Normal Probability Distributions: Finding Probabilities

Normal Probability Distributions

Finding Probabilities for Normally Distributed Variables

The normal distribution is a fundamental concept in statistics, describing many natural phenomena and measurement errors. When a random variable x is normally distributed, probabilities for intervals of x can be found by calculating the area under the normal curve for those intervals. This process often involves converting x-values to z-scores and using the standard normal distribution table or technology.

• Normal Distribution: A continuous probability distribution characterized by a symmetric, bell-shaped curve.

• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1.

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

Steps to Find Probabilities:

1. Convert the upper and lower bounds of the interval to z-scores using $z = \frac{x - \mu}{\sigma}$.

2. Use the standard normal distribution table (or technology) to find the area under the curve for the interval.

3. The area represents the probability that x lies within the specified interval.

Key Point: The transformation from x to z does not change the area under the curve, so probabilities are preserved.

Example 1: Probability Less Than a Given Value

Scenario: U.S. workers (ages 16+) who do not work from home travel an average of 26.8 minutes one-way to work each day, with a standard deviation of 17.1 minutes. Find the probability that a randomly selected worker travels less than 15 minutes to work each day. Assume travel times are normally distributed.

• Mean (\mu): 26.8 minutes

• Standard deviation (\sigma): 17.1 minutes

• Find: $P(x < 15)$

Solution:

1. Calculate the z-score for 15 minutes: $z = \frac{15 - 26.8}{17.1}$

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

3. The result gives the probability that a worker travels less than 15 minutes.

Interpretation: If the probability is low (e.g., less than 5%), this is an unlikely event.

Example 2: Probability Within and Above Intervals

Scenario: For each trip to a supermarket, a shopper spends an average of 47 minutes with a standard deviation of 12 minutes. Find:

• (a) The probability that a shopper spends between 26 and 56 minutes in the store.

• (b) The probability that a shopper spends more than 35 minutes in the store.

• (c) For 200 shoppers, how many would you expect in each interval?

Solution:

1. Calculate z-scores for the interval endpoints: For 26 minutes: $z_1 = \frac{26 - 47}{12}$ For 56 minutes: $z_2 = \frac{56 - 47}{12}$ For 35 minutes: $z = \frac{35 - 47}{12}$

2. Use the standard normal table to find the probabilities for each interval.

3. Multiply the probability by 200 to estimate the expected number of shoppers in each interval.

Example: If the probability for 26–56 minutes is 0.68, then $0.68 \times 200 = 136$ shoppers are expected in that interval.

Using Technology to Find Normal Probabilities

Modern statistical software and calculators can compute normal probabilities directly, without manual z-score conversion. Common tools include Minitab, Excel, StatCrunch, and the TI-84 Plus.

• Specify the mean (\mu), standard deviation (\sigma), and the x-values for the interval.

• The software returns the probability for the specified interval.

Example 3: In the U.S., 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 randomly selected state has fewer than 300 active physicians per 100,000 residents.

• Input: mean = 286, standard deviation = 57, x = 300

• Result: Probability ≈ 0.601 (or 60.1%)

Interpretation: About 60.1% of states have fewer than 300 active physicians per 100,000 residents.

Application Example: Batting Averages in Baseball

Batting averages of Major League Baseball players are approximately normally distributed with a mean of 0.245 and a standard deviation of 0.017.

• Find: What percent of players have a batting average of 0.260 or greater?

• For a roster of 40 players: How many would you expect to have a batting average of 0.260 or greater?

Solution:

1. Calculate the z-score for 0.260: $z = \frac{0.260 - 0.245}{0.017}$

2. Use the standard normal table or technology to find the probability for $x \geq 0.260$.

3. Multiply the probability by 40 to estimate the expected number of players.

Summary Table: Steps for Finding Normal Probabilities

Step	Description
1	Identify the mean (\mu) and standard deviation (\sigma) of the distribution.
2	Convert x-values to z-scores: $z = \frac{x - \mu}{\sigma}$
3	Use the standard normal table or technology to find the area (probability).
4	Interpret the probability in context (e.g., expected number in a sample).

Additional info:

• Technology can streamline probability calculations, especially for complex intervals.

• Normal distribution assumptions should be checked before applying these methods.

• For more on determining normality, see Appendix C (not included here).