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Lecture 01: Variable Types, Parameters, & Statistics

• Statistical inference is framed as detecting a signal amid noise
• Observational units are the entities on which data is recorded
• Variables are characteristics recorded for each observational unit, and can be quantitative or categorical
• Parameters and statistics must be distinguished using appropriate notation

The 1969 American Draft Lottery

• The last draft lottery was on December 1, 1969
• Draft was based on birthdays to eliminate bias
• 366 capsules, each with a date, were put into a bin
• Capsules were drawn one at a time, assigning draft numbers
• September 14 was assigned draft number 1, signifying young men born in the date would be drafted

Was the Lottery Fair?

• Scatterplot axes: birthdays' sequential date on the x-axis, draft number on the y-axis

Detecting the Signal Amid Noise

• Statistics help detect meaningful patterns in collected data
• The "signal" is meaningful information
• "Noise" is unwanted variation interfering with the signal
• Examples illustrate detecting a signal, consider signals amid noise in various contexts

Observational Units & Variables

• Data must be organized and cleaned for effective description
• Observational units (cases) are entities about which information is recorded, which affects your sample size.
• A variable describes a characteristic of an observational unit that can assume different values

Types of Variables

• The variable type guides what kinds of summaries are appropriate
• Variables are quantitative or categorical

Quantitative Variables

• Quantitative variables, also called numerical or measurement variables, take on numerical values
• Arithmetic operations can be applied to to quantitative variables

Categorical Variables

• Categorical, or qualitative variables, place individuals or items into groups or categories, called levels

Populations, Samples, Parameters, and Statistics

• The population of interest is the group about which the researcher aims to draw a conclusion
• A sample is a subset of the population observed for the study

Parameters and Statistics

• Parameters and statistics are numerical quantities describing attributes of data
• A parameter describes an attribute of a population of interest
• A statistic describes an attribute of an observed sample

Quantitative Mean/Standard Deviation

• µ represents the true population mean.
• û represents the observed sample mean
• σ represents standard deviation of the population
• ô represents sample standard deviation

Categorical Proportion

• π = the true proportion of all women in Ann Arbor who are taller than 6 ft.
• n = the observed proportion of a sample of 100 women in Ann Arbor who are taller than 6 ft.

Lecture 02: Exploring Sampling Distributions: Sample Proportions and Sample Means

• Sampling distributions are used to help make predictions and inferences about population parameters based on sample data

Data Set Descriptions

• Institution dataset from US Department of Education has demographic information from postsecondary education institutions with Title IV funding
• Data includes 1842 US colleges and universities that grant bachelor's degrees in 2019

The Sampling Distribution

• Statistics are interested in the distribution of sample statistics

Distribution Attributes

• These distributions are characterized by central tendency, shape, and variability
• The sample proportion (îî) and the sample mean (û) are sample statistics

Variable Distributions

• A distribution summarizes how data points spread across possible values
• 3 main distribution attributes that are considered central tendency, shape, and variability (spread, dispersion)

Key Component of the Sampling Distribution

• A sampling distribution displays all possible values of a statistic from repeated samples of the same size from a population

Simulation Study

• The rate at which population of interest takes on a particular categorical outcome depends on a simulation study to understand its properties

Common Characteristics of Sampling Distributions

• Center: Should be true pop. proportion π
• Variability: Can contribute if samples aren't
• Shape: Skewness may be caused by factors like non random sampling.

Lecture 03: Approximating the Sampling Distribution of Using a Normal Curve

• Meeting specific assumptions allows a Normal curve to be a model for the sampling distribution of îî
• It helps allow for calculating the likelihood of observing specific sample proportion values

Characteristics of Sampling Distributons

• Symmetry and unimodality are common
• Adjusted using two quantities: a mean and a standard deviation

Central Limit Theorem

• The CLT has several versions, each describing how a sampling distribution of a statistic will behave, given two assumptions are met

Assumptions

• The sample must be collected randomly from the population
• A sufficiently large sample must be gathered for center limit theorem to take effect

Variable Type and Theorem Results

• For a categorical variable, a large enough sample is n·π≥ 10 and n(1-π)≥10
• If assumptions are met, by the CLT, sampling the distribution of many sample statistics can be approximated by a normal curve

Variable and Normal Distribution

• For a population parameter, we can define a sample size (n) and a sample mean (Û).

