Unit 5 Guide: The Foundations of Statistical Inference

Parameter

A number that describes some characteristic of the population.

Statistic

A number that describes some characteristic of a sample.

Sampling Distribution

The distribution of values taken by a statistic in all possible samples of the same size.

Biased Estimator

A statistic whose sampling distribution's mean is not equal to the true value of the parameter.

Unbiased Estimator

A statistic whose sampling distribution's mean is equal to the true value of the parameter.

Sample Proportion

The proportion of successes in a sample, denoted as (\hat{p}) (p-hat).

Mean of Sampling Distribution (Proportion)

The mean of the sampling distribution of (\hat{p}) is equal to the population proportion (p).

Standard Deviation of Sampling Distribution (Proportion)

The standard deviation is given by (\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}).

10% Condition

For sampling without replacement, if (n\le 0.10N), the standard deviation formula for proportions is accurate.

Large Counts Condition

For a Normal approximation of the sampling distribution of a proportion, both (np\ge10) and (n(1-p)\ge10) must hold.

Sampling Distribution of a Sample Mean

The distribution of values taken by the sample mean from repeated samples.

Mean of Sampling Distribution (Mean)

The mean of the sampling distribution of (\bar{x}) is equal to the population mean (\mu).

Standard Deviation of Sampling Distribution (Mean)

The standard deviation is given by (\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}).

Central Limit Theorem (CLT)

If the sample size is large enough ((n \ge 30)), the sampling distribution of the sample mean is approximately Normal.

Population Distribution Normality

If the population distribution is Normal, the sampling distribution of (\bar{x}) is also Normal.

Shape of Sampling Distribution

The Central Limit Theorem states the shape becomes Normal as sample size increases.

Common Mistake: Population vs. Sampling Distribution

Confusing changes in the sample size with changes in the population distribution.

Law of Large Numbers (LLN)

As the sample size increases, the sample mean (\bar{x}) approaches the population mean (\mu).

Correct Z-score for Sampling Distribution

Z-score for sampling distribution is calculated as (z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}).

Notation Errors

Greek letters denote parameters (e.g., (p, \mu)), and English letters denote statistics (e.g., (\hat{p}, \bar{x})).

Center of Sampling Distribution (Mean)

Both sample proportion and sample mean are unbiased estimators of their respective population parameters.

Standard Deviation of Estimator (Mean)

Is less variable than individual observations and is affected by sample size.

Normality Condition for Proportions

Large Counts Condition must be applied, not CLT.

Magic Number for Central Limit Theorem

A sample size of at least 30 is required for the Central Limit Theorem to apply.

Sampling Distribution Characteristics

Includes Center, Spread, and Shape for both means and proportions.

Approximate Normality

The sampling distribution can be modeled by a Normal curve under certain conditions.

Spread of Sample Proportion

Describes variability in sample proportions based on sample size.