AP STATS

What is a Cluster Sample?

The population is divided into groups, and some of these groups are randomly selected. Every individual in the chosen groups is surveyed.

What is a Stratified Random Sample?

The population is divided into groups and a SRS of individuals is chosen from every group.

What is Undercoverage Bias?

Occurs when some members of the population are less likely to be included in the sample than others.

What is Nonresponse Bias?

Individuals chosen for the sample are unable to be contacted or refuse to participate.

What is Response Bias?

A systematic pattern of incorrect responses in a sample survey which can lead to inaccurate conclusions.

What is a Codebook?

Each variable has an explanation of what the numbers represent.

What is a Multistage sample?

A sample design that uses a variety of sampling methods.

Observational Study

A study is observational when the researcher doesn't assign choices; they simply observe them.

Experiment

The researcher assigns treatments to participants.

What is Random Assignment?

Individuals are randomly assigned treatment groups.

How do you perform random selection?

Giving every research participant a number and choosing a sample using a random number generator.

What is Comparison?

Comparing two or more treatments.

What is Control?

Controlling extraneous variables by keeping other possible factors the same for all treatment groups.

What is Random Assignment (as a Principle of Experimental Design)?

Assigning experimental units to treatment groups at random.

What is Replication?

Using enough experimental units in each group so that any differences in the effects of the treatments can be distinguished from chance differences between the groups.

What is a Randomized Block Design?

Experimental units are blocked into groups based on a similar characteristic, then treatments are randomly assigned within each block.

What is a Matched Pairs Design?

Subjects are matched in pairs based on similar characteristics. One subject in each pair receives one treatment, and the other subject receives a different treatment.

What is a Double-Blind Study?

Neither the subjects nor those who interact with them and measure the response variable know which treatment a subject received.

What is a Single-Blind Study?

A study where only the subjects do not know which treatment they are receiving.

What is a Confounding Variable?

Occurs when a variable that is not among the explanatory or response variables in a study may influence the relationship between these variables.

What is a Control Group?

A group used in an experiment that does not receive the treatment.

What is a Factor?

A variable whose levels are controlled by the experimenter.

What is a Treatment?

A specific value of a factor.

What is a Placebo?

A fake treatment given to the control group.

What is the Placebo Effect?

When knowing you are part of an experiment, you have a perceived or actual improvement in medical condition.

What is Qualitative Data?

Data that consists of names, labels, or nonnumerical entries

What is Quantitative Data?

Data that is numerical and consists of counts/measurements.

What is an Explanatory Variable?

The variable whose effect you are examining.

What is a Response Variable?

The variable that is being predicted.

What is Random Selection?

Each member of the population has an equal chance of being chosen.

What is Inferential Statistics?

Used to make claims about a population.

What is Descriptive Statistics?

Organizing, summarizing, and displaying data.

What is a Statistic?

A number that describes a sample.

What is a Parameter?

A number that describes a population.

Define Population

The entire group of individuals that we want information about.

Define Sample

A subset of individuals in the population from which we collect data.

What is a Random Number Generator?

Every possible subject is assigned a number, and those numbers are randomly chosen until the desired sample size is achieved.

What is a Sampling Frame?

A list of every individual in the population.

What is Lack of Realism?

Drawing conclusions that do not generalize to the population of interest.

What is the General Addition Rule?

\P(A \cup B) = P(A) + P(B) - P(A \cap B)

What is Conditional Probability?

\P(A|B) = \frac{P(A \cap B)}{P(B)}

What is the definition of Independence?

Events A and B are independent if \P(A|B) = P(A) or \P(B|A) = P(B)

When is independence assumed?

With replacement = independent. Without replacement = not independent (unless sample is less than 10% of population).

What is the Multiplication Rule for Independent Events?

\P(A \cap B) = P(A) \cdot P(B), if A and B are independent.

What is a Geometric Random Variable?

A discrete random variable where X = the number of trials needed to obtain the first success.

What are the conditions for a Geometric Random Variable?

1. Each trial is independent.
2. There are only two possible outcomes.
3. The probability of success is the same for each trial.

What is the Mean of a Geometric Random Variable?

\mu_x = \frac{1}{p}, where p is the probability of success on each trial.

