Statistics Chapter 8: Sampling Variability

Questions and Answers

What is a statistic?

Any quantity computed from values in a sample

What does sampling variability refer to?

The observed value of a statistic depends on the particular sample selected from the population; typically, it varies from sample to sample.

What is a sampling distribution?

The distribution of a statistic

What is the first rule of the properties of the sampling distribution of the sample mean?

Mean of the distribution = population mean Signup and view all the answers

What does the Central Limit Theorem state?

When n is sufficiently large, the sample distribution of x bar is approximately normally distributed even when the population distribution is not normal. Signup and view all the answers

What is the minimum sample size for applying the Central Limit Theorem?

30 Signup and view all the answers

What is the formula for the Z distribution when the Central Limit Theorem is applied?

z = (x bar - mean of the distribution) / standard error Signup and view all the answers

What is the mean of proportions in the properties of the sampling distribution of p?

Mean of proportions = proportion (pi) Signup and view all the answers

What is the rule of thumb for using a normal approximation?

If both np >= 10 and n(1-p) >= 10, then it is safe to use a normal approximation. Signup and view all the answers

What is a variable?

Any characteristic whose value may change from one individual to another Signup and view all the answers

What defines a univariate data set?

Consists of observations on a single variable made on individuals in a sample or proportion Signup and view all the answers

What is selection bias?

The tendency for samples to differ from the corresponding population as a result of systematic exclusion. Signup and view all the answers

What does measurement or response bias refer to?

The tendency for samples to differ from the corresponding population because the method of observation tends to produce values that differ from the true value. Signup and view all the answers

What is nonresponse bias?

The tendency for samples to differ from the corresponding population because data is not obtained from all individuals selected. Signup and view all the answers

What is a simple random sample of size n (SRS)?

A sample that is selected in a way that ensures that every different possible sample of the desired size has the same chance of being selected. Signup and view all the answers

What is a sampling frame?

Each item on a list can be identified with a number and a table of random digits or a generator may be used to select the sample. Signup and view all the answers

What is meant by 'replacement' in sampling?

Mostly without replacement; assume without replacement if n. Signup and view all the answers

Study Notes

Key Concepts in Sampling Variability and Distributions

Statistic: A computed quantity from sample values, essential for summarizing data characteristics.
Sampling Variability: Reflects how the observed statistic is influenced by the specific sample chosen, leading to variations between different samples.
Sampling Distribution: The overall distribution of a statistic, providing insights into the expected behavior of sample statistics.

Properties of Sampling Distribution of the Sample Mean

The mean of the sampling distribution equals the population mean.
Standard error is calculated as the population's standard deviation divided by the square root of the sample size (n).
If the population is normally distributed, the sampling distribution of the sample mean (x bar) will also be normal, regardless of sample size.

Central Limit Theorem (CLT)

The sample distribution of the sample mean (x bar) approaches a normal distribution as the sample size (n) becomes sufficiently large, regardless of the population's distribution shape.
CLT applies accurately when the sample size exceeds 30.

Z Distribution in Central Limit Theorem

The z-score formula for sample means: z = (x bar - mean of the distribution) / standard error, which approximates a normal distribution.

Properties of the Sampling Distribution of Sample Proportions (p)

The mean of the sample proportion equals the population proportion (pi).
Standard error of the sample proportion is calculated using the formula: √(pi(1 - pi) / n).
When n is large and pi is not close to 0 or 1, the sampling distribution for proportions approximates a normal distribution.

Rule of Thumb for Normal Approximation

Use normal approximation if both np ≥ 10 and n(1 - p) ≥ 10 to validate the sample size adequacy for analytical procedures.

Definitions and Biases

Variable: Any characteristic that can vary among individuals in a population.
Data: Collected observations, which can involve one or multiple variables.
Univariate Data Set: Focuses on a single variable's observations from a sample.
Selection Bias: Occurs when certain segments of a population are systematically excluded, leading to skewed samples.
Measurement/Response Bias: Arises when the observation method leads to inaccuracies in data collection.
Nonresponse Bias: Develops when not all selected individuals provide data, affecting sample representation.

Sampling Techniques

Simple Random Sample (SRS): Each possible sample of a specified size has an equal chance of selection, ensuring unbiased representation.
Sampling Frame: A numbered list of items used to construct the sample, often relied on a random digit table or generator for selection.
Replacement: Generally assumes sampling without replacement unless stated otherwise, maintaining sample integrity.

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Explore key concepts in Sampling Variability and Sampling Distributions through this flashcard quiz. Each term is defined to help you understand the variability in statistics based on different sample selections. Perfect for mastering Chapter 8 of your statistics course.