Calculate The Following Probabilities Using The Standard Normal Distribution.

Introduction

The standard normal distribution, often referred to as the z-distribution, is a normal distribution with a mean of 0 and a standard deviation of 1. It's widely used in statistics because it allows us to compare different normal distributions to each other.

Calculating probabilities using the standard normal distribution involves converting raw data to z-scores and then using z-tables or statistical software to find the corresponding probabilities.

Standard Normal Distribution

The standard normal distribution is defined by the probability density function:

Where:

• z is the z-score
• π is the mathematical constant pi (approximately 3.1416)
• e is the base of the natural logarithm (approximately 2.7183)

The standard normal distribution is symmetric about the mean (z = 0) and has a total area under the curve of 1. The probability of any value falling within a certain range can be found by calculating the area under the curve between the corresponding z-scores.

Calculating Probabilities

To calculate probabilities using the standard normal distribution, follow these steps:

1. Convert your data to z-scores using the formula:
   z = (X - μ) / σ
   Where:
   • X is the raw score
   • μ is the population mean
   • σ is the population standard deviation
2. Use the z-score to find the corresponding probability in the standard normal distribution table or using statistical software.
3. Interpret the probability based on the context of your data.

For example, if you want to find the probability that a value is less than a certain z-score, you would look up the cumulative probability for that z-score in the standard normal distribution table.

Example Calculations

Let's look at an example to illustrate how to calculate probabilities using the standard normal distribution.

Example 1: Finding P(Z ≤ 1.25)

To find the probability that a z-score is less than or equal to 1.25:

1. Look up the cumulative probability for z = 1.25 in the standard normal distribution table.
2. The table shows that P(Z ≤ 1.25) = 0.8944.

This means there is an 89.44% probability that a z-score will be less than or equal to 1.25.

Example 2: Finding P(0.5 ≤ Z ≤ 1.5)

To find the probability that a z-score is between 0.5 and 1.5:

1. Find P(Z ≤ 1.5) = 0.9332
2. Find P(Z ≤ 0.5) = 0.6915
3. Subtract the two probabilities: 0.9332 - 0.6915 = 0.2417

This means there is a 24.17% probability that a z-score will be between 0.5 and 1.5.