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Finding Probabilities for Normally Distributed Variables Using Tables and Technology

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Normal Probability Distributions

Finding Probabilities for Normally Distributed Variables

In this section, we explore how to determine probabilities for variables that are normally distributed. This involves using the standard normal probability table (z-table) and various forms of technology. Mastery of these methods is essential for statistical inference and data analysis.

  • Normal Distribution: A continuous probability distribution that is symmetric about the mean, characterized by its bell-shaped curve.

  • Parameters: The normal distribution is defined by its mean (μ) and standard deviation (σ).

  • Standard Normal Distribution: A special case of the normal distribution with mean 0 and standard deviation 1. Probabilities are found using z-scores.

Converting to the Standard Normal Distribution (Z-Scores)

To use the standard normal table, raw data values (x) must be converted to z-scores. The z-score represents the number of standard deviations a value is from the mean.

  • Z-Score Formula:

  • Interpretation: A positive z-score indicates a value above the mean; a negative z-score indicates a value below the mean.

Example 1: Probability Above a Given Value

Problem: The average speed of vehicles on a highway is 67 mph (μ = 67) with a standard deviation of 3.5 mph (σ = 3.5). What is the probability that a randomly selected vehicle exceeds 70 mph?

  • Step 1: Calculate the z-score for x = 70

  • Step 2: Find the cumulative probability for z = 0.86 (rounded to two decimals for the z-table)

  • From the z-table: P(Z < 0.86) = 0.8051

  • Step 3: Find the probability of exceeding 70 mph

  • Interpretation: There is approximately a 19.5% chance that a randomly selected vehicle exceeds 70 mph.

  • Using technology (without rounding z): For z = 0.8571, cumulative probability is 0.8043, so P(X > 70) = 0.1957.

Example 2: Probability Below a Given Value

Problem: The number of physicians per state is normally distributed with μ = 280 and σ = 78 (per 100,000 residents). What is the probability that a randomly selected state has fewer than 300 physicians per 100,000 residents?

  • Step 1: Calculate the z-score for x = 300

  • Step 2: Round z to two decimals: z = 0.26

  • Step 3: Find cumulative probability from the z-table

  • From the z-table: P(Z < 0.26) = 0.6026

  • Interpretation: There is approximately a 60% chance that a randomly selected state has fewer than 300 physicians per 100,000 residents.

  • Using technology: Probability is approximately 0.6012.

Example 3: Probability Between Two Values

Problem: Using the same physician data (μ = 280, σ = 78), what is the probability that a state has between 300 and 350 physicians per 100,000 residents?

  • Step 1: Find cumulative probability for x = 350

  • Step 2: Find cumulative probability for x = 300

  • Step 3: Subtract the two probabilities

Using technology (Excel):

  • P(X < 350) = 0.8153

  • P(X < 300) = 0.6012

  • P(300 < X < 350) = 0.8153 - 0.6012 = 0.2141

Interpretation: There is a 21.4% chance that a randomly selected state has between 300 and 350 physicians per 100,000 residents.

Using Technology to Find Normal Probabilities

Statistical software and calculators can compute normal probabilities directly. Common tools include Excel, TI-84 Plus, StatCrunch, and Minitab. These tools require input of the mean, standard deviation, value of interest, and whether a cumulative probability is desired.

  • Excel Function: NORM.DIST(x, mean, standard_dev, cumulative)

  • Set cumulative = TRUE for cumulative probability (P(X < x)), FALSE for probability density function value.

  • Other software (Minitab, StatCrunch, TI-84 Plus) have similar input requirements.

Summary Table: Steps for Finding Normal Probabilities

Step

Description

Formula/Action

1

State the problem and identify mean (μ) and standard deviation (σ)

Given in problem

2

Convert x to z-score

3

Use z-table or technology to find cumulative probability

Look up P(Z < z) or use software

4

For probabilities above a value, subtract from 1

5

For probabilities between two values, subtract cumulative probabilities

Key Points

  • Always convert raw scores to z-scores before using the standard normal table.

  • Round z-scores to two decimals for table lookup; technology can use more precise values.

  • For probabilities above a value, subtract the cumulative probability from 1.

  • For probabilities between two values, subtract the cumulative probabilities.

  • Technology simplifies calculations and reduces rounding errors.

Example Applications

  • Determining the proportion of vehicles exceeding a speed limit.

  • Estimating the likelihood of a state having a certain number of physicians.

  • Calculating the probability that a measurement falls within a specified range.

Additional info: The examples provided illustrate both the use of the standard normal table and technology for finding probabilities. Mastery of both methods is recommended for comprehensive understanding and exam preparation.

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