Normal Probability Distributions: Finding Probabilities

Normal Probability Distributions

Section 5.2: Finding Probabilities for Normal Distributions

This section explores how to determine probabilities for variables that follow a normal distribution. The process involves calculating the area under the normal curve for a specified interval, which represents the probability that a random variable falls within that interval. Both traditional table methods and technology-based approaches are discussed.

• Normal Distribution: A continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean (μ) and standard deviation (σ).

• Standard Normal Distribution: A special case of the normal distribution where the mean is 0 and the standard deviation is 1. Values are represented as z-scores.

• Z-score: The number of standard deviations a value x is from the mean. Calculated as: $z = \frac{x - \mu}{\sigma}$

• Probability Calculation: The probability that x falls within a certain interval is found by calculating the area under the normal curve for that interval.

Example: Probability for a Normally Distributed Variable

Scenario: U.S. workers who do not work from home travel an average of 26.8 minutes (μ) to work each day, with a standard deviation of 17.1 minutes (σ). Find the probability that a randomly selected worker travels less than 15 minutes to work each day.

• Step 1: Calculate the z-score for x = 15: $z = \frac{15 - 26.8}{17.1}$

• Step 2: Use the standard normal table to find the probability corresponding to this z-score.

• Result: The probability is 0.2451, or 24.51%.

• Interpretation: About 25% of workers travel less than 15 minutes, making this an unlikely event.

Example: Probability for Intervals and Expected Counts

Scenario: The average time spent in a supermarket is 47 minutes (μ) with a standard deviation of 12 minutes (σ). Find the probability for two intervals:

• Between 26 and 56 minutes: Probability is 0.7333. For 200 shoppers, expect about 147 shoppers in this interval.

• More than 35 minutes: Probability is 0.8413. For 200 shoppers, expect about 168 shoppers in this interval.

Example: Using Technology to Find Normal Probabilities

Modern statistical software (e.g., Minitab, Excel, StatCrunch, TI-84 Plus) can compute normal probabilities directly. You must specify the mean, standard deviation, and the x-values for the interval.

• Scenario: The number of active physicians per state is normally distributed with a mean of 286 and a standard deviation of 57 (per 100,000 residents). Find the probability that a state has fewer than 300 active physicians per 100,000 residents.

• Result: The probability is approximately 0.597, or 59.7%.

Key Concepts and Formulas

• Normal Probability: Area under the curve between two points represents the probability that a variable falls within that interval.

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

• Using Tables: Standard normal tables provide cumulative probabilities for z-scores.

• Using Technology: Software can compute probabilities without manual z-score conversion.

Applications

• Estimating the likelihood of events (e.g., commute times, shopping durations).

• Predicting expected counts in populations based on probability intervals.

• Utilizing technology for efficient probability calculations.