Statistics Final Review PDF

Preview

Summary

Download Statistics Final Review PDF

Description

Probability • • • • •

Experiment/Trial – any activity from which an outcome, measurement, or result is obtained Random Experiment/Trial – when outcomes cannot be predicted with certainty Sample Space – the set of all the possible “outcomes” for a given experiment Outcome – one possible result of an experimental trial Event – a set containing one or more of the outcomes from the sample space

Example: Coin Toss • Experiment/Trial – Repeatedly toss a coin in the air, catch it, and call it • Random Experiment/Trial – Yes: we generally cannot predict the outcome of a coin toss • Sample Space – {heads, tails} • Outcome – heads or tails • Event – the results of a single coin toss (either heads or tails… there are no “groups” or outcomes, because we’re only tossing one coin, and you can only have one outcome at once) Some Basic Rules of Probabilities • 0  P(outcome)  1 The probability of any outcome is between 0 and 1 inclusive •  P(outcomei) = 1 The sum of probabilities for all mutually exclusive outcomes = 1 How do you Calculate Probability? P(outcomei )

frequency of outcomei count of desired outcomes size of relavant population count of all possible outcomes

Probability of Compound Events • Conjoint probability (“intersection”) o The probability of A and B → P(AB) o The probability that two events will occur simultaneously. o Multiplication rule • If A and B are independent events, P(A B) = P(A) * P(B) • If A and B are dependent events, P(A B) = P(A) * P(BA) = P(B) * P(AB) •

Combined probability (“union”) o The probability of A or B (or both) → P(AB) o The probability that either A or B (or both) will occur. Visually, this is represented by areas A + B + C A

C

B

o Additive rule ▪ If A and B are not mutually exclusive (i.e. they can occur at the same time), P(A B) = P(A) + P(B) − P(AB) ▪ If A and B are mutually exclusive, P(A B) = P(A) + P(B)

▪

Events are “mutually exclusive” if they cannot occur at the same time. B

Mutually exclusive P(AB) = 0

A

Conditional Probability  B) • The probability of A given B → P(A • Conditional probability is the probability that something will happen, given the occurrence of another event – e.g. probability of having health insurance given that you are unemployed. • In computing conditional probability, it can be helpful to construct a frequency table of variables. Create rows for each subject category and create columns for the treatment or condition. Sum the columns and rows in order to determine total frequencies and the size of the sample space. Independent Events vs. Dependent Events • Two events are independent if P(AB) = P(A) • Two events are dependent if P(A|B) = P(AB)/P(B) • If two events are independent, the probability of one occurring does not change the probability of the other occurring. It is important to note whether events are independent or dependent because the methods of calculating probabilities for independent and dependent events are different. However, you will never go wrong with the dependent formula!

A 4-Piece Puzzle – From Populations to Confidence Intervals There are four concepts that we have been talking about during this course that are all related. It helps to understand how they are related. The following table shows the evolution of ideas from populations to confidence intervals. PROPERTIES & COMMENTS



frequency (# observations)

POPULATION

DISTRIBUTION



 

▪

Mean

▪ ▪

Standard Deviation Population Size

▪

These are the values we really want to know, but they are also the values we typically don’t know. We estimate them from sample data.

▪

Mean

▪ ▪

Standard Deviation Sample Size

▪

Because the sample is a subset of the population, it typically has a smaller variance.

▪

We can take n samples from a population. For each sample, the sample statistics, and hence our population estimates, are different.

▪

Mean

▪

Standard Error

N

frequency (# observations)

SAMPLE

x Take a Sample

x

s

Take Many Samples

x

frequency (# samples)

SAMPLING DISTRIBUTION

x

Convert to Z Variable

probability

Z VARIABLE DISTRIBUTION

x

▪

=x

x = 

ns

n

1.96

z

Area under the z-distribution between two critical values is the probability the population mean lies between x − E and x + E .

We can record x for each sample in a separate histogram; this is a Sampling Distribution. The Central Limit Theorem: if n ≥ 30, the Sampling Distribution is a normal distribution.

x −  n

→  = x−z



▪

z=

▪

If x has a normal distribution (which the Central Limit Theorem tells us it does), then z has a unit normal distribution. If z = 0, our estimate of the population mean, x , is exactly equal to the actual population mean,  . This is not generally the case, so we use Confidence Intervals to specify a range of values in which the population mean is likely to be: x − E    x + E , E = zcrit   n

▪ 1

Sample mean: x

(aka, the standard deviation of the sampling distribution)

▪

-1

E (x ) =  s  nN

(mean x )

x

-1.96

x s n

▪

E

n

Constructing confidence intervals In case case in the following, E is the margin of error on each side of the mean. • Confidence interval for a population mean (n  30) o Write down the sample mean x and the sample size n o Write down the population standard deviation , if known; otherwise, use the sample standard deviation s as an estimate of  E = Zc • ( ) ≈ Zc • ( s ) o Find the critical value Zc for the given level of confidence n n o Calculate the margin of error E and create the confidence interval •

Confidence interval for a population mean (n < 30) o Write down the sample mean x and the sample size n o Write down the sample standard deviation s E = tc • ( s ) o Write down the degrees of freedom (n – 1) n o Find the critical value tc for the given level of confidence o Calculate the margin of error E and create the confidence interval

•

Confidence interval for a population proportion E Zc o Write down the sample size n o Write down the sample proportion pˆ and find qˆ =1 - pˆ o Find the critical value Zc for the given level of confidence o Calculate the margin of error E and create the confidence interval

pˆ qˆ n

Stating your conclusion After you have calculated a confidence interval (e.g. for a population mean), the best way to state your conclusion is “I am __% confident that the population mean is within the range ___ to ___.”...