04 Cómo usar la tabla de distribución normal

Key Concepts

Tabla de distribución normal 01 (Standard Normal Distribution Table)
Media cero y desviación típica uno (Mean of zero and standard deviation of one)
Área por debajo de un valor Z (Area under a Z-value)
Área por encima de un valor Z (Area above a Z-value)
Simetría de la distribución normal (Symmetry of the normal distribution)
Probabilidad (Probability)
Valor Z positivo y negativo (Positive and negative Z-values)

Using the Standard Normal Distribution Table

The video explains how to use the standard normal distribution table (tabla de distribución normal 01) to find probabilities associated with a normal distribution that has a mean of 0 and a standard deviation of 1. The table provides the area under the curve to the left of a given Z-value.

Limitation of the Table: The table only directly provides the area (probability) below a positive Z-value.
Example: To find the area below Z = 0.34, locate 0.3 in the table's first column and 0.04 in the first row. The intersection gives the area (probability) as 0.6331. This means P(Z < 0.34) = 0.6331.

Calculating Areas Above a Positive Z-Value

Since the total area under the normal distribution curve is 1, the area above a Z-value can be calculated by subtracting the area below it from 1.

Formula: P(Z > z) = 1 - P(Z < z)
- Where 'z' is a positive Z-value.
Explanation: The probability of being above a certain Z value is equal to 1 minus the probability of being below that Z value.

Calculating Areas Above a Negative Z-Value

The video leverages the symmetry of the normal distribution to calculate areas above negative Z-values.

Method: The area above a negative Z-value (-z) is equal to the area below the corresponding positive Z-value (z).
Formula: P(Z > -z) = P(Z < z)
"Trick": Change the sign of Z and flip the inequality sign. If you have P(Z > -z), change it to P(Z < z).

Calculating Areas Below a Negative Z-Value

This scenario combines the previous two concepts.

Method: The area below a negative Z-value (-z) is equal to the area above the corresponding positive Z-value (z). Then, the area above the positive Z-value is calculated as 1 minus the area below it.
Formula: P(Z < -z) = P(Z > z) = 1 - P(Z < z)
Explanation: First, transform the probability of being below a negative Z to the probability of being above the positive counterpart. Then, calculate the probability of being above using the formula described earlier.

Calculating Areas Between Two Z-Values

To find the area between two Z-values (Z1 and Z2), subtract the area below the smaller Z-value (Z1) from the area below the larger Z-value (Z2).

Formula: P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
Explanation: This calculates the area under the curve between Z1 and Z2 by subtracting the cumulative probability up to Z1 from the cumulative probability up to Z2.

Summary of Formulas

The video provides a summary of the formulas for calculating probabilities in different scenarios:

P(Z < z): Look up directly in the table (for positive z).
P(Z > z): 1 - P(Z < z)
P(Z > -z): P(Z < z)
P(Z < -z): 1 - P(Z < z)
P(Z1 < Z < Z2): P(Z < Z2) - P(Z < Z1)

Conclusion

The video provides a practical guide to using the standard normal distribution table to calculate probabilities in various scenarios, including areas above, below, and between Z-values, both positive and negative. The key is to understand the table's limitations and use the properties of the normal distribution (symmetry and total area = 1) to transform the problem into a form that can be solved using the table. The video promises examples in the next video to solidify these concepts.