Normal Probability Distributions

Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution with the following properties:

• Bell-shaped: The graph is symmetric and bell-shaped.

• Mean (μ) = 0: The mean of the distribution is zero.

• Standard deviation (σ) = 1: The standard deviation is one.

The equation for a normal distribution is:

For the standard normal distribution, μ = 0 and σ = 1.

Uniform Distributions

A uniform distribution is a continuous probability distribution where all outcomes are equally likely within a certain range. The graph is a rectangle, and the area under the curve represents probability.

• Area under the curve = 1

• Area = height × width

Density Curve

The density curve is the graph of any continuous probability distribution. The total area under the curve is always 1, establishing a direct correspondence between area and probability.

Notation and Cautions

• P(a < z < b): Probability that z is between a and b.

• P(z > a): Probability that z is greater than a.

• P(z < a): Probability that z is less than a.

• Caution: Do not confuse z-scores (horizontal axis values) with areas (probabilities under the curve).

Critical Values

A critical value is a z-score that separates significantly low or high values from the rest of the distribution.

Applications of the Normal Distribution

Z-Score Formula

The z-score for a value x is calculated as:

Guidelines for Using Normal Distributions

• Use graphs to visualize problems.

• Distinguish between z-scores and areas.

• Choose the correct side of the graph (right for upper percentiles, left for lower percentiles).

• Negative z-scores are on the left half of the distribution.

• Probabilities (areas) are always between 0 and 1.

Significantly High and Low Values

• Significantly high: P(x or greater) ≤ 0.05

• Significantly low: P(x or less) ≤ 0.05

Other thresholds (e.g., 0.01) may be used depending on context.

Sampling Distributions and Estimators

Sampling Distribution

The sampling distribution of a statistic is the distribution of that statistic for all possible samples of a given size from the same population.

• Sample proportion (p̂): Distribution of sample proportions from all samples of size n.

• Sample mean (̄x): Distribution of sample means from all samples of size n.

• Sample variance (s2): Distribution of sample variances from all samples of size n.

Behavior of Sample Statistics

• Sample proportions and means tend to be normally distributed as n increases.

• The mean of sample proportions equals the population proportion (unbiased estimator).

• The mean of sample means equals the population mean (unbiased estimator).

• The mean of sample variances equals the population variance (unbiased estimator).

• Sample standard deviation (s) is a biased estimator of population standard deviation (σ), but the bias is small for large samples.

Unbiased and Biased Estimators

• Unbiased estimators: Sample proportion, sample mean, sample variance.

• Biased estimators: Sample median, range, sample standard deviation.

The Central Limit Theorem (CLT)

Statement of the CLT

For all samples of size n > 30, the sampling distribution of the sample mean can be approximated by a normal distribution with mean μ and standard deviation σ/√n.

Formulas:

• Mean of sample means:

• Standard deviation of sample means (standard error):

• Z-score for sample mean:

Finite Population Correction

If sampling without replacement and n > 0.05N, adjust the standard error:

Rare Event Rule

If the probability of an observed event is very small under a given assumption, and the event occurs, the assumption is likely incorrect.

Assessing Normality

Normal Quantile Plot

A normal quantile plot (or normal probability plot) graphs sample data values against expected z-scores from the standard normal distribution. If the points are close to a straight line, the data are approximately normal.

• Construct a histogram: If not bell-shaped, data are not normal.

• Check for outliers: More than one outlier suggests non-normality.

• Use a normal quantile plot for further assessment.

Normal Approximation to the Binomial Distribution

Conditions for Normal Approximation

• np ≥ 5 and nq ≥ 5 (some recommend 10 instead of 5).

• Sample is a simple random sample or from independent binomial trials.

Parameters for the normal approximation:

• Mean:

• Standard deviation:

Apply a continuity correction by representing a discrete value x as the interval x - 0.5 to x + 0.5.

Procedure for Using the Normal Approximation

1. Check that np ≥ 5 and nq ≥ 5.

2. Find μ and σ for the normal distribution.

3. Apply continuity correction to x.

4. Graph and shade the appropriate area under the normal curve.

Finding Areas Under the Normal Curve

Cumulative Area from the Left

The cumulative area from the left up to a vertical line above a specific value of z can be found using statistical tables or software.

• Table A-2

• Statdisk

• Minitab

• Excel

• StatCrunch

Area Between Two Boundaries

The area bounded on the left and right by vertical lines above specific values can be found using calculators or software.

• TI-83/84 Plus calculator

• StatCrunch