Study Guide: Real Applications of Normal Probability Distributions

Normal Probability Distributions

Standard and Nonstandard Normal Distributions

Normal distributions are fundamental in statistics for modeling continuous data. A standard normal distribution has a mean of 0 and a standard deviation of 1, while a nonstandard normal distribution can have any mean (μ) and standard deviation (σ). To apply standard normal methods to nonstandard distributions, we use a conversion formula to transform values.

• Standardization Formula: Converts any value x from a normal distribution to a z-score: $z = \frac{x - \mu}{\sigma}$

• z-score: Represents the number of standard deviations a value x is from the mean μ.

• Purpose: Allows comparison and probability calculations using the standard normal table (Table A-2).

Finding Areas with a Nonstandard Normal Distribution

To find probabilities for nonstandard normal distributions, follow a systematic procedure:

1. Sketch the normal curve, label the mean and specific x values, and shade the region representing the desired probability.

2. Convert boundary x values to z-scores using the standardization formula.

3. Use technology (e.g., StatCrunch, Excel) or Table A-2 to find the area under the curve.

Example: Proportion of Men Taller Than 72 Inches

Heights of men are normally distributed with μ = 68.6 in. and σ = 2.8 in. Find the percentage of men taller than 72 in.

• Step 1: Sketch and shade the region to the right of x = 72 in.

• Step 2: Convert 72 in. to a z-score: $z = \frac{72 - 68.6}{2.8} = 1.21$

• Step 3: Use technology or Table A-2 to find the area to the right of z = 1.21. Area = 0.1123 (or 11.23%)

Using Table A-2

Table A-2 provides cumulative areas from the left for z-scores. To find the area to the right, subtract the cumulative area from 1.

• Example: For z = 1.21, cumulative area to the left is 0.8869. Area to the right: 1 - 0.8869 = 0.1131

Finding Values from Known Areas (Percentiles)

Sometimes, the area (probability or percentage) is known, and the corresponding value x must be found. This is common in design applications (e.g., cockpit or door height).

• Step 1: Sketch the normal curve and shade the region representing the given area.

• Step 2: Use Table A-2 or technology to find the z-score corresponding to the area.

• Step 3: Convert the z-score to the x value using: $x = \mu + z \cdot \sigma$

Example: Aircraft Cockpit Design (95th Percentile)

Heights of women are normally distributed with μ = 63.7 in. and σ = 2.9 in. Find the maximum height for the shortest 95% of women (the 95th percentile).

• Step 1: Sketch and shade the region for the shortest 95%.

• Step 2: Find z for area 0.9500. Table A-2 gives z = 1.645.

• Step 3: Calculate x: $x = 63.7 + 1.645 \times 2.9 = 68.5$ in.

• Interpretation: A height less than 68.5 in. would allow 95% of women to be eligible as pilots.

Example: Front Door Design (95th Percentile)

Heights of adults are normally distributed with μ = 66.2 in. and σ = 3.8 in. Find the door height that allows 95% of adults to walk through without bending.

• Step 1: Sketch and shade the region for the shortest 95%.

• Step 2: Find z for area 0.9500 (z = 1.645).

• Step 3: Calculate x: $x = 66.2 + 1.645 \times 3.8 = 72.5$ in.

• Interpretation: A door height of 72.5 in. accommodates 95% of adults, which is less than the standard door height of 80 in.

Key Concepts and Definitions

• Standard Normal Distribution: Mean = 0, Standard Deviation = 1

• Nonstandard Normal Distribution: Any mean μ, any standard deviation σ

• z-score: $z = \frac{x - \mu}{\sigma}$

• Percentile: The value below which a given percentage of observations fall

• Table A-2: Standard normal table listing cumulative areas from the left for z-scores

Comparison Table: Standard vs. Nonstandard Normal Distributions

Distribution Type	Mean (μ)	Standard Deviation (σ)
Standard Normal	0	1
Nonstandard Normal	Any value	Any value

Summary of Procedures

• To find probability for x: Convert x to z, use Table A-2 or technology to find area.

• To find x for a given area: Find z for the area, convert z to x using $x = \mu + z \cdot \sigma$.

Applications

• Designing equipment to accommodate a certain percentage of the population

• Evaluating suitability of standards (e.g., door heights, cockpit requirements)

• Interpreting real-world data using normal distribution models

Additional info:

• Technology tools (StatCrunch, Excel) can automate calculations for normal distributions.

• Always check that the solution makes sense in the context of the problem and the graph.