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The Z-Table is an essential tool in statistics. It helps us find probabilities related to the standard normal distribution. The standard normal distribution is a special type of normal distribution that has a mean of 0 and a standard deviation of 1. When we use a Z-table, we can determine how likely it is for a certain value of the random variable, often represented as \(z\), to occur.

The Z-Table works by providing the probability that the \(z\) score will be less than or equal to a particular value. Each entry in the Z-table corresponds to a specific \(z\) score and gives the cumulative probability up to that score. For example in our context, to find \(P(z \leq 1.62)\), we look for 1.6 in the left column and 0.02 at the top row of the Z-Table, and the intersecting value gives us the probability we need.

• This table helps in problems like finding \(P(0 \leq z \leq 1.62)\), where multiple \(z\) scores are involved.
• Cumulative means it adds up all probabilities up to that point.

The task of calculating probability in a standard normal distribution involves combining information from Z-tables. The main aim is to find the likelihood that a random variable \(z\) will land within a specified range, here between 0 and 1.62.

To find \(P(0 \leq z \leq 1.62)\):

• First, use the Z-Table to determine \(P(z \leq 1.62)\). This gives the entire probability of \(z\) being less than or equal to 1.62.
• Because the standard normal distribution is perfectly symmetrical about the mean of 0, the probability that \(z\) is less than or equal to 0 is always 0.5.
• To get the probability between 0 and 1.62, subtract the left part, \(P(z \leq 0)\), from the total \(P(z \leq 1.62)\).
• This gives us \(0.9474 - 0.5 = 0.4474\). Thus, \(P(0 \leq z \leq 1.62) = 0.4474\).

Visualizing problems with shading on the standard normal distribution curve helps in understanding the concepts better. When tasked with shading, it simply means marking the area under the curve that corresponds to the calculated probability.

For the probability \(P(0 \leq z \leq 1.62)\):

• You would shade the region under the curve starting from \(z = 0\) and ending at \(z = 1.62\).
• The shaded area visually represents the probability that \(z\) is greater than or equal to 0 and less than or equal to 1.62, which we've calculated as 0.4474.
• This technique is useful because it provides a quick method to grasp how probabilities work in the standard normal distribution and what part of the data they represent.

Shading is a powerful visualization tool in statistics, aiding learners in connecting abstract numbers to real concepts.