Chapter 6: Social Science Sampling and Inferential Statistics

Why is measuring an entire population usually considered unrealistic or infeasible?

Researchers are constrained by time, energy, and financial resources.

What is the primary purpose of using sampling in social science research?

It allows researchers to generalize about the entire population.

What are the two broad categories of sampling methods?

Non-random (non-probability) sampling and random (probability) sampling.

Accidental Sampling

A non-random sampling method that results in a convenience sample.

In which non-random sampling method are samples drawn to match the population on specific characteristics?

Quota sampling.

Which sampling method relies on the researcher's judgment to select a representative sample?

Purposive sampling.

What is the defining feature of random (probability) sampling?

All elements of the targeted population have known probabilities of being selected.

List the four major types of random sampling methods mentioned in the text.

Simple random, systematic, stratified, and cluster (multistage) sampling.

What terms refer to the true mean and standard deviation of a random variable X in a population?

Population parameters.

What symbols are used to denote the population mean and population standard deviation?

μ and σ.

What terms refer to the mean and standard deviation calculated from a specific sample?

Sample statistics.

What symbols are used to denote the sample mean and sample standard deviation?

X̄ and s.

Sampling Error

The difference or deviation of sample statistics from the population parameters.

When does sampling error necessarily occur?

Whenever data from a sample are used to estimate population parameters.

What is the primary task of inferential statistics regarding sampling error?

To find a range where sample statistics are likely to fall within population parameters with certain probability.

Sampling Distribution

A hypothetical distribution of all possible values a statistic can take from all possible samples of the same size.

Why is the notion of a sampling distribution considered a pivotal concept in statistics?

It is essential for population parameter estimation and statistical testing.

According to the Law of Large Numbers, what is the relationship between the mean of sample means and the population mean?

The mean of the sample means in the sampling distribution equals the population parameter μ.

How does the standard deviation of the sampling distribution compare to the population standard deviation?

It is always smaller than the standard deviation of the population parameter.

Standard Error of the Mean

The standard deviation of the mean in a sampling distribution.

What is the formula for the standard error of the mean (σ_X̄)?

σ_X̄ = σ / √n

According to the Central Limit Theorem, what happens to the sampling distribution as the sample size becomes reasonably large (>20)?

The sampling distribution approximates a normal distribution regardless of the original population distribution.

How is the normal distribution of the sample mean denoted in notation?

X̄ ~ N(μ, σ / √n)

Where do most sample means fall in a sampling distribution relative to the true population mean?

Within 3 standard deviation units above and below μ.

What is the formula to compute a z-score using the sampling distribution of means?

z = (X̄ - μ) / σ_X̄

Confidence Interval (CI)

An interval estimate of population parameters based on information collected from a sample.

What does a Confidence Interval represent in terms of probability?

A range within which the population parameter will fall with a certain probability.

What is the formula for a 68% Confidence Interval when the true standard deviation is known?

68% CI = X̄ ± σ_X̄

What is the formula for a 95% Confidence Interval when the true standard deviation is known?

95% CI = X̄ ± 1.96σ_X̄

What is the formula for a 99% Confidence Interval when the true standard deviation is known?

99% CI = X̄ ± 2.58σ_X̄

Why must we use an estimate of the standard error of the mean in most practical research situations?

Usually, the true population standard deviation is unknown.

What correction is used in the denominator to find an unbiased estimate of the standard error?

N - 1

What is the formula for the unbiased estimate of the standard error (s_X̄)?

s_X̄ = s / √(N - 1)

What distribution must be used for sampling means if the population standard deviation is unknown?

The t-distribution.

What is the formula for the t-ratio when estimating the population mean?

t = (X̄ - μ) / s_X̄

What statistical value determines which specific t-distribution applies to a sample?

Degrees of freedom (df).

What is the formula for degrees of freedom (df) when estimating a single mean?

df = N - 1

How is the alpha value (α) calculated from a level of confidence?

α = 1 - level of confidence

What is the formula for a 95% Confidence Interval when the true standard deviation is unknown?

95% CI = X̄ ± t_c,.05 · s_X̄

What is the formula for a 99% Confidence Interval when the true standard deviation is unknown?

99% CI = X̄ ± t_c,.01 · s_X̄

What symbol is used for the standard error of the proportion?

s_P

What is the formula to calculate the standard error of the proportion (s_P)?

s_P = √[P(1 - P) / N]

What is the formula for a 95% Confidence Interval for a proportion?

95% CI = P ± 1.96s_P

What is the formula for a 99% Confidence Interval for a proportion?

99% CI = P ± 2.58s_P

In the formula for s_P, what does the letter P represent?

The sample proportion.

What determines the specific t-value used in a Confidence Interval calculation?

The degrees of freedom and the alpha level.

Which table should be used to find critical values when the population standard deviation is unknown?

Table B (the t-distribution table).

The __ refers to the variation among all possible sample means.

Standard deviation of the sampling distribution.

Does the sampling distribution approximate a normal distribution if the original population is not normally distributed?

Yes, provided the sample size is reasonably large.

What happens to the standard error of the mean as the sample size n increases?

The standard error decreases.

True or False: The sample mean is rarely identical to the population mean.

True.

In systematic sampling, how are elements of the population selected?

Elements are selected from the targeted population at a fixed, periodic interval.

What is another name for cluster sampling?

Multistage sampling.

If a researcher wants a 95% Confidence Interval, what is the corresponding α level?

.05

If a researcher wants a 99% Confidence Interval, what is the corresponding α level?

.01

What is the unbiased estimate of the standard error of the mean through algebraic reordering?

s_X̄ = s / √(N - 1)

Which sampling method results in a 'convenience sample'?

Accidental sampling.

What is the focus of the task of inferential statistics?

To find ways to estimate population parameters within a certain range and probability.

How many standard deviation units above and below μ do most sample means fall?

Three units.

What does the symbol σ represent in the standard error formula?

The population standard deviation.