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AP Statistics Study Guide

1
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) based on many sample
samples
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The concept of a sampling distribution and the central limit theorem are crucial in
statistical analysis and decision-making
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The
company wants to know the average length of its cells
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Instead, they take a sample of cells and measure their distances
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Now, the central limit theorem comes into play
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This means that for large enough
sample sizes, we can assume that the sample means to follow a normal
distribution, simplifying the statistical analysis
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For instance, in opinion polls, the central limit theorem estimates the
proportion of people who hold a certain view by taking a population sample and
calculating the mean
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In summary, the sampling distribution and the central limit theorem form the
basis of inferential statistics, allowing us to generalize a population using a sample
of data
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Probability and Probability Distributions
Probability and Probability Distributions: Probability is the likelihood of an event
occurring
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Examples
of common probability distributions include the normal, binomial, and Poisson
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It is expressed as a number between 0 and 1, with
0 indicating that an event is impossible and one meaning that an event is certain
to happen
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Random variables are values subject to randomness and can take different
values for different outcomes of a random process
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Discrete probability distributions include the binomial distribution and the
Poisson distribution
...
An example of a binomial distribution is the
number of heads in 10 coin flips
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Continuous probability distributions include the normal distribution, a
symmetrical bell-shaped curve commonly used to model many continuous
variables such as height, weight, and IQ scores
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In summary, probability and probability distributions are important tools for
understanding and predicting the behavior of random processes
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It gives us an idea of the
uncertainty in our assessment and is calculated from a sample
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Confidence intervals are a crucial statistical inference tool used to estimate
population parameters based on sample data
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For example, consider a survey of 1000 people to determine the proportion of
people who support a certain political candidate
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55, and the confidence interval for the
population proportion is calculated to be (0
...
58) with a 95% confidence level
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52, 0
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The confidence level represents the degree of certainty that the interval contains
the true population parameter
...
Conversely, lower
confidence levels correspond to narrower gaps, reflecting greater certainty in the
forecast
...
They allow researchers and practitioners to

make inferences about populations based on sample data and to make informed
decisions based on the data
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They are widely
used in statistical inference and decision-making, providing a measure of the
uncertainty in the estimate and allowing us to make informed conclusions based
on the data
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Hypothesis Testing
Hypothesis Testing: Hypothesis testing is a statistical procedure that uses sample
data to test a claim or hypothesis about a population parameter
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Hypothesis testing is a statistical procedure that uses sample data to test claims
or assumptions about a population parameterHypothesisis testing aims to
determine whether there is enough evidence in the sample data to support or
reject a particular hypothesis about the population
...

The null hypothesis represents the default assumption or status quo and usually
states no difference or effect
...


For example, in a medical study, the null hypothesis might be that a new drug
does not affect blood pressure, while the alternative view is that the new
medication does influence blood pressure
...
The significance level determines the threshold for

accepting or rejecting the null hypothesis
...


A commonly used test statistic in hypothesis testing is the p-value, which
represents the probability of observing the sample data if the null hypothesis is
true
...


To make data-based decisions, hypothesis testing is widely used in various fields,
including medicine, psychology, economics, and engineering
...


In summary, hypothesis testing is a statistical procedure that uses sample data to
test claims or assumptions about a population parameter
...
The p-value is a commonly used test statistic in
hypothesis testing, and it helps researchers and practitioners to make informed
decisions based on the data
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Design of Experiments
Design of Experiments: The design of experiments (DOE) is the systematic and
planned collection of data to test hypotheses and make informed decisions
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There are several common

experimental designs, including completely randomized, randomized block, and
factorial designs
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It involves carefully selecting and
manipulating variables of interest, controlling extraneous variables, and
randomizing the assignment of treatment levels to subjects to reduce the impact
of confounding variables
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In this study, the variables of interest
are the type of fertilizer and the amount applied
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Randomization ensures that any observed effects are due to the fertilizer,
not chance
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In a completely randomized design, the treatments are randomly assigned to the
subjects, with no control over extraneous variables
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In a randomized block design, the subjects are first divided into blocks based on
some extraneous variable, and then the treatments are randomly assigned within
each block
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In a factorial design, two or more variables are manipulated at the same time
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DOE is widely used in many fields, including agriculture, medicine, engineering,
and psychology, to make informed decisions based on data
...


In summary, the design of experiments (DOE) is a systematic approach to
collecting data that involves carefully selecting and manipulating variables,
controlling extraneous variables, and randomizing the assignment of treatment
levels to subjects
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DOE is widely
used in many fields to make informed decisions based on data and to isolate the
effects of variables of interest
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Regression and Correlation
Regression and Correlation: Regression analysis is a statistical method used to
model the relationship between a dependent variable and one or more
independent variables
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Correlation measures the
strength and direction of the relationship between two variables
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Regression analysis models the relationship between a dependent variable and
one or more independent variables
...
Regression
analysis can be used to make predictions, build models for causal inference, and
determine which independent variables are significant predictors of the
dependent variable
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The dependent variable is salary, and the independent variable is age
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Correlation measures the strength and direction of the relationship between two
variables
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A negative correlation means that as one variable
increases, the other variable decreases
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For example, consider a study examining the relationship between height and
weight
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A correlation coefficient close to 1 would indicate a
strong positive correlation, while a correlation coefficient close to 0 would
indicate no correlation
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These methods allow for data analysis and outcomes

prediction, helping researchers and practitioners make informed decisions based
on data
...
Regression analysis models the relationship
between a dependent variable and one or more independent variables, while
correlation measures the strength and direction of the relationship between two
variables
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They are
often used when the assumptions of parametric tests still need to be met
...

