Introduction to Statistical Inference
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Questions and Answers

What is the name given to a random variable or random vector when there is a sample of observations?

Statistic

What is the name given to the probability distribution of a statistic Y?

Sampling distribution of Y

What does Bias (W) represent?

The difference between the expected value of W and the true value of the parameter θ

What is the formula for mean squared error (MSE)?

<p>MSE (W) = V(W) + Bias (W)</p> Signup and view all the answers

What is the name given to the function of the sample (X1, ..., Xn) that is used to estimate the parameter θ?

<p>Point estimator</p> Signup and view all the answers

What is the range of values 1-a in a minimum level 1-a confidence interval?

<p>0 &lt; a &lt; 1</p> Signup and view all the answers

The length of a confidence interval is a measure of its precision.

<p>True (A)</p> Signup and view all the answers

What is the name given to the statement being tested that F∈ I = {F:0-∈}?

<p>Null hypothesis</p> Signup and view all the answers

What is the name given to the rival statement that we suspect is true, instead of the null hypothesis?

<p>Alternative hypothesis</p> Signup and view all the answers

What is the name given to a hypothesis if the distribution of (X1, ..., Xn) is completely specified by the hypothesis?

<p>Simple hypothesis</p> Signup and view all the answers

What is the name given to the subset C of X where if (x1, ..., xn)∈C, we reject the null hypothesis & if (x1, ..., xn)∉C, we accept the null hypothesis?

<p>Critical region</p> Signup and view all the answers

What is the name given to a statistic that is used in the specification of a critical region?

<p>Test statistic</p> Signup and view all the answers

What is the name given to the error that is committed when a true null hypothesis is rejected?

<p>Type I error</p> Signup and view all the answers

What is the name given to the error that is committed when a false null hypothesis is not rejected?

<p>Type II error</p> Signup and view all the answers

What is the name given to the probability of observing under Ho a sample outcome at least as extreme as the one observed?

<p>P-value</p> Signup and view all the answers

Flashcards

Identically distributed random variables

Random variables with the same probability distribution function.

Random sample

A set of independent and identically distributed random variables from a population.

Sample statistic

A numerical characteristic of a sample, such as mean or variance.

Population

The entire group of individuals or units from which a sample is drawn.

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Parametric family

A collection of probability distributions with a parameter that determines the shape of their distribution.

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Nonparametric family

A collection of probability distributions that cannot be completely described by a few parameters.

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Point estimator

A single value used to estimate an unknown parameter in a population.

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Statistic

A function of sample data used for estimation or testing.

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Sampling distribution

The probability distribution of a statistic.

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Bias

The difference between the expected value of an estimator and the true parameter value.

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Unbiased estimator

An estimator whose expected value equals the true parameter value.

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Mean squared error

A measure of the average squared distance between an estimator and the parameter it estimates.

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Confidence interval

A range of values likely to contain the true parameter value.

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Confidence level

The probability that a confidence interval contains the true parameter value.

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Statistical hypothesis

A statement about the probability distribution of a random variable.

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Null hypothesis (H0)

The hypothesis being tested, often a statement of 'no effect'.

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Alternative hypothesis (H1)

The hypothesis that supports evidence against H0.

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Critical region

The set of sample values that lead to rejection of the null hypothesis.

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Type I error

Rejecting a true null hypothesis.

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Type II error

Failing to reject a false null hypothesis.

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Size of a test

The probability of rejecting a true null hypothesis.

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p-value

The probability of observing a test statistic as extreme or more extreme than the one computed from the sample, assuming the null hypothesis.

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Study Notes

Introduction to Statistical Inference

  • Random Sampling: Two random variables are identically distributed if they share the same distribution function. A random sample of size n from a distribution F consists of independent and identically distributed (iid) random variables X₁, ..., Xₙ, all following the same distribution F.
  • Statistic: Given a random sample X₁, ..., Xₙ, a statistic is a function T(X₁, ..., Xₙ) of the sample. A statistic can be a single value or a vector. The probability distribution of a statistic is its sampling distribution.
  • Parametric Families: A parametric family of density functions is indexed by a parameter θ, which specifies the form of the distribution. The set of all possible values of θ is the parameter space Θ.
  • Nonparametric Families: Nonparametric families are collections of distribution functions that cannot be completely specified or indexed by a finite number of numerical parameters. They are also known as distribution-free families.

Point Estimation

  • Point Estimator: A point estimator is any function W(X₁, ..., Xₙ) of a sample. Any statistic is a point estimator,
  • Estimate: A numerical value of the estimator is called an estimate.
  • Bias: The bias of an estimator W of a parameter θ is the difference between the expected value of W and θ: Bias(W) = E(W) - θ. An unbiased estimator has zero bias, meaning E(W) = θ.
  • Mean Squared Error (MSE): The MSE of an estimator W is E[(W - θ)²]. It measures the average squared difference between the estimator and the parameter. Lower MSE implies greater precision. MSE = Variance + Bias².

Interval Estimation

  • Confidence Interval: A confidence interval for a parameter θ is a random interval [L(X₁, ..., Xₙ), U(X₁, ..., Xₙ)] such that for all distributions F in a family I, P[L(X₁, ..., Xₙ) ≤ θ ≤ U(X₁, ..., Xₙ)] = 1 - α, where 0 < α < 1. The value 1-α is the confidence level.
  • Confidence Level: The confidence level represents the probability that the interval will contain the true parameter value.
  • Equal Tails: If the distribution of a statistic T is symmetric, the length of a confidence interval based on T is minimized by choosing equal-tailed confidence intervals that maximize the probability of the parameter falling within.

Hypothesis Testing

  • Null Hypothesis (H₀): The statement being tested, typically representing a "no difference" condition.
  • Alternative Hypothesis (H₁): The rival statement that is suspected to be true.
  • Critical Region: A subset C of the sample space X such that if the sample falls in C, we reject H₀; otherwise, we accept H₀.
  • Test Statistic: A statistic T used to determine whether to reject or accept H₀.
  • Type I Error: Rejecting a true H₀ (false positive).
  • Type II Error: Accepting a false H₀ (false negative).
  • Size of a Test (α): The maximum probability of Type I error over all possible values of the parameter under H₀.
  • p-value: The probability of observing a more extreme result than the observed one, assuming H₀ is true. Smaller p-values provide stronger evidence against H₀.

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STAT 144: Introductory Statistical Theory I - Chapter 5 - Introduction to Statistical Inference PDF

Description

Explore the fundamentals of statistical inference, including concepts such as random sampling, statistics, and the distinctions between parametric and nonparametric families of distributions. This quiz will test your understanding of these key ideas and their applications in data analysis.

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