Non-Standard Normal Distributions

Finding Z-Scores for Non-Standard Normal Variables

In statistics, the Z-score is a measure that describes how many standard deviations a data point is from the mean of a distribution. Z-scores are essential for converting values from any normal distribution to the standard normal distribution, allowing for probability calculations using standard normal tables.

• Z-score formula: where x is the observed value, μ is the mean, and σ is the standard deviation.

• Application: Z-scores allow us to use the standard normal table to find probabilities for non-standard normal distributions.

• Example: If commute times are normally distributed with a mean of 20 minutes and a standard deviation of 5 minutes, the probability that a randomly selected person commutes for less than 15 minutes can be found by: Using the standard normal table, .

Probabilities for Non-Standard Normal Distributions

To find probabilities for values in a non-standard normal distribution, convert the value to a Z-score and use the standard normal table.

• Step-by-step process:

  1. Calculate the Z-score for the value of interest.

  2. Use the standard normal table to find the probability associated with that Z-score.

• Example: The arrival times of a bus follow a normal distribution with mean 12 minutes and standard deviation 3 minutes. To find the probability the bus arrives on time (less than 10 minutes late):

Finding Values from Probabilities (Inverse Normal Calculations)

Sometimes, we need to find the value x in a normal distribution that corresponds to a given probability. This involves finding the Z-score from the probability and then solving for x.

• Inverse Z-score formula:

• Application: If 5% of people have a commute time less than x, find x such that . From the standard normal table, . minutes.

• Example: Heights of adult women are normally distributed with mean 160 cm and standard deviation 7 cm. Find the height x such that 5% of women are shorter than x: cm.

Summary Table: Z-Score Applications

Additional info:

• These concepts are foundational for understanding sampling distributions and statistical inference, including confidence intervals and hypothesis testing.

• Standard normal tables (Z-tables) are used extensively for these calculations.