AP Statistics Content Review

AP Statistics Content Review

Official content review for 2020's AP Statistics Exam (with alterations in regards to the COVID-19 outbreak)

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Probability

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Chance Experiment

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Date created

Apr 29, 2020

Cards (25)

Probability

(11 cards)

Chance Experiment

Front

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

Back

Classical Approach to Probability

Front

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...)

Back

Conditional Probability

Front

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!

Back

Law of Large Numbers

Front

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!)

Back

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

Front

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). $$A \cap B$$

Back

Sample Space

Front

The collection of all possible outcomes of a chance experiment.

Back

Addition Rule (for Mutually Exclusive Events)

Front

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

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

Back

Mutually Exclusive

Front

!IMPORTANT!

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

Venn Diagram of mutually exclusive events -- they have no events in common!
Back

Event

Front

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

Back

Fundamental Properties of Probability

Front
  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.

Back

Simple Event

Front

An event consisting of exactly one outcome.

Back

Single Sample Hypothesis Testing

(14 cards)

Test Statistic

Front

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.

Back

Hypothesis Test

Front

In carrying out a test of H0 versus Ha, the null hypothesis H0 will be rejected in favor of Ha 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.

Back

Tailed Tests

Front
how we set up the hypothesis testing
Back

P-value

Front

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.

Back

Hypothesis Forms

Front
Summary of a hypothesis procedure
Back

One-Sample t Test for Population MEAN

Front

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

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.

Back

Assumptions for One-Sample t test for Population Mean

Front
  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!!

Back

Assumptions for Large-Sample z test

Front
  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!!

Back

Large-Sample z test for p

Front

This is for p, where p is a proportion.

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

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

 

Back

Null Hypothesis

Front

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.

Back

Power of a Test

Front

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, $$\text{power} = 1-\beta$$

Back

Type II Error

Front

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

Back

Type I Error

Front

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

Back

Alternative Hypothesis

Front

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

Back