6.4

Normal Probability Distributions

Overview

Normal probability distributions are fundamental in statistics for modeling real-world phenomena and for making inferences about populations based on sample data. This section focuses on the Central Limit Theorem (CLT), its practical applications, and considerations for using normal distributions in statistical analysis.

The Central Limit Theorem (CLT)

Definition and Importance

• Central Limit Theorem (CLT): For all samples of the same size n with n > 30, the sampling distribution of the sample mean $\bar{x}$ can be approximated by a normal distribution, regardless of the shape of the population distribution.

• The mean of the sampling distribution is $\mu$ (the population mean).

• The standard deviation of the sampling distribution is $\frac{\sigma}{\sqrt{n}}$ (where $\sigma$ is the population standard deviation).

This theorem allows statisticians to use normal probability methods for inference, even when the original population is not normally distributed, provided the sample size is sufficiently large.

Mathematical Statement

• Given a population with mean $\mu$ and standard deviation $\sigma$, and simple random samples of size $n$:

$\mu_{\bar{x}} = \mu$

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Practical Rules for Applications Involving a Sample Mean $\bar{x}$

Requirements

• The population must be normally distributed, or the sample size must be large ($n > 30$).

Key Properties

• Mean of all values of $\bar{x}$: $\mu_{\bar{x}} = \mu$

• Standard deviation of all values of $\bar{x}$: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Z-score conversion for $\bar{x}$:

$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

When the Population is Not Normally Distributed and $n \leq 30$

• The distribution of $\bar{x}$ cannot be well-approximated by a normal distribution, and normal-based methods do not apply.

Considerations for Practical Problem Solving

1. Check Requirements

• Verify that the normal distribution can be used by confirming that the original population is normal or the sample size is $n > 30$.

2. Individual Value or Mean from a Sample?

• Determine whether you are analyzing a single value $x$ or the mean $\bar{x}$ from a sample of $n$ values.

• For an individual value from a normal population:

$z = \frac{x - \mu}{\sigma}$

• For a sample mean from $n$ values:

$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Example: Boeing 737 Airline Seats

Problem Context

• American Airlines uses Boeing 737 jets with 126 seats in the main cabin. An engineer considers reducing the seat width from 16.6 in. to 16.0 in.

• Adult males have hip widths that are normally distributed with mean $\mu = 14.3$ in. and standard deviation $\sigma = 0.9$ in.

Questions

1. Find the probability that a randomly selected adult male has a hip width greater than 16.0 in.

2. Suppose all 126 seats are occupied by males. Find the probability that the mean hip width is greater than 16.0 in.

3. For seat design, which result is more relevant? What do the results suggest about reducing the seat width?

Solution (a): Probability for an Individual

• Convert 16.0 in. to a z-score:

$z = \frac{16.0 - 14.3}{0.9} = 1.89$

• From the standard normal table, the cumulative area to the left of $z = 1.89$ is approximately 0.9706.

• Probability that $x > 16.0$ is $1 - 0.9706 = 0.0294$ (or about 2.94%).

• Excel calculation gives a more accurate result: 0.0295.

Solution (b): Probability for the Mean of 126 Males

• Parameters for the sampling distribution:

$\mu_{\bar{x}} = 14.3$

$\sigma_{\bar{x}} = \frac{0.9}{\sqrt{126}} = 0.1$

• Convert 16.0 in. to a z-score for the sample mean:

$z = \frac{16.0 - 14.3}{0.1} = 17.00$

• From the standard normal table, the cumulative area to the left of $z = 17$ is approximately 0.9999.

• Probability that $\bar{x} > 16.0$ is $1 - 0.9999 = 0.0001$ (or 0.01%).

• It is highly unlikely that the mean hip width of 126 adult males will exceed 16.0 in.

Solution (c): Interpretation and Application

• The result from part (a) is more relevant for seat design, as individual seats are occupied by individual passengers.

• Approximately 3% of adult males would have hip widths greater than the seat width, which could result in several passengers per flight requiring special accommodation.

• Reducing the seat width to 16.0 in. does not appear feasible due to the potential discomfort and logistical challenges.

Summary Table: Z-Score Formulas

Situation	Z-Score Formula	Standard Deviation Used
Individual value from normal population	$z = \frac{x - \mu}{\sigma}$	$\sigma$
Sample mean from sample of size $n$	$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$	$\frac{\sigma}{\sqrt{n}}$

Additional info: The Central Limit Theorem is foundational for inferential statistics, enabling the use of normal probability methods for hypothesis testing and confidence intervals, even when the population distribution is unknown, provided the sample size is sufficiently large.