Non-parametric tests are statistical tests that do not assume a specific population
distribution, such as a normal distribution
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Non-parametric tests are useful in a variety of situations, including:
When the data is not normally distributed
When the sample size is small
When the data is ordinal or nominal (categorical) rather than continuous
Examples of non-parametric tests include:

Wilcoxon rank-sum test: This test compares the medians of two independent
groups
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Kruskal-Wallis test compares the medians of more than two independent groups
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Friedman test compares the medians of repeated measures across multiple
dependent groups
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Non-parametric tests are commonly used in medicine, psychology, and
environmental science
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In psychology, non-parametric tests might be used to compare the scores of
different groups on a mental ability test when the data is not normally
distributed
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Examples of non-parametric tests include the
Wilcoxon rank-sum test, the Kruskal-Wallis test, and the Friedman test
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8
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They are used to test
hypotheses about the standard of a people
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One-sample tests are statistical tests that compare the mean of a sample to a
known or hypothesized population mean
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Examples of one-sample tests include:

T-test: This test is used to compare the mean of a sample to a known population
means or a hypothesized mean
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Z-test: This test is used to compare the mean of a sample to a known population
mean
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One-sample tests are used to answer questions such as:

Is the average height of a group of people different from the average size of the
general population?
Is the average SAT score of students in a particular school different from the
national average SAT score?
The sample mean is compared to a known population means or a hypothesized
compromise in these examples
...


In summary, one-sample tests are statistical tests that compare the mean of a
sample to a known or hypothesized population mean
...
These tests are used to test hypotheses

about the mean of a people and are commonly used in fields such as business,
education, and social sciences
...
Two-Sample Tests
Two-Sample Tests: Two-sample tests are statistical tests that compare the means
of two independent samples
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Examples of two-sample tests
include the independent t-test, the paired t-test, and the Mann-Whitney U test
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These tests are used to test hypotheses about the
difference between the means of two populations
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Examples of two-sample tests include:

Independent t-test: This test compares the means of two independent samples
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Paired t-test: This test compares the means of two dependent samples
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Mann-Whitney U test is used to compare the means of two independent samples
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Two-sample tests are used to answer questions such as:

Is there a difference in the average weight loss between the two different diet
programs?
Is there a difference in the average IQ between two different ethnic groups?
The means of two independent samples are compared in these examples, and the
test statistic is calculated
...


In summary, two-sample tests are statistical tests that compare the means of two
independent samples
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These tests are used to test
hypotheses about the difference between the means of two populations and are
commonly used in fields such as medicine, psychology, and marketing
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Analysis of Variance (ANOVA)
Analysis of Variance (ANOVA): Analysis of variance (ANOVA) is a statistical method
used to test the equality of means of two or more groups
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There are several types of
ANOVA, including one-way ANOVA, two-way ANOVA, and repeated measures
ANOVA
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It is used to determine if there is a
significant difference between the means of two or more groups and to identify
which group(s) are different
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For
example, suppose you are a manufacturer of light bulbs and want to determine if
there is a difference in the lifespan of three different brands of bulbs
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Two-way ANOVA is used when you have two independent variables, each with
multiple levels
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You could collect data on the
lifespan of bulbs from multiple brands and temperatures and perform a two-way
ANOVA to test the hypothesis
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For example, you might measure the height of a group of individuals at multiple
time points and use repeated measures ANOVA to determine if there is a
significant difference in height growth between the individuals
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Chi-Square Tests
Chi-Square Tests: The chi-square test is a non-parametric statistical test used to
determine if there is a significant association between two categorical variables
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For example, a chi-square test could be
used to test if there is a significant association between smoking and lung cancer
or if the observed number of heads in a series of coin flips differs from the
expected number
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Goodness of Fit Tests
The goodness of Fit Tests: Goodness of fit tests are used to determine how well a
model fits a data set
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The goal is

to determine if the observed and expected frequencies are close enough that the
model can be considered a good fit for the data
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13
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Unlike linear regression, which models the relationship as a
straight line, non-linear regression can model more complex relationships
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Sales increase with temperature up to
a certain point and decrease with further temperature increases
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Time Series Analysis
Time Series Analysis: Time series analysis is a statistical method used to analyze
and model time-based data
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For example, a time series analysis could be used to
model the relationship between inflation and the unemployment rate over some
time to make predictions about future values
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Bayesian Statistics
Bayesian Statistics: Bayesian statistics is a branch of statistics that uses Bayes'
theorem to update probabilities based on new information
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For example, a Bayesian statistician might use
prior assumptions about the probability of a disease being caused by a certain risk
factor and update those beliefs based on new data collected from a study
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