Statistics Overview and Distributions

Numerical Descriptive Measures

• Used to describe population size.

Normal Distribution Overview

• Location and shape described by mean (m) and standard deviation (s).

Binomial Distribution Overview

• Consists of n trials.
• Location and shape determined by probability (p).

Parameters in Distribution

• Unknown values often define the distribution's form.

Sample Reliance on Parameters

• Crucial for understanding parameters.

Statistics Overview

• Calculated numerical descriptive measures.
• Descriptive measures calculated from samples.

Sample Variability in Statistics

• Variations across samples.
• Random variables.

Sampling Distributions in Statistics

Repeated Sampling Overview

• Shows possible values and their frequencies.

Sampling Distribution of Statistics

• Defines the probability distribution of possible statistic values.
• From random samples of size n.

Central Limit Theorem

• Random samples from a non-normal population (with a finite mean and standard deviation).
• Large sample sizes (n) lead to an approximately normal distribution of the sample mean.
• Approximation improves as sample size (n) increases.

Central Limit Theorem (Alternative)

• Sum of n measurements is approximately normal.
• Involves mean (nm) and standard deviation.

Statistical Inference Statistics

• Deals with sums or averages of sample measurements.

Nearly Normal Distributions in Large Statistics

Understanding Behavior and Inference Reliability

• Describe behavior and evaluate inference reliability.

Normal Sample Distribution

• Guarantees a normal sampling distribution, regardless of sample size.

Sample Population Distribution

• Approximately symmetrical for sample populations.
• Becomes normal for small sample sizes (n).

Skewed Sample Population Requirement

• Sample size must be at least 30.
• Distribution must approach normality.

Random Sample Selection

• n-sized sample from a population with mean (m) and standard deviation (s).

Sample Sampling Distribution

• Mean: m
• Standard deviation: -1 (note: this is likely a typo/missing information)

Normal Population Distribution

• Normal sampling distribution for all sample sizes.

Sampling Distribution Normality in Nonnormal Populations

• Normal distribution observed with large sample sizes (n).

Standard Deviation of x-Bar

• Also known as Standard Error (SE).

Standardizing Interval of Interest

• If sampling distribution is normal or similar.
• Rescale the interval of interest.

Selecting Random Sample from Binomial Population

• Sample size (n).
• Parameter (p).

Sample Distribution Overview

• Distribution of sample proportion.

Mean & Standard Deviation

Sampling Distribution Overview

• Large sample sizes (n).
• Probability (p) not close to 0 or 1.
• Approximately normal distribution.

Standard Deviation of P-hat

• Also known as Standard Error (SE).

Standardizing or Rescaling Interval of Interest

• If sampling distribution is normal or similar.
• Rescaling the interval of interest.

Assignable Variable Change Cause

• Cause can be identified and corrected.

Random Variation Overview

• Uncontrolled variation.

Process Control Overview

• Random variation in process variable.
• Process is in control.

Controlling Process Variance

• Reducing variation.
• Keeping process variable measurements within specified limits.

Production Process

• Taking n-samples.
• Calculating sample mean.

CLT Sampling Distribution

• Approximately normal distribution.
• Most values fall within an interval.

Process Out of Control

• Values outside specified interval.

Control Chart Creation

• Collect k samples of size n.
• Use sample data to estimate mean (m) and standard deviation (s).

Mean Estimation in Process Variables

• Uses grand average of sample statistics.
• Calculates nk measurements on process variable.

Standard Deviation Estimation

• Estimated by s (standard deviation of nk measurements).

Control Chart Creation (cont.)

• Utilize centerline and control limits.

Production Sample Calculation

• Taking n-size sample.
• Calculating proportion of defective items.

CLT Sampling Distribution (in this context)

• Approximately normal distribution.
• Most values fall within interval.

Process Out of Control (in this context)

• Values outside specified interval.

Control Chart Creation (proportion context)

• Collect k samples of size n.
• Estimate p (proportion defective) for each sample using sample data.

Population Proportion Defective Estimation

• Estimated with ... (formula missing)

Grand Average of Sample Proportions

• Calculated for k samples.

Control Chart Creation (proportion context cont.)

• Utilize centerline and control limits.