AP Statistics Content Review

Chance Experiment

Any activity or situation in which there is uncertainty about which of two or more possible outcomes will result.

Classical Approach to Probability

P(E) signifies the probability of an event E. It's the ratio of successes over total outcomes, and officially, is:

$$P(E) = \frac{\text{number of outcomes favorable to E}}{\text{number of outcomes in the entire sample space}}$$

Probability is ALWAYS between 0 and 1, where 0 means it's impossible and 1 means certain. (If it's not, then you did something wrong...)

Conditional Probability

Suppose that E and F are two events with P(F) > 0. The conditional probability of the event E given F has happened is: $$P(E|F) = \frac{P(E\cap F)}{P(F)}$$

Note: check out the syntax!

Law of Large Numbers

As the number of repetitions of a chance experiment increases, the chance that the relative frequency of occurrence for an event will differ from the true probability of the event by more than any small number approaches 0.

Basically, the more times you do the experiment, the experimental probability slowly approaches the theoretical probability. (If a problem involves this thinking, write out the name of this law!)

Boolean Expressions (Not A, A or B, A and B)

Not A: All outcomes that don't contain A, can also be expression $$A^c,\,A',\,\bar A$$

A or B: All outcomes in A, B, and both (so the sum of the two). $$A \cup B$$

A and B: All outcomes that are in both A and B (the intersection). $$A \cap B$$

Sample Space

The collection of all possible outcomes of a chance experiment.

Addition Rule (for Mutually Exclusive Events)

$$P(\text{E or F}) = P(E\cup F) = P(E) + P(F)$$ 

We touched upon this earlier, but this is an AXIOM of probability. This can be applied to multiple events as well!

Mutually Exclusive

!IMPORTANT!

Two events are mutually exclusive if they have NO OUTCOMES IN COMMON. The term disjoint is also used. 

Event

Any collection of outcomes from the sample space of a chance experiment.

Fundamental Properties of Probability

1. For any event E, 0 ≤ P(E) ≤ 1
2. If S is the sample space for an experiment, P(S) = 1.
3. If two events E and F are mutually exclusive, P(E or F) = P(E) + P(F).
4. For any event E, P(E) + P(not E) = 1. Therefore $$P(\text{not E}) = 1 - P(E)$$ and $$P(E) = 1 - P(\text{not E})$$

These are pretty simple, but remember to use these to do a sanity check when doing problems.

Simple Event

An event consisting of exactly one outcome.

Test Statistic

A value computed using sample data. It's the value that is used to make the final decision to reject or fail to reject H₀.

Hypothesis Test

In carrying out a test of H0 versus Ha, the null hypothesis H₀ will be rejected in favor of H_a only if sample evidence strongly suggests that H0 is false. 

If the sample does not provide such evidence, H0 will not be rejected. 

The two possible conclusions are reject H0 or fail to reject H0.

Tailed Tests

P-value

The measure of inconsistency between the hypothesized value and the observed sample. It's the probability of assuming H₀ to be true, and if it is less than or equal to alpha, then we reject H₀. Otherwise, we fail to reject. (There are exceptions to this, however, but we don't need to think about those.) 

It is also sometimes called the observed significance level.

Hypothesis Forms

One-Sample t Test for Population MEAN

$$H_0: \mu = \text{hypothesized value (hv)}$$

Test Statistic: $$t = \frac{\bar x - hv}{\frac{s}{\sqrt n}}$$

The degrees of freedom \(\text{df} = n-1\). We find the value of \(P\) from a t-table. It's the area to the right or left of whatever \(t\) value we get from the test statistic.

Assumptions for One-Sample t test for Population Mean

1. x-bar and s are the sample mean and sample standard deviation from a random sample.
2. The sample size is large (generally n ≥ 30) or the population distribution is approximately normal.

REMEMBER TO WRITE THESE DOWN IN THE FRQs WHENEVER POSSIBLE!!

Assumptions for Large-Sample z test

1. p-hat is the sample proportion from a random sample
2. The sample size is large. This can be used if n satisfied both (n)(hv) ≥ 10 and n(1-hv)≥10. 
3. If sampling is without replacement, the sample size can be no more than 10% of the actual population size.

REMEMBER TO WRITE THESE DOWN IN THE FRQs WHENEVER POSSIBLE!!

Large-Sample z test for p

This is for p, where p is a proportion.

$$H_0: p = \text{hypothesized value}$$

Test Statistic: $$z = \frac{\hat p - \text{hypo. value}}{\sqrt{\frac{(hv)(1-hv)}{n}}}$$

Null Hypothesis

A claim about a population characteristic that is initially assumed to be true. We don't assume otherwise unless there is strong evidence to prove it to be so. It is represented as H₀.

Power of a Test

The probability of rejecting the null hypothesis.

Effect of Various Factors on the power of the test:

1. The larger the size of the discrepancy between the HV and the actual value, the greater the power
2. The larger the significance level (alpha), the greater the power of the test
3. The larger the sample size, the greater the power

When H₀ is false, $$\text{power} = 1-\beta$$

Type II Error

Failing to reject H₀ when H₀ is actually false. This is a false negative, signified by beta. 

Type I Error

Rejecting H₀ when H₀ is true. False positive. Signified by alpha, which is also known as the significance level. 

Alternative Hypothesis

A competing claim about a population characteristic. Denoted by H_a.