AP Stat Overall

Parameter

Some number which pertains to the part of the POPULATION that has a certain quality.

Examples of a parameter

The proportion of people in the United States who have a car is 86%. This is an example of…

Statistic

Some number which pertains to the part of the SAMPLE that has a certain quality.

Sampling Variability

Value of statistic varies with repeated random sampling. Not a bias, just random chance.

Population

All the things, the group in which the sample is derived from

Sample

The chosen group which is derived from the population for tests and statistics.

p

proportion of the POPULATION that the parameter represents

p hat

Proportion of the SAMPLE that the statistic represents

µ

Mean of the POPULATION.

x bar

mean of the sample

sigma(σ)

standard deviation of the POPULATION

S

Standard deviation of the SAMPLE

Sampling distribution

Distribution created from the data of samples

Population distribution

Distribution created from the data of a population

Unbiased Estimator

A sample distribution where a certain aspect is consistently the same as the population’s.

Biased estimator

A sample distribution where a certain aspect is consistently different in the same direction when compared to the population’s.

Accuracy

Proximity to the actual population proportion

Precision

Proximity to the other results.

less variability/smaller std dev

A larger sample size leads to…

µ of p hat

Mean of the sampling distribution proportions

µ of x bar

Mean of the sampling distribution means

10% condition

If the sample size is less than 10% of the population, it shows if the p can be calculated from σ.

Note: this is on the equation sheet

Large Counts Condition

If np≥10 and n(1-p)≥10, then is satisfied. Shows that for a PROPORTION, the sample distribution of p hats will be approximately normal.

Note: this is on the equation sheet

1. Define p or x in words.

2. State that the mean of the sample distributions is equal to the population mean

3. Use the 10% rule to find standard deviation of the sample means from the population standard deviation.

4. Use large counts condition, central limit theorem, or that the population is normal to show that the mean of sample statistics is equal to the parameter.

5. plug into calculator and use normal distr or inv norm and do your calculations.

Note: REMEMBER UNITS, and you HAVE to write out “Central Limit Theorem“

Notation Steps

Central limit theorem

If the population is distributed non-normally, then the mean of the sample distributions is distributed normally only if the sample size is greater than 30.

Proving sample normality if the population is distributed normally

If the population is distributed normally, then the sample is naturally distributed normally too.

How to calculate a factorial using calculator

Math → Prob → !, with the number before the !

How to calculate a combination using a calculator

Math → Prob → NCr, number before the NCr is the total number choosing from and the number after being the number chosen.

Chi square

Test that quantifies the amount observed values differ from expected ones.

Chi-square distributions as df goes up

Always positive, skewed right but become increasingly symmetrical

df for chi-square

number of categories needed before you can autofill the rest.

How to see how much each observed value contributes to the X²

Ctrb → scroll right, orders by first to last category.

Large counts for Chi-square

All expected values are greater than 5.

What kind of variables are needed to be able to perform chi-square.

Must be categorical

How to create matrices

2nd → matrix → edit → A/B

How to use matrices for chi-square

Observed → 2nd → matrix → select applicable matrix.

X² - GOF

Checks to see if the claimed distribution of a categorical variable is correct

X² - homogeneity

Tests to see if a categorical variable has the same distribution across different populations

X² test for independence

Checks if there is association between two categorical variables in a population

Requirements for ANY chi-square

Random Sample/Experiment

10% condition

Large counts

First step for solving a chi-square problem

State the population

Second step for solving a chi-square problem

Define the null and alternative. Always talk about association/difference in categorical variables

Third step for solving a chi-square problem

Clear conditions

(Random, 10%, large counts)

Fourth step for solving a chi-square problem

State formula: x² = Σ(o-e)²/e

Fifth step for solving a chi-square problem

Calculator dump

Sixth step for solving a chi-square problem

Interpret the p-value

Proving normality for two distributions

Use p₁ and p₂ for large counts during confidence intervals, and when using a test which just states that the two p values are equal, you use phat_c

Mean of sampling distribution for two distributions

Difference in means, µ₁ - µ₂ or p₁ - p₂

Degrees of freedom without calculator

Lower sample size - 1

Checking Independence for two populations

10% condition for each sample seperately.

2 prop z tests

stat → tests
used to test whether there is a statistically significant difference between two sample proportions.

p hat_c

The combined two sample proportion, proportion when H₀ doesn’t say if one value is necessarily true just that the two values are equal or smth

This is on the equation sheet but found by (x₁ + x₂) / (n₁ + n₂)

2 prop z interval

stat → tests
Used to create a interval of values for p₁ - p₂.

2 sample t test

stat → tests

Used to test if the difference in two means is statistically significant.

1 sample t test

stat → tests

Used to test if the chance of getting the statistic mean value is statistically significant or not.

