable

-To find the mean of XX, multiply each possible value by its probability, then add all the products: μX=x1p1+x2p2+⋯=∑xipi

Managing Bias and Variability

-To reduce bias, use random sampling. When we start with a list of the entire population, simple random sampling produces unbiased estimates—the values of a statistic computed from an SRS neither consistently overestimate nor consistently underestimate the value of the population parameter. -To reduce the variability of a statistic from an SRS, use a larger sample. You can make the variability as small as you want by taking a large enough sample

Independent

-Two events A and B are called this if knowing that one occurs does not change the probability that the other occurs

Multiplication Rule for Independent Events

-Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs -If A and B are independent, P(A and B) = P(A)P(B)

Disjoint

-Two events A and B are this if they have no outcomes in common

Independent Events

-Two events A and B that both have positive probability are independent if P(B|A)=P(B)

Random

-We call a phenomenon random if individual outcomes are uncertain but there is, nonetheless, a regular distribution of outcomes in a large number of repetitions

Definition of Conditional Probability

-When P(A)>0, the conditional probability of B given A is P(B|A)=P(A and B)/P(A)

Addition rule for disjoint events

-f events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(A or B or C) = P(A) + P(B) + P(C)

90%

1.645

95%

1.96

99%

2.576

General Addition Rule for Unions of Two Events

For any two events A and B, P(A or B )=P(A)+P(B)-P(A and B)

The standard deviation for Poisson

mean**.5

Binomial Mean

np