Normal Distribution Rules

• The normal distribution only works when you can expect at least 10 successes and at least 10 failures in a random sample of size n

Lecture 04: Exploring Sampling Distributions: Sample Means

• A sampling distribution shows the sample statistic itself, not just the sample data

Quantitative and Categorical

• Focusing on quantitative data you can visualize the distribution of the sample itself, whereas with categorical data you are working with the distribution produced counts

CLT and Distribution Behavior

• if large enough sample size, CLT says regardless of unimodal it must behave if ran from 0 to infinity

The effect of sample size

• The sample size directly impacts the behavior of a sample statistic, specifically, as the sample size n increases the variability of a statistic declines
• Statistics produced from larger samples tend to be more precise estimators of population parameters

1.2 Population distribution is not unimodal and symmetric

• Data was right skewed with 4,647.319 sample space

Central tendency of distribution

• Bell shapes but if don't use it gets skewed if not large enough
• population must be @ random

Sampling the data and analyzing to see properties and tendencies

• at small sample sizes, the sampeling dist. of A is also right skaud. the severity leads to declinc as sample size increases

The sampling distribution of u key characteristics

• the variability(spread) declines as n increases
• If unimodal and symmetric, the shape of the distribution of all possible sample means would test to being a normal aproxamently
• it gets a large sample if isn't symmetric to use the normal distribution

Lecture 05: Simulation-Based Hypothesis Tests

• Steps include specifying hypotheses, recognizing a simulated null distribution, interpreting a p-value, estimating a p-value, and recognizing p-values

Key Terms And Ideas For The Sampling Distribution

• These steps can assist in visualized how sample statistics behave across repeated samples.

Sampling

• Collecting a random sample of size n, assuming the null hypothesis is true
• Summarize sample results by calculating sample proportion stats and comparing these values
• Note the difference between Ho and statistic

Null Distribution

• Resulting distribution is referred to as null distribution, giving p = (y/n) can give the null sample hypothesis by random chance

P Values

• Can estimate P values by assessing proportions using certain steps, P value has a certain calculation

P Value and The Steps It Performs

• Perform a set of steps and you receive good evidence
• You provide a list and you receive an argument
• If it was lower it will say something small values in the sample
• The sample value in that case there will be a sample size

PValue and When You Assume

• Use a test whether is true
• You can try it
• Note: This example illustrates concepts applicable to various types of date, not just voter turnout in the US

Lecture 06: Hypothesis Tests for (π)

• Tests the objective is the measure the value of the p value
• To know which sample population to use

From Simulation to Theory-Based Methods of Inference

• The average of all the possible statistic values (e.g., their expected value) is population parameter.
• the standard deviation of at the possitive sample statistic decrease
• the sample is small enough to produce the data from lecture

The Normal Model for the Sampling Distribution of

• the sampling distribution of a population

Steps

• state the two claims about the unknow population proportion(unknown parameter)
• simulate could have been outcomes, you obtain after simulate or find more
• the number value by the population side and then find how the shape

Hypothesis Test

• In a test of competing hypotheses, H"0 is by default, assume to be true
• Competing claims have a goal but it the evidence

Lecture 07: Hypothesis Tests for μ

• Explores hypothesis testing focusing on a single population mean
• It outlines the t-distribution and its usage such as when normal model don't work and exploring the logic and steps of hypothesis testing

Goal of a Hypothesis

• To address claims about population
• How similar we can approximate the sampling distribution by a Normal Distibution.

CLT Characteristics

• If is from 0 to some number
• A lot of random sample with a different side of the sampling from

Tests and Errors

• There has to be so error rate or it has to make sense , that is the main goal of there tests

Steps For The Hypothesis

• The sample is collected with random from is population, the test value will be approximately

Lecture 08: Estimated Effect Sizes

• Objective is to explain the limitation of test which, when conclusion is drawn null
• This computes and interrupt
• Estimated effect sizes are important.

Limitations of Hypothesis test

• Test state and p values a influenced by two factors of a hypothesis. test 1 has a lot that samples which is also to estimate what going to be using