STATISTICS 188 NOTES
Chapter 7: Sampling Distributions
● Key Concepts:
○ Sampling Distribution: A sampling distribution is the distribution(spread) of
results that would occur had one selected all possible samples from a population.
In practice, a single random sample of a predetermined size is selected, and the
result obtained is just one of the results in the sampling distribution.
○ Sampling Distribution of the mean: This is the distribution of all possible
sample means calculated from all possible samples of a given size taken from a
population.
■ Unbiased Property of the Sample Mean: The sample mean (X̄) is an
unbiased estimator of the population mean (μ). This means that the mean
of the sampling distribution of the mean will be equal to the population
mean.
■ Standard Error of the Mean (σX̄ or SE(X̄)): This value is the standard
deviation of all possible sample means. It expresses how much sample
means vary from sample to sample. As the sample size (n) increases,
the standard error of the mean decreases by a factor equal to the square
root of the sample size.
■ Formula: σX̄ = σ / √n
○ Sampling from Normally Distributed Populations: If the population from
which samples are drawn is normally distributed, then the sampling distribution of
the mean will also be normally distributed, regardless of the sample size.
○ Sampling from Non-normally Distributed Populations - The Central Limit
Theorem (CLT): The CLT is crucial for making inferences about the population
mean without knowing the specific shape of the population distribution.
■ The CLT states that as the sample size (n) gets large enough, the
sampling distribution of the mean is approximately normally distributed.
This applies regardless of the shape of the individual values in the
population. A commonly used rule of thumb for "large enough" is n ≥ 30.
○ Sampling Distribution of the Proportion: Similar to the mean, there is a
sampling distribution for the proportion of events of interest in a population.
, ○ Sampling from Finite Populations: The CLT and the standard errors of the
mean and proportion are based on samples selected with replacement.
However, most survey research samples without replacement from populations
with a finite size (N). In such cases, a finite population correction factor
(FPCF) may be applied to the standard error ( Chapter 10).
● Formulae:
○ Standard Error of the Mean : σX̄ = σ / √n
○ Finite Population Correction Factor (FPCF) for the standard error of the mean:
√((N - n) / (N - 1)) (Introduced in Chapter 10)
Chapter 8: Confidence Interval Estimation
● Key Concepts:
○ Point Estimate: A single value of a sample statistic used to estimate a
population parameter (e.g., sample mean (X̄) as a point estimate for population
mean (μ)).
○ Interval Estimate (Confidence Interval): A range of numbers constructed
around a point estimate. The interval is constructed such that there is a known
probability that the interval includes the population parameter.
○ Sampling Error (e): The variation that occurs due to selecting a single sample
from the population. The size of the sampling error depends on the amount of
variation in the population (σ) and the sample size (n). Large samples tend to
have less sampling error.
■ Formula (for the mean): e = |X̄ - μ| (This is the definition, not a
calculation formula for determining sample size).
○ Level of Confidence (1 - α) X 100%: The probability that the constructed
confidence interval includes the population parameter. α is the proportion in the
tails of the distribution that is outside the confidence interval (α/2 in each tail for a
two-sided interval).
○ Critical Value (Zα/2 or tα/2): The value from a standard normal (Z) or t
distribution that determines the width of the confidence interval based on the
level of confidence. For a (1 - α) confidence level, the critical value leaves α/2 in
each tail of the distribution.
○ Confidence Interval Estimate for the Mean (σ Known): When the population
standard deviation (σ) is known, the confidence interval for the population mean
(μ) is calculated using the Z distribution.
■ Formula: X̄ ± Zα/2 * (σ / √n)
○ Confidence Interval Estimate for the Mean (σ Unknown): When the
population standard deviation (σ) is unknown and is estimated by the sample
Chapter 7: Sampling Distributions
● Key Concepts:
○ Sampling Distribution: A sampling distribution is the distribution(spread) of
results that would occur had one selected all possible samples from a population.
In practice, a single random sample of a predetermined size is selected, and the
result obtained is just one of the results in the sampling distribution.
○ Sampling Distribution of the mean: This is the distribution of all possible
sample means calculated from all possible samples of a given size taken from a
population.
■ Unbiased Property of the Sample Mean: The sample mean (X̄) is an
unbiased estimator of the population mean (μ). This means that the mean
of the sampling distribution of the mean will be equal to the population
mean.
■ Standard Error of the Mean (σX̄ or SE(X̄)): This value is the standard
deviation of all possible sample means. It expresses how much sample
means vary from sample to sample. As the sample size (n) increases,
the standard error of the mean decreases by a factor equal to the square
root of the sample size.
■ Formula: σX̄ = σ / √n
○ Sampling from Normally Distributed Populations: If the population from
which samples are drawn is normally distributed, then the sampling distribution of
the mean will also be normally distributed, regardless of the sample size.
○ Sampling from Non-normally Distributed Populations - The Central Limit
Theorem (CLT): The CLT is crucial for making inferences about the population
mean without knowing the specific shape of the population distribution.
■ The CLT states that as the sample size (n) gets large enough, the
sampling distribution of the mean is approximately normally distributed.
This applies regardless of the shape of the individual values in the
population. A commonly used rule of thumb for "large enough" is n ≥ 30.
○ Sampling Distribution of the Proportion: Similar to the mean, there is a
sampling distribution for the proportion of events of interest in a population.
, ○ Sampling from Finite Populations: The CLT and the standard errors of the
mean and proportion are based on samples selected with replacement.
However, most survey research samples without replacement from populations
with a finite size (N). In such cases, a finite population correction factor
(FPCF) may be applied to the standard error ( Chapter 10).
● Formulae:
○ Standard Error of the Mean : σX̄ = σ / √n
○ Finite Population Correction Factor (FPCF) for the standard error of the mean:
√((N - n) / (N - 1)) (Introduced in Chapter 10)
Chapter 8: Confidence Interval Estimation
● Key Concepts:
○ Point Estimate: A single value of a sample statistic used to estimate a
population parameter (e.g., sample mean (X̄) as a point estimate for population
mean (μ)).
○ Interval Estimate (Confidence Interval): A range of numbers constructed
around a point estimate. The interval is constructed such that there is a known
probability that the interval includes the population parameter.
○ Sampling Error (e): The variation that occurs due to selecting a single sample
from the population. The size of the sampling error depends on the amount of
variation in the population (σ) and the sample size (n). Large samples tend to
have less sampling error.
■ Formula (for the mean): e = |X̄ - μ| (This is the definition, not a
calculation formula for determining sample size).
○ Level of Confidence (1 - α) X 100%: The probability that the constructed
confidence interval includes the population parameter. α is the proportion in the
tails of the distribution that is outside the confidence interval (α/2 in each tail for a
two-sided interval).
○ Critical Value (Zα/2 or tα/2): The value from a standard normal (Z) or t
distribution that determines the width of the confidence interval based on the
level of confidence. For a (1 - α) confidence level, the critical value leaves α/2 in
each tail of the distribution.
○ Confidence Interval Estimate for the Mean (σ Known): When the population
standard deviation (σ) is known, the confidence interval for the population mean
(μ) is calculated using the Z distribution.
■ Formula: X̄ ± Zα/2 * (σ / √n)
○ Confidence Interval Estimate for the Mean (σ Unknown): When the
population standard deviation (σ) is unknown and is estimated by the sample