What is the Standard Deviation of a Geometric Random Variable?

\sigma_x = \sqrt{\frac{1-p}{p^2}}, where p is the probability of success on each trial.

What is the formula for Geometric Probability?

\P(X = k) = (1-p)^{k-1} \cdot p, where P is the probability of success on each trial. X is the number of trials, k.

What is a Binomial Random Variable?

A discrete random variable where X = the number of successes in a fixed number of trials.

What are the conditions for a Binomial Random Variable?

1. There are a fixed number of trials (n).
2. Each trial is independent.
3. There are only two possible outcomes.
4. The probability of success is the same for each trial.

What is the Mean of a Binomial Random Variable?

\mu_x = np, where n is the number of trials and p is the probability of success on each trial.

What is the Standard Deviation of a Binomial Random Variable?

\sigma_x = \sqrt{np(1-p)}, where n is the number of trials and p is the probability of success on each trial.

What is the formula for Binomial Probability?

\P(X = k) = {n \choose k} \cdot p^k \cdot (1-p)^{n-k}, where n is the number of trials and p is the probability of success on each trial. X is the number of successes, k.

When can a Binomial distribution be considered approximately normal?

If n is large enough such that np \geq 10 and n(1-p) \geq 10, then a binomial distribution will be approximately normal.

What is the Central Limit Theorem?

If n is large enough such that n \geq 30, the sampling distribution of sample means will be approximately normal.

What is Standard Error?

The standard deviation of the sampling distribution of the sample mean.

What is the formula for Standard Error?

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}, where \sigma is the standard deviation of the population and n is the sample size.

What is a Confidence Interval?

An interval of plausible values for a parameter.

How does a confidence interval estimate a population parameter?

Estimates the population parameter by using a sample statistic and margin of error.

What is the general formula for a Confidence Interval?

statistic\pm (critical value) \cdot (standard error)

What is Confidence Level?

The probability that the interval will capture the true population parameter in repeated samples.

What is Margin of Error?

The amount added and subtracted from the sample statistic to create the interval.

What is the Margin of Error for a population mean?

z* (critical value) \cdot \frac{\sigma}{\sqrt{n}}, where \sigma is the standard deviation of the population and n is the sample size.

What happens to the confidence interval as you increase confidence level?

Critical value increases. Interval becomes wider.

What happens to the confidence interval as you increase sample size?

Standard error decreases. Interval becomes narrower.

How do you use a confidence interval to determine if a hypothesized value is plausible?

The hypothesized value for the population parameter is not contained in the confidence interval. The hypothesized value is plausible if it is contained in the interval.

What is a Significance Test?

A formal procedure for using observed data to decide between two competing claims.

What is a Hypothesis?

A statement about the population parameter.

What is a Null Hypothesis?

The claim we weigh evidence against in a significance test.

What is an Alternative Hypothesis?

The claim about the population that we are trying to find evidence for.

What is the process of performing a hypothesis test?

Calculate a test statistic, find the p-value, and make a conclusion (reject or fail to reject the null hypothesis).

What is a Test Statistic?

A statistic calculated from sample data that measures how far the sample data diverges from what we would expect if the null hypothesis were true.

What is a P-Value?

The probability of obtaining the sample results (or more extreme) if the null hypothesis is true.

How do you draw a conclusion from a significance test?

If the p-value is less than alpha, we reject the null hypothesis and conclude that there is convincing evidence for the alternative hypothesis.

What is a Type I Error?

A type I error is when you reject the null hypothesis when the null hypothesis is true.

What is a Type II Error?

A type II error is when you fail to reject the null hypothesis when the alternative hypothesis is true.

When does a Type I error occur?

If we reject H0 when H0 is true.

When does a Type II error occur?

If we fail to reject H0 when Ha is true.

What is Power?

The probability of rejecting the null hypothesis when it is false.

When testing for one categorical variable…

Use the chi-square test for goodness of fit.

When testing for two categorical variables…

Use the chi-square test for independence.

When comparing the distribution of a categorical variable across multiple populations…

Use the chi-square test for homogeneity.

What is the formula for the Chi-Square Test Statistic?

\chi^2 = \sum\frac{(Observed - Expected)^2}{Expected}.