Φ(z)

Definition

Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution. It represents the probability that a random variable from a standard normal distribution is less than or equal to a given value z.

5 Must Know Facts For Your Next Test

1. Φ(z) is used to calculate probabilities for a standard normal random variable, which is denoted as Z ~ N(0, 1).
2. The value of Φ(z) represents the area under the standard normal curve to the left of the z-score, which is the probability that a random variable is less than or equal to that z-score.
3. Φ(0) = 0.5, meaning that the probability of a standard normal random variable being less than or equal to 0 is 0.5 or 50%.
4. Φ(-z) = 1 - Φ(z), which allows for the calculation of probabilities for negative z-scores.
5. Φ(z) can be used to find the value of z given a specific probability, which is useful for hypothesis testing and confidence interval construction.

Review Questions

• Explain the relationship between Φ(z) and the standard normal distribution.
  • Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution. It represents the probability that a random variable from a standard normal distribution, denoted as Z ~ N(0, 1), is less than or equal to a given value z. The value of Φ(z) corresponds to the area under the standard normal curve to the left of the z-score, which is the probability that the random variable is less than or equal to that z-score.
• Describe how Φ(z) can be used to calculate probabilities for standard normal random variables.
  • Φ(z) is used to calculate probabilities for standard normal random variables. For example, the probability that a standard normal random variable is less than or equal to a specific z-score is given by Φ(z). Additionally, the probability that a standard normal random variable is greater than a specific z-score can be calculated as 1 - Φ(z), and the probability that a standard normal random variable is between two z-scores can be calculated as Φ(z2) - Φ(z1).
• Analyze the significance of the property Φ(-z) = 1 - Φ(z) and how it can be utilized in statistical analyses.
  • The property Φ(-z) = 1 - Φ(z) is significant because it allows for the calculation of probabilities for negative z-scores. This is particularly useful in statistical analyses, such as hypothesis testing and confidence interval construction, where we often need to evaluate probabilities for both positive and negative z-scores. By using this property, we can efficiently calculate the probabilities for standard normal random variables without the need to refer to a standard normal distribution table for negative z-scores.

Related terms

Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1, which is the most commonly used normal distribution in statistics.

Cumulative Distribution Function (CDF): A function that gives the probability that a random variable is less than or equal to a given value.

z-score: A standardized score that represents the number of standard deviations a data point is from the mean of a normal distribution.