2 sample t interval

stats → tests

Used to create a interval of plausible values of µ₁ - µ₂.

1 sample t interval

stats → tests

Used to create a interval of plausible mean values based on a statistic mean and sample size.

1 prop z interval

stats → tests

Creates a interval of plausible values given a phat value.

1 prop z test

stats → tests

tests to see if a specific statistic is statistically significant given a population p value.

Significance Tests

Assess the evidence to reject or fail to reject a null hypothesis

Null Hypothesis

A hypothesis where the original stated claim is true, has to be a equal

Alternative Hypothesis

The claim that refutes the null hypothesis, states that something else is happening

First step towards writing out a full question

State the population and parameter, with parameter in variable

Second step towards solving a full question

State the null and alternative hypotheses

P-Value

The probability of getting a more or equally extreme statistic as observed

What happens if P-Value < alpha

Too rare of a chance, you have convincing evidence to reject the null

What happens if P-value > alpha

The probability is possible, you don’t have convincing evidence to reject the null

Evaluating P-Value sentence structure

Assuming that (H₀ in context), there is a P-Value probability of getting a sample (parameter within context) of (statistic in number) or more extreme by just chance.

Significance Level(alpha)

Cutoff point for probability tests, determines whether your P-Value is convincing evidence or not. Usually 0.05

Final conclusion sentence structure

Our data (is/is not) statistically significant. Since our P-value is (greater than/less than) our alpha value, we (reject/fail to reject) the null. We (do/do not) have convincing evidence that our (alternative hypothesis is true, in context)

What variable should H₀ be about

Parameters. µ or p.

Type 1 Error

H₀ is true but we rejected the null.

Type 2 Error

H₀ is false but we failed to reject the null.

Probability of a type 1 error

alpha, or 1 - chance of correctly accepting null.

Note: this is because the alpha is the cutoff, so the probability you get convincing evidence purely by chance is just the chance you hit a value within the alpha threshold, or just the alpha value in terms of probability

third step towards solving a full problem

Check for random sample, selection or assignment

fourth step towards solving a problem

check independence

fifth step towards solving a problem

check for normality through CLT, large counts, x being distributed normally, or normal probability plot.

Sixth step for solving a full problem

Write out your formula for z or t score.

Choosing between p or p hat when solving for z score

Use p if given, otherwise use p hat.

When a double sided test is applicable

When your H_A states that your parameter is simply not equal to a value, rather than just greater or less than that.

seventh step for solving a problem

calculator dump all the information you get onto your paper, including degrees of freedom if doing a t-test.

eighth step for solving a problem

Draw conclusions using sentence structures using alpha and the p value found.

Power

Chance that the null is false and you correctly reject the null.

probability of a type 2 error

Beta, or 1 - probability of power

How changing alpha changes power

The higher the alpha, the higher the chance you reject the null and thus the higher the power and vice versa

How to change power

changing alpha, sample size, or choosing a more extreme H_A

How changing sample size changes power

A higher sample size correlates to a lower standard deviation, which increases the probability that you reject the null and thus increases the power and vice versa.

How changing the H_A changes power

A more severe H_A shifts the distribution for actual values further from that of the H₀. That means that you would have a higher chance of rejecting the null, as more of the distribution lies past the alpha threshold. That leads you to be more likely to reject the null, increasing power. The opposite is true.

Critical Value

The Z score whose positive side shows the upper bound of the proportion and the negative the lower bound. Use inv norm to find this for z^* or inv t for t^*

Don’t do this when interpreting a confidence interval

It is not true that there is a _% chance that the true parameter is within the interval. It’s either 100% chance or 0% chance.

Using ME restrictions to determine sample size needed

Use Z^* ± sqrt((p hat) x (1-p hat) / n ≤ ME and solve or Z^* ± σ/sqrt n ≤ ME

t distributions

distributions based on the sample itself, which approach normality as the degrees of freedom increase

Degrees of freedom

Found by number of independent categories - 1

How to find t*

2nd → distr → invT

How to find z*

2nd → distr → invNorm

What to use for p^ in z star confidence intervals

use the p^ given or 0.5 if not given.

Steps for solving a “build a confidence interval“ problem

1. Show what the population and parameter are.

2. Check if random sampled. If not, state “proceed with caution“, but you cannot draw conclusions later on using your interval because SRS not stated.

3. See if stated to be independent or check with 10% condition

4. Check normality using CLT, knowing x bar is normally distributed if x is normal, or using normal probability plot.

5. Write the formula you are using for your confidence interval.

6. Use stat → test → 1 Prop Z interval or TInterval.

7. Write down everything you give the calculator(besides the critical value)

8. Evaluate your interval.

Interpreting confidence levels

We are _% confident that our interval captures the true (parameter within context)

When you can use chance to talk about confidence intervals

There is a _% chance that before generating our interval, it will capture the true (parameter within context).