Finding Areas Using the Standard Normal Table (for tables that give the area between 0 and z)

Summary

This video explains how to find areas under the standard normal curve using Z-tables. It demonstrates how to calculate probabilities between specific Z-values by referencing standard normal tables, interpreting areas between 0 and the Z-value. The video also discusses how to handle symmetric properties of the normal distribution, such as determining probabilities for negative Z-values. Practical examples, including calculations for specific Z-values like 1.43 and -1.28, help illustrate these concepts, with clear explanations on how to use symmetry and table lookups to determine probabilities.

Takeaways

• 😀 Z-scores are used to find probabilities in a standard normal distribution, which is symmetric around zero.
• 😀 The standard normal table provides the area between 0 and the Z-value, which is crucial for probability calculations.
• 😀 The probability that Z is exactly equal to a constant (like 1.43) is zero because Z is a continuous random variable.
• 😀 To find the probability that Z lies between two values, the area under the curve between those values is calculated.
• 😀 Symmetry of the standard normal distribution is useful for calculating probabilities involving negative Z-scores.
• 😀 When calculating areas for negative Z-scores, we can use the symmetry of the distribution to find the equivalent area for positive Z-scores.
• 😀 For any Z-value, the total area under the curve is 1, with half of the area on either side of zero.
• 😀 To find the probability that Z is greater than a certain value (like 1.43), subtract the area found from 0.5 (the total area to the right of zero).
• 😀 The standard normal table does not contain negative values, but the symmetry about zero allows us to find areas for negative Z-values by referencing positive values.
• 😀 In cases where the area to the left of a Z-value is given, we use the standard normal table to find the corresponding Z-value that yields that area.
• 😀 The standard normal distribution is often used to solve problems that involve finding the area between two Z-scores, or the probability of a Z-score falling in a given range.

Q & A

What is the main purpose of the standard normal table in the script?

The main purpose of the standard normal table is to help find the area under the standard normal curve for a given Z value, which corresponds to the probability that the random variable Z lies between zero and that Z value.

Why does the probability of Z being exactly 1.43 equal zero?

The probability that Z is exactly equal to any constant, such as 1.43, is zero because Z is a continuous random variable, and the probability of a continuous random variable taking any specific value is always zero.

How is the probability that Z lies between 0 and 1.43 calculated?

The probability that Z lies between 0 and 1.43 is calculated by finding the corresponding area in the standard normal table. In this case, the value of 1.43 is looked up, and the area is found to be 0.4236.

What is the probability that Z is greater than 1.43?

The probability that Z is greater than 1.43 is calculated by subtracting the area under the curve from 0.5 (the area to the right of 0), since the distribution is symmetric. The result is 0.0764.

How is symmetry about zero used in the standard normal distribution?

Symmetry about zero is used to simplify calculations. If the area for a positive Z value is known, the same area applies for the corresponding negative Z value. This allows us to use the table for only positive Z values.

How do you find the probability that Z lies between -1.28 and 0.72?

To find the probability that Z lies between -1.28 and 0.72, the areas between 0 and 1.28, and between 0 and 0.72, are looked up in the table. The symmetry about zero argument is used for the negative value of Z, so the area between 0 and -1.28 is the same as between 0 and 1.28.

What is the method for calculating the probability of Z being greater than or equal to -0.37?

To calculate the probability of Z being greater than or equal to -0.37, the symmetry argument is used. The area to the right of -0.37 is the same as the area to the left of 0.37. The area between 0 and 0.37 is looked up in the table, and the total probability is the sum of the area to the left of zero (0.5) and the area from the table (0.1443), resulting in a total probability of 0.6443.

Why is the standard normal distribution table limited to only positive Z values?

The standard normal distribution table is limited to only positive Z values because the distribution is symmetric about zero. This symmetry allows us to deduce the areas for negative Z values without needing a separate table.

How does the Z value of 1.28 relate to the area under the curve?

The Z value of 1.28 corresponds to an area of 0.3997 under the curve, which means that approximately 39.97% of the distribution lies to the left of Z = 1.28.

What would you do if you were given an area and asked to find the corresponding Z value?

If given an area and asked to find the corresponding Z value, you would typically look up the area in the standard normal table, which provides the Z values corresponding to specific cumulative areas. If the area is not directly in the table, interpolation might be